This paper extends multilevel Picard algorithms to high-dimensional semilinear Black-Scholes PDEs, demonstrating polynomial computational effort and overcoming the curse of dimensionality in derivative pricing with default risks.
Contribution
It introduces a new MLP algorithm for semilinear Black-Scholes equations and proves polynomial complexity, a first in this context.
Findings
01
MLP algorithms overcome the curse of dimensionality for these PDEs.
02
Computational effort grows polynomially with dimension and accuracy.
03
First proof of polynomial tractability for semilinear Black-Scholes PDEs.
Abstract
Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man made complex systems. In particular, parabolic PDEs are a fundamental tool to determine fair prices of financial derivatives in the financial industry. The PDEs appearing in financial engineering applications are often nonlinear and high dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue is that most approximation methods for nonlinear PDEs in the literature suffer under the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods have not been shown not to suffer under the curse of…
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Full text
Overcoming the curse of dimensionality
in the approximative pricing of
financial derivatives with default risks
Martin Hutzenthaler,
Arnulf Jentzen, and Philippe von Wurstemberger
Abstract
Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man made complex systems.
In particular, parabolic PDEs are a fundamental tool to determine fair prices of financial derivatives in the financial industry.
The PDEs appearing in financial engineering applications are often nonlinear (e.g. PDE models which take into account the possibility of a defaulting counterparty) and high dimensional since the dimension typically corresponds to the number of considered financial assets.
A major issue is that most approximation methods for nonlinear PDEs in the literature suffer under the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy
grows exponentially in the dimension
of the PDE or in the reciprocal
of the prescribed approximation accuracy and nearly all approximation methods have not been shown not to suffer under the curse of dimensionality.
Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven
that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations.
In this paper we extend those findings to a more general class of semilinear PDEs including as special cases semilinear Black-Scholes equations used for the pricing of financial derivatives with default risks.
More specifically, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity is globally Lipschitz continuous, that the computational effort of our method grows at most polynomially both in the dimension and the reciprocal of the prescribed approximation accuracy.
This is, to the best of our knowledge, the first result showing that the approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem.
Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man made complex systems.
In particular, parabolic PDEs are a fundamental tool to determine fair prices of financial derivatives in the financial industry.
The use of PDEs for option pricing originated in the work of Black, Scholes, & Merton (see [9, 76]) which suggested that the price of a financial derivative satisfies a linear parabolic PDE, nowadays known as Black-Scholes equation.
The derivation of their theory is based on several assumption which are not met in the financial practice and consequently various changes and extensions to the original pricing model have been developed.
One key modification of the initial Black-Scholes model is to include the possibility of a defaulting counterparty (cf., e.g., Burgard & Kjaer [14], Crepey et al. [24], Duffie et al. [33], and Henry-Labordere [53]).
Such extended models suggest that the price process of a financial derivative satisfies a certain semilinear PDE (cf. (1) in Theorem 1.1 below and Subsections 4.2–4.3 below).
Typically, such PDEs can not be solved explicitly and it is therefore a very active topic of research to solve such PDEs approximatively;
cf., e.g.,
[30, 94, 95, 97]
for deterministic approximation methods for PDEs,
cf., e.g.,
[2, 6, 7, 10, 12, 13, 17, 18, 19, 20, 25, 26, 27, 28, 38, 39, 31, 32, 40, 41, 42, 43, 44, 45, 46, 57, 69, 70, 71, 72, 73, 74, 78, 79, 82, 83, 84, 85, 89, 90, 91, 96, 102, 103, 104]
for probabilistic approximation methods for PDEs using discretizations of the associated backward stochastic differential equations (BSDEs),
cf., e.g.,
[11, 21, 37, 49, 68, 105]
for probabilistic approximation methods for PDEs using temporal discretizations of the associated second-order BSDEs
cf., e.g.,
[16, 53, 55, 56, 75, 88, 93, 98, 101]
for probabilistic approximation methods for PDEs using branching diffusions processes,
cf., e.g.,
[99, 100]
for probabilistic approximation methods for PDEs using nested Monte Carlo simulations,
cf., e.g.,
[35, 36, 59, 60]
for full history recursive multilevel Picard (MLP) approximation methods for PDEs,
and cf., e.g.,
[3, 4, 8, 15, 34, 51, 52, 54, 58, 65, 80, 87, 92]
for approximation methods for PDEs which are based on reformulations of PDEs as a deep learning problems.
The PDEs appearing in financial engineering applications are often high dimensional since the dimension corresponds to the number of financial assets (such as stocks, commodities, interest rates, or exchange rates) in the involved hedging portfolio.
A major issue is that most approximation methods suffer under the so-called curse of dimensionality (see Bellman [5]) in the sense that the computational effort to compute an approximation with a prescribed accuracy ε>0 grows exponentially in the dimension d∈N of the PDE or in the reciprocal \nicefrac1ε of the prescribed approximation accuracy (cf., e.g., E et al. [36, Section 4] for a discussion of the curse of dimensionality in the PDE approximation literature) and nearly all approximation methods have not been shown not to suffer under the curse of dimensionality.
Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced in E et al. [35, 36] and it was proven, under restrictive assumptions on the regularity of the solution of the PDE that they overcome the curse of dimensionality for semilinear heat equations.
Building on this work, [59] proposed for semilinear heat equations an adaption of the original MLP scheme in [35, 36].
Under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous [59, Theorem 1.1] proves that the proposed scheme does indeed overcome the curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy ε>0 grows at most polynomially in both the dimension d∈N of the PDE and the reciprocal \nicefrac1ε of the prescribed approximation accuracy.
In this paper we generalize the MLP algorithm of [59] and the main result of this article, Theorem 3.20 below, proves that the MLP algorithm proposed in this paper overcomes the curse of dimensionality for a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used for the pricing of financial derivatives with default risks.
In particular, we show for the first time that the solution of a semilinear Black-Scholes PDE with a globally Lipschitz continuous nonlinearity can be approximated
with a computational effort which grows at most polynomially in both the dimension and the reciprocal of the prescribed approximation accuracy.
Put differently, we show that the approximation of solutions of such semilinear Black-Scholes equations is a polynomially tractable approximation problem (cf., e.g., Novak & Wozniakowski [81]).
To illustrate the main result of this paper, Theorem 3.20 below, we present in the following theorem, Theorem 1.1 below, a special case of Theorem 3.20.
Theorem 1.1 demonstrates that the MLP algorithm proposed in this article overcomes the curse of dimensionality for the approximation of solutions of certain semilinear Black-Scholes equations.
Theorem 1.1**.**
Let
T∈(0,∞), p,P,q∈[0,∞), α,β∈R,
Θ=∪n=1∞Zn,
let f:R→R be a Lipschitz continuous function,
let ξd∈Rd, d∈N, and gd∈C2(Rd,R), d∈N, satisfy that
\sup_{d\in\mathbb{N},x\in\mathbb{R}^{d}}\big{(}\tfrac{|g_{d}(x)|}{d^{\mathfrak{P}}(1+\left\|x\right\|_{\mathbb{R}^{d}}^{p})}+\frac{\left\|\xi_{d}\right\|_{\mathbb{R}^{d}}}{d^{q}}\big{)}<\infty,
let ud∈C1,2([0,T]×Rd,R), d∈N, be polynomially growing functions which satisfy
for all d∈N, t∈(0,T), x=(x1,x2,…,xd)∈Rd that
ud(T,x)=gd(x)
and
[TABLE]
let (Ω,F,P) be a probability space,
let
Rθ:Ω→[0,1],
θ∈Θ,
be independent U[0,1]-distributed random variables,
let
Rθ=(Rtθ)t∈[0,T]:[0,T]×Ω→[0,T],
θ∈Θ,
be the stochastic processes which satisfy
for all t∈[0,T], θ∈Θ that
Rtθ=t+(T−t)Rθ,
let
Wd,θ=(Wd,θ,i)i∈{1,2,…,d}:[0,T]×Ω→Rd,
θ∈Θ, d∈N,
be independent standard Brownian motions,
assume that
(Wd,θ)d∈N,θ∈Θ
and
(Rθ)θ∈Θ
are independent,
for every d∈N, θ∈Θ, t∈[0,T], s∈[t,T], x=(x1,x2,…,xd)∈Rd
let
Xt,sd,θ,x=(Xt,sd,θ,x,i)i∈{1,2,…,d}:Ω→Rd
be the function which satisfies
for all
i∈{1,2,…,d} that
[TABLE]
let
VM,nd,θ:[0,T]×Rd×Ω→R, M,n∈Z, θ∈Θ, d∈N,
be functions which satisfy
for all d,M,n∈N, θ∈Θ, t∈[0,T], x∈Rd that
VM,−1d,θ(t,x)=VM,0d,θ(t,x)=0
and
[TABLE]
and for every d,n,M∈N, t∈[0,T], x∈Rd let Cd,M,n∈N0 be the number of realizations of standard normal random variables which are used to compute one realization of VM,nd,0(t,x) (see (336) below for a precise definition).
Then
there exist functions
N=(Nd,ε)d∈N,ε∈(0,1]:N×(0,1]→N
and
C=(Cδ)δ∈(0,∞):(0,∞)→(0,∞)
such that
for all d∈N, ε∈(0,1], δ∈(0,∞) it holds that
Cd,Nd,ε,Nd,ε≤Cδd1+(P+qp)(2+δ)ε−(2+δ)
and
[TABLE]
Theorem 1.1 is an immediate consequence of Theorem 4.4 below. Theorem 4.4 in turn is a consequence of Theorem 3.20 below, the main result of this paper.
We now provide some explanations for Theorem 1.1.
In Theorem 1.1 we present a stochastic approximation scheme (cf. (VM,nd,0)M,n,d∈N in Theorem 1.1 above) which is able to approximate in the strong L2-sense the initial value ud(0,ξd) of the solution of an uncorrelated semilinear Black-Scholes equation (cf. (1) in Theorem 1.1 above) with a computational effort which grows at most polynomially in both the dimension d∈N and the reciprocal \nicefrac1ε of the prescribed approximation accuracy ε>0.
The time horizon T∈(0,∞), the drift parameter α∈R, the diffusion parameter β∈R, as well as the Lipschitz continuous nonlinearity f:R→R of the semilinear Black-Scholes equations in Theorem 1.1 above (cf. (1) in Theorem 1.1 above) are fixed over all dimensions
(cf. Theorem 4.3 for a more general result with dimension-dependent drift and diffusion coefficients and dimension-dependent nonlinearities which may additionally depend on the time and the space variable).
The approximation points (ξd)d∈N and the terminal conditions (gd)d∈N of the PDE (1) in Theorem 1.1 above are both allowed to grow in a certain polynomial fashion determined by the constants p,P,q∈[0,∞).
The idea for the full history multilevel Picard scheme (cf. (VM,nd,θ)M,d∈N,n∈N0,θ∈Θ in Theorem 1.1 above) is based on a reformulation of the semilinear PDE in (1) as a stochastic fixed point equation.
For this we consider the independent solution fields (Xd,θ)d∈N,θ∈Θ of the stochastic differential equation (SDE) associated to the PDE in (1) and for every t∈[0,T] we consider independent U[t,T]-distributed random variables (Rtθ)θ∈Θ.
As a consequence of the Feynman-Kac formula
we obtain that (ud)d∈N are the unique at most polynomially growing functions which satisfy
for all d∈N, θ∈Θ, t∈[0,T], x∈Rd that
[TABLE]
Note that
for all d,M,n∈N, θ∈Θ, t∈[0,T], x∈Rd it holds that
[TABLE]
Thus for every d,M∈N, θ∈Θ the sequence of random fields (VM,nd,θ)n∈N0 behave, in expectation, like Picard iterations for the stochastic fixed point equation in (5) above. In each iteration in (3) the expectation of the Picard iteration for the stochastic fixed point equation in (5) is approximated with a multilevel Monte Carlo approach on a telescope expansion over the full history of the previous iterations.
According to the multilevel Monte Carlo paradigm the number of samples in each level is chosen such that computationally inexpensive summands (corresponding to small k∈{0,1,2,…,n} in (6)) of the telescope expansion get sampled more often than computationally expensive ones (corresponding to large k∈{0,1,2,…,n} in (6)).
Roughly speaking, the conclusion of Theorem 1.1 above states (cf. Theorem 1.1 above for the precise formulation) that
for every d∈N, ε∈(0,1] there exists a natural number N∈N
such that VN,Nd,0(0,ξd) approximates ud(0,ξd) in the L2-sense with accuracy ε and such that the computational effort to compute VN,Nd,0(0,ξd) is essentially of the order d1+2(P+pq)ε−2.
Remarkably this is exactly the computational complexity of the standard Monte Carlo approximation of the solution of the PDE (1) in the case that the nonlinearity f vanishes (cf., e.g., Graham & Talay [47]).
The remainder of this paper is structured as follows.
In Section 2 we prove a well-known distributional flow property for the composition of independent solutions fields of a stochastic differential equation (SDE) (see Lemma 2.19 below), which will be a key assumption in the abstract treatment of stochastic fixed point equations in Section 3.
Several auxiliary results which are needed for the proof of the flow property (see Lemma 2.19 below) in Section 2 will be used again in Section 3.
Section 3 introduces the MLP algorithm, provides a complexity analysis in the setting of stochastic fixed point equations in Subsections 3.1–3.5, and then carries over those results to semilinear Kolmogorov PDEs in Subsection 3.6 leading to Theorem 3.20 below, the main result of this article.
In the last section, Section 4, we apply the result for general semilinear Kolmogorov PDEs of Theorem 3.20 to semilinear heat equations (see Subsection 4.1) and semlinear Black-Scholes equations (see Subsection 4.2 and Subsection 4.3) which are notably used to compute prices for financial derivatives in the presence of counterparty credit risks (see Subsection 4.3).
2 On a distributional flow property for stochastic differential equations (SDEs)
In our analysis of the proposed MLP algorithm in Section 3 below, we will make use of random fields which satisfy a certain flow-type condition (see (154) in Setting 3.1 below).
The main intent of this section is to establish that solution processes of SDEs enjoy, under suitable conditions (see Lemma 2.19 below for details), this flow-type property.
To rigorously prove this result we need a series of elementary and well-known results, presented in Subsections 2.1–2.7 below, many of which will be reused in Section 3.
2.1 Time-discrete Gronwall inequalities
In this subsection we present elementary and well-known Gronwall inequalities (cf., e.g., Agarwal [1]).
Lemma 2.1**.**
Let N∈N, α∈[0,∞), (βn)n∈{0,1,2,…,N−1}⊆[0,∞), (ϵn)n∈{0,1,2,…,N}⊆[0,∞] satisfy
for all n∈{0,1,2,…,N} that
Throughout this proof let (un)n∈{0,1,2,…,N}⊆[0,∞] be the extended real numbers which satisfy
for all n∈{0,1,2,…,N} that
[TABLE]
We claim that
for all n∈{0,1,2,…,N} it holds that
[TABLE]
We now prove (10) by induction on n∈{0,1,2,…,N}.
For the base case n=0 observe that (9) ensures that
[TABLE]
This proves (10) in the base case n=0.
For the induction step {0,1,2,…,N−1}∋(n−1)→n∈{1,2,…,N} observe that (9) implies that
for all n∈{1,2,…,N} with
un−1=α[∏k=0n−2(1+βk)]
it holds that
[TABLE]
Induction thus establishes (10).
Moreover, note that (7), (9), and induction prove that
for all n∈{0,1,2,…,N} it holds that
[TABLE]
This and (10) establish that
for all n∈{0,1,2,…,N} it holds that
[TABLE]
The fact that
for all x∈R it holds that
(1+x)≤exp(x) therefore ensures that
for all n∈{0,1,2,…,N} it holds that
Note that Lemma 2.1 establishes Corollary 2.2.
The proof of Corollary 2.2 is thus completed.
∎
2.2 A priori moment bounds for solutions of SDEs
In this subsection we establish in the elementary result in Lemma 2.6 below for every p∈[0,∞) a bound on the p-th absolute moment of the solution of an SDE with deterministic initial value, a one-sided linear growth condition on the drift coefficient of the SDE, and a linear growth condition on the diffusion coefficient of the SDE (cf. (43) in Lemma 2.6 below).
Our proof of Lemma 2.6 employs standard Lyapunov-type techniques from the literature to establish the desired a priori moment bound
(cf., e.g., Cox et al. [22, Section 2.2]).
Lemma 2.3**.**
Let d,m∈N, T,C1,C2∈[0,∞),
let ⟨⋅,⋅⟩:Rd×Rd→R be the Euclidean scalar product on Rd,
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let ∣∣∣⋅∣∣∣:Rd×m→[0,∞) be the Frobenius norm on Rd×m,
and
let
μ:[0,T]×Rd→Rd,
σ:[0,T]×Rd→Rd×m, and
Vp:Rd→(0,∞), p∈[2,∞),
be functions which satisfy
for all t∈[0,T], x∈Rd, p∈[2,∞) that
[TABLE]
Then
(i)
it holds
for all p∈[2,∞) that
Vp∈C∞(Rd,(0,∞))
and
2. (ii)
Throughout this proof let
σi,j:[0,T]×Rd→R,
i∈{1,2,…,d},
j∈{1,2,…,m},
be the functions which satisfy for all
t∈[0,T],
x∈Rd
that
[TABLE]
Note that the chain rule,
the fact that the function
Rd∋x↦1+∥x∥2∈(0,∞) is infinitely often differentiable,
and
the fact that for every p∈[2,∞) the function
(0,∞)∋s↦s2p∈(0,∞)
is infinitely often differentiable
establish item (i).
It thus remains to prove
item (ii).
For this, observe that the chain rule ensures that
for all
x=(x1,…,xd)∈Rd,
i,j∈{1,2,…,d},
p∈[2,∞)
it holds that
[TABLE]
and
[TABLE]
This implies that for all
t∈[0,T],
x=(x1,…,xd)∈Rd,
p∈[2,∞)
it holds
that
[TABLE]
In addition, note that the Cauchy Schwarz inequality assures that
for all
t∈[0,T],
x=(x1,…,xd)∈Rd
it holds that
[TABLE]
This, (18), and (23)
demonstrate that
for all t∈[0,T], x∈Rd, p∈[2,∞) it holds that
[TABLE]
Young’s inequality (with p=\nicefracp2, q=\nicefracp(p−2)=\nicefracp2−1\nicefracp2 for p∈(2,∞) in the usual notation of Young’s inequality) hence proves that
for all t∈[0,T], x∈Rd, p∈(2,∞) it holds that
[TABLE]
Moreover, note that (25) ensures that
for all t∈[0,T], x∈Rd it holds that
[TABLE]
Combining this and (26) establishes item (ii).
The proof of Lemma 2.3 is thus completed.
∎
Lemma 2.4**.**
Let d,m∈N, T,ρ∈[0,∞), ξ∈Rd,
let ⟨⋅,⋅⟩:Rd×Rd→R be the Euclidean scalar product on Rd,
let
μ∈C([0,T]×Rd,Rd),
σ∈C([0,T]×Rd,Rd×m),
V∈C2(Rd,(0,∞))
satisfy
for all t∈[0,T], x∈Rd that
[TABLE]
let (Ω,F,P,(Ft)t∈[0,T]) be a filtered probability space which satisfies the usual conditions,
let W:[0,T]×Ω→Rm be a standard (Ω,F,P,(Ft∈[0,T]))-Brownian motion,
and let
X:[0,T]×Ω→Rd
be an (Ft)t∈[0,T]/B(Rd)-adapted stochastic process with continuous sample paths which satisfies
that
for all t∈[0,T] it holds P-a.s. that
Let d,m∈N, T,ρ1,ρ2∈[0,∞), ξ∈Rd,
let ⟨⋅,⋅⟩:Rd×Rd→R be the Euclidean scalar product on Rd,
let
μ∈C([0,T]×Rd,Rd),
σ∈C([0,T]×Rd,Rd×m),
V∈C2(Rd,(0,∞))
satisfy
for all t∈[0,T], x∈Rd that
[TABLE]
let (Ω,F,P,(Ft)t∈[0,T]) be a filtered probability space which satisfies the usual conditions,
let W:[0,T]×Ω→Rm be a standard (Ω,F,P,(Ft∈[0,T]))-Brownian motion,
and let
X:[0,T]×Ω→Rd
be an (Ft)t∈[0,T]/B(Rd)-adapted stochastic process with continuous sample paths which satisfies
that
for all t∈[0,T] it holds P-a.s. that
Throughout this proof assume w.l.o.g. that ρ1>0 (cf. Lemma 2.4) and that T>0 and
let
V:[0,T]×Rd→(0,∞)
be the function which satisfies for all
t∈[0,T],
x∈Rd
that
[TABLE]
Note that
the fact that V∈C2(Rd,(0,∞))
ensures that for all
t∈[0,T],
x∈Rd
it holds that
(I)
V∈C2([0,T]×Rd,(0,∞)),
2. (II)
(∂t∂V)(t,x)=−ρ1e−ρ1t(V(x)+ρ1ρ2),
3. (III)
(∇xV)(t,x)=e−ρ1t(∇V)(x), and
4. (IV)
(HessxV)(t,x)=e−ρ1t(HessV)(x).
Observe that
items (II)–(IV)
and
(35)
assure that
for all
t∈[0,T],
x∈Rd
it holds
that
[TABLE]
Combining this with
Itô’s formula
demonstrates that
for all t∈[0,T] it holds that
[TABLE]
Therefore, we obtain that
for all t∈[0,T] it holds that
[TABLE]
The fact that
for all a∈R it holds that
ea−1≤aea hence ensures that
for all t∈[0,T] it holds that
Let d,m∈N, T,C1,C2∈[0,∞), ξ∈Rd,
let ⟨⋅,⋅⟩:Rd×Rd→R be the Euclidean scalar product on Rd,
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let ∣∣∣⋅∣∣∣:Rd×m→[0,∞) be the Frobenius norm on Rd×m,
let μ∈C([0,T]×Rd,Rd), σ∈C([0,T]×Rd,Rd×m) satisfy
for all t∈[0,T], x∈Rd that
[TABLE]
let (Ω,F,P,(Ft)t∈[0,T]) be a filtered probability space which satisfies the usual conditions,
let W:[0,T]×Ω→Rm be a standard (Ω,F,P,(Ft∈[0,T]))-Brownian motion,
and let
X:[0,T]×Ω→Rd
be an (Ft)t∈[0,T]/B(Rd)-adapted stochastic process with continuous sample paths which satisfies
that
for all t∈[0,T] it holds P-a.s. that
Throughout this proof let
(ρ1(p))p∈[2,∞),(ρ2(p))p∈[2,∞)⊆[0,∞) satsify
for all p∈[2,∞) that
[TABLE]
and
let Vp:Rd→(0,∞), p∈[2,∞), be the functions which satisfy
for all p∈[2,∞), x∈Rd that
[TABLE]
Observe that Lemma 2.3 and (43) assure that
for all t∈[0,T], x∈Rd, p∈[2,∞) it holds that Vp∈C∞(Rd,(0,∞)) and
[TABLE]
Lemma 2.5 hence implies that
for all t∈[0,T], p∈[2,∞) it holds that
[TABLE]
This, Jensen’s inequality, and the fact that
for all p∈[0,2] it holds that
3\nicefracp2≤p+1
assure that
for all t∈[0,T], p∈[0,2) it holds that
[TABLE]
Combining this with (49) implies (45).
The proof of Lemma 2.6 is thus completed.
∎
2.3 Temporal regularity properties for solutions of SDEs
For the proof of our strong L2-error estimates for Euler-Maruyama approximations in Subsection 2.4 we need Corollary 2.8 below, which asserts that, under suitable conditions (see Corollary 2.8 below for details), solutions of SDEs have a certain temporal regularity property. To prove Corollary 2.8 we employ (without providing a proof) a well-known temporal regularity property for solutions of SDEs from the literature stated in Lemma 2.7 below (cf., e.g., Da Prato et al. [29, Proposition 3], Cox et al. [23, Corollary 3.8], and Jentzen et al. [63, Proposition 5.1]).
Additionally, we offer in Lemma 2.10 below a self contained proof of an explicit temporal regularity estimate for solutions of SDEs with deterministic initial values which will be used in Subsection 2.8.
Lemma 2.7** (Temporal regularity of solutions of time-homogeneous SDEs).**
Let d,m∈N,
T∈(0,∞),
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let
(Ω,F,P,(Ft)t∈[0,T])
be a filtered probability space which satisfies the usual conditions,
let
W:[0,T]×Ω→Rm
be a standard
(Ω,F,P,(Ft)t∈[0,T])-Brownian motion,
let μ:Rd→Rd, σ:Rd→Rd×m
be globally Lipschitz continuous functions,
and let X:[0,T]×Ω→Rd be an (Ft)t∈[0,T]/B(Rd)-adapted stochastic processes with continuous sample paths
which satisfies that E[∥X0∥2]<∞ and
which satisfies
that for all t∈[0,T]
it holds P-a.s. that
[TABLE]
Then it holds that
[TABLE]
Lemma 2.8** (Temporal regularity of solutions of time-inhomogeneous SDEs).**
Let d,m∈N,
T∈(0,∞),
L∈[0,∞),
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let
(Ω,F,P,(Ft)t∈[0,T])
be a filtered probability space which satisfies the usual conditions,
let
W:[0,T]×Ω→Rm
be a standard
(Ω,F,P,(Ft)t∈[0,T])-Brownian motion,
let μ:[0,T]×Rd→Rd
and σ:[0,T]×Rd→Rd×m
be globally Lipschitz continuous functions,
and let X:[0,T]×Ω→Rd be an (Ft)t∈[0,T]/B(Rd)-adapted stochastic processes with continuous sample paths
which satisfies that E[∥X0∥2]<∞ and
which satisfies
that for all t∈[0,T]
it holds P-a.s. that
Throughout this proof
let ∣∣∣⋅∣∣∣:Rd+1→[0,∞) be the Euclidean norm on Rd+1,
let Y:[0,T]×Ω→Rd+1 be the stochastic process
which satisfies for all t∈[0,T] that
[TABLE]
and let μ~:Rd+1→Rd+1
and σ~:Rd+1→R(d+1)×m
be the functions which satisfy
for all y=(y1,y2,…,yd+1)∈Rd+1 that
[TABLE]
[TABLE]
Observe that the hypothesis that μ and σ are globally Lipschitz continuous functions
and the fact that
R∋y↦min{max{y,0},T}∈R is a globally Lipschitz continuous function
assure that μ~ and σ~ are globally Lipschitz continuous functions.
Moreover, note that it holds
for all t∈[0,T], x∈Rd that
[TABLE]
This and (53) assure
that for all t∈[0,T]
it holds P-a.s. that
[TABLE]
The fact that μ~ and σ~ are globally Lipschitz continuous functions and Lemma 2.7
(with d=d+1, m=m, T=T, μ=μ~, σ=σ~, X=Y
in the notation of Lemma 2.7)
hence prove that
The following very elementary and well-known result will be helpfull in the proof of Lemma 2.10 below and will be repeatedly used throughout this paper.
Lemma 2.9** (A consequence of Hölders inequality).**
Let (Ω,F,μ) be a measure space and
let f:Ω→[0,∞] be an F/B([0,∞])-measurable function.
Then
Lemma 2.10** (Explicit temporal regularity for solutions of SDEs with deterministic initial values).**
Let d,m∈N,
T∈(0,∞),
L∈[0,∞),
ξ∈Rd,
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let ∣∣∣⋅∣∣∣:Rd×m→[0,∞) be the Frobenius norm on Rd×m,
let
(Ω,F,P,(Ft)t∈[0,T])
be a filtered probability space which satisfies the usual conditions,
let
W:[0,T]×Ω→Rm
be a standard
(Ω,F,P,(Ft)t∈[0,T])-Brownian motion,
let μ:[0,T]×Rd→Rd, σ:[0,T]×Rd→Rd×m
be functions
which satisfy for all t,s∈[0,T], x,y∈Rd
that
[TABLE]
and let X:[0,T]×Ω→Rd be an (Ft)t∈[0,T]/B(Rd)-adapted stochastic processes with continuous sample paths
which satisfies
that for all t∈[0,T]
it holds P-a.s. that
Throughout this proof let
⟨⋅,⋅⟩:Rd×Rd→R be the Euclidean scalar product on Rd
and let C∈(0,∞) be given by
[TABLE]
Note that (64) and the triangle inequality assure that
for all t∈[0,T], x∈Rd it holds that
[TABLE]
[TABLE]
This assures that
for all t∈[0,T], x∈Rd it holds that
[TABLE]
In addition, note that (69) implies that
for all t∈[0,T], x∈Rd it holds that
[TABLE]
Moreover, observe that (65), Lemma 2.9, Tonelli’s theorem, and Itô’s isometry demonstate that
for all t∈[0,T], s∈[t,T] it holds that
[TABLE]
The triangle inequality, (68), and (69) therefore ensure that
for all t∈[0,T], s∈[t,T] it holds that
[TABLE]
Furthermore, note that
(70),
(71),
(65),
and
Lemma 2.6
(with d=d, m=m, T=T, C1=C, C2=C, ξ=ξ, μ=μ, σ=σ, X=X in the notation of Lemma 2.6)
assure that
for all t∈[0,T] it holds that
[TABLE]
This,
(73),
the fact that C≥1,
the fact that
for all x∈[0,∞) it holds that
max{x,1+x}≤ex,
and
the fact that
for all x,y∈[0,∞) it holds that
x+y≤x+y
demonstrate that
for all t∈[0,T], s∈[t,T] it holds that
[TABLE]
This implies (66).
The proof of Lemma 2.10 is thus completed.
∎
2.4 Strong error estimates for Euler-Maruyama approximations
Our proof of the flow-type property of solutions of SDEs in Subsection 2.8 below makes use of Euler-Maruyama approximations of solutions. For that reason we present in this subsection explicit strong L2-error estimates for Euler-Maruyama approximations in Proposition 2.11 and Corollary 2.12 below.
The results in this subsection are essentially well-known (cf., e.g., Kloeden & Platen [67, Chapter 10] and Milstein [77]).
Proposition 2.11** (Strong convergence of the Euler-Maruyama method).**
Let d,m,N∈N,
T∈(0,∞),
L∈[0,∞),
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let ∣∣∣⋅∣∣∣:Rd×m→[0,∞) be the Frobenius norm on Rd×m,
let
(Ω,F,P,(Ft)t∈[0,T])
be a filtered probability space which satisfies the usual conditions,
let
W:[0,T]×Ω→Rm
be a standard
(Ω,F,P,(Ft)t∈[0,T])-Brownian motion,
let ζ:Ω→Rd be an F0/B(Rd)-measurable function which satisfies that E[∥ζ∥2]<∞,
let μ:[0,T]×Rd→Rd, σ:[0,T]×Rd→Rd×m
be functions
which satisfy for all t,s∈[0,T], x,y∈Rd
that
[TABLE]
let X:[0,T]×Ω→Rd be an (Ft)t∈[0,T]/B(Rd)-adapted stochastic processes with continuous sample paths
which satisfies that E[∥X0∥2]<∞
and which satisfies
that for all t∈[0,T]
it holds P-a.s. that
[TABLE]
let t0,t1,…,tN∈[0,T] satisfy that
[TABLE]
and let
X:{0,1,…,N}×Ω→Rd be the stochastic process which satisfies
for all n∈{1,2,…,N} that
Throughout this proof assume w.l.o.g. that
t0<t1<t2<…<tN,
let (hn)n∈{1,2,…,N}⊆(0,T], H∈(0,T], K∈[0,∞] satisfy
for all n∈{1,2,…,N} that
[TABLE]
let
t:[0,T]→{t0,t1,t2,…,tN} be the function which satisfies
for all s∈[0,T] that
[TABLE]
and let
n:[0,T]→{0,1,2,…,N} be the function which satisfies
for all s∈[0,T] that
[TABLE]
Note that
the hypothesis that E[∥X0∥2]<∞,
the fact that μ and σ are globally Lipschitz continuous functions,
(77),
and
Lemma 2.8 imply that K<∞.
Next observe that (79) and induction assure that
for all
n∈{0,1,2,…,N}
it holds P-a.s. that
[TABLE]
This and (77) imply that
for all
n∈{0,1,2,…,N}
it holds P-a.s. that
[TABLE]
The triangle inequality hence proves that for all
n∈{0,1,2,…,N}
it holds that
[TABLE]
Lemma 2.9, Tonelli’s Theorem, and Itô’s isometry
therefore imply
that for all
n∈{0,1,2,…,N}
it holds that
[TABLE]
This and (76) show that
for all n∈{0,1,2,…,N}
it holds that
[TABLE]
This, the triangle inequality, and the fact that for all s∈[0,T] it holds that ∣s−t(s)∣≤H imply that
for all n∈{0,1,2,…,N}
it holds that
[TABLE]
The fact that
for all x,y∈[0,∞) it holds that
(x+y)2≤2x2+2y2
hence proves that
for all n∈{0,1,2,…,N}
it holds that
[TABLE]
The discrete Gronwall-type inequality in Lemma 2.1
(with
N=N,
\alpha=2\big{(}\left(\mathbb{E}\!\left[\|X_{0}-\zeta\|^{2}\right]\right)^{\!\nicefrac{{1}}{{2}}}\allowbreak+L(1+\sqrt{T})[\sqrt{T}H+(\int_{0}^{T}\mathbb{E}\!\left[\|X_{s}-X_{\mathfrak{t}(s)}\|^{2}\right]ds)^{\!\nicefrac{{1}}{{2}}}]\big{)}^{2},
(βn)n∈{0,1,2,…,N−1}=(2L2(1+T)2hn+1)n∈{0,1,2,…,N−1},
(ϵn)n∈{0,1,2,…,N}=(E[∥Xtn−Xn∥2])n∈{0,1,2,…,N}
in the notation of Lemma 2.1)
and the fact that ∑k=1Nhk=T
therefore show
that
[TABLE]
This and the fact that
for all s∈[0,T] it holds that
∣s−t(s)∣≤H
imply that
[TABLE]
The fact that H≤TH hence assures that
[TABLE]
This implies (80).
The proof of Proposition 2.11
is thus completed.
∎
Corollary 2.12**.**
Let d,m,N∈N,
T∈(0,∞), t∈[0,T], s∈[t,T],
L∈[0,∞),
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let ∣∣∣⋅∣∣∣:Rd×m→[0,∞) be the Frobenius norm on Rd×m,
let
(Ω,F,P,(Ft)t∈[0,T])
be a filtered probability space which satisfies the usual conditions,
let
W:[0,T]×Ω→Rm
be a standard
(Ω,F,P,(Ft)t∈[0,T])-Brownian motion,
let ζ:Ω→Rd be an Ft/B(Rd)-measurable function with E[∥ζ∥2]<∞,
let μ:[0,T]×Rd→Rd, σ:[0,T]×Rd→Rd×m
be functions
which satisfy for all r,h∈[0,T], x,y∈Rd
that
[TABLE]
let X:[t,s]×Ω→Rd be an (Fr)r∈[t,s]/B(Rd)-adapted stochastic processes with continuous sample paths
which satisfies that E[∥Xt∥2]<∞
and which satisfies
that for all r∈[t,s]
it holds P-a.s. that
[TABLE]
let r0,r1,…,rN∈[0,T] satisfy that
[TABLE]
and let
X:{0,1,…,N}×Ω→Rd be the stochastic process which satisfies
for all n∈{1,2,…,N} that
Throughout this proof assume w.l.o.g. that s>t.
Observe that Proposition 2.11
(with
d=d,
m=m,
N=N,
T=s−t,
L=L,
(Ω,F,P,(Fr)r∈[0,T])=(Ω,F,P,(Ft+r)r∈[0,s−t]),
(Wr)r∈[0,T]=(Wt+r−Wt)r∈[0,s−t],
ζ=ζ,
(μ(r,x))r∈[0,T],x∈Rd=(μ(t+r,x))r∈[0,s−t],x∈Rd,
(σ(r,x))r∈[0,T],x∈Rd=(σ(t+r,x))r∈[0,s−t],x∈Rd,
(Xr)r∈[0,T]=(Xt+r)r∈[0,s−t],
(tn)n∈{0,1,…,N}=(rn−t)n∈{0,1,…,N},
(Xn)n∈{0,1,…,N}=(Xn)n∈{0,1,…,N}
in the notation of Proposition 2.11)
establishes that
[TABLE]
This implies (98).
The proof of Corollary 2.12 is thus completed.
∎
2.5 On identically distributed random variables
The next elementary and well-known result, Lemma 2.13 below, provides a sufficient condition for two random variables to have the same distribution.
Lemma 2.13**.**
Let (Ω,F,P) be a probability space,
let (E,d) be a metric space,
let X,Y:Ω→E be random variables which satisfy that
for all globally bounded and Lipschitz continuous functions g:E→R it holds that
[TABLE]
Then
it holds that
X and Y are identically distributed random variables.
Throughout this proof
for every n∈N
let hn:[0,∞)→[0,1] be the function which satisfies
for all r∈[0,∞) that
[TABLE]
for every closed and non-empty set A⊆E let
DA:E→[0,∞)
be the function which satisfies
for all e∈E that
[TABLE]
and for every n∈N and every closed and non-empty set A⊆E let
fA,n:E→[0,1]
be the function which satisfies
for all e∈E that
[TABLE]
Note that the triangle inequality assures that
for all closed and non-empty sets A⊆E and all e1,e2∈E, a∈A, ε∈(0,∞)
with DA(e1)≥DA(e2) and d(e2,a)≤DA(e2)+ε it holds that
[TABLE]
The fact that for all closed and non-empty sets A⊆E and all e∈E, ε∈(0,∞)
there exists a∈A such that d(e,a)≤DA(e)+ε hence assures that
for all closed and non-empty sets A⊆E and all e1,e2∈E it holds that
[TABLE]
Moreover note that
for all n∈N, r1,r2∈[0,∞) with r1≤r2 it holds that
[TABLE]
Combining this with (105) establishes that
for all closed and non-empty sets A⊆E and all n∈N, e1,e2∈E it holds that
[TABLE]
This demonstrates that
for every closed and non-empty set A⊆E and every n∈N it holds that fA,n:E→[0,1] is a globally bounded and Lipschitz continuous function.
Next observe that
the fact that
for all closed and non-empty sets A⊆E and all e∈A it holds that
DA(e)=0 assures that
for all closed and non-empty sets A⊆E and all n∈N, e∈A it holds that
[TABLE]
Moreover, note
the fact that
for all closed and non-empty sets A⊆E and all e∈E∖A
there exists n∈N such that DA(e)>n1
and
the fact that
for all n∈N it holds that hn is a non-increasing function
assure that
for all closed and non-empty sets A⊆E and all e∈E∖A
there exist n∈N such that
for all m∈{n,n+1,…} it holds that
[TABLE]
Combining this and (108) establishes that
for all closed and non-empty sets A⊆E and all e∈E it holds that
[TABLE]
The theorem of dominated convergence,
the fact that for all closed and non-empty sets A⊆E and all n∈N it holds that fA,n:E→[0,1] is a globally bounded and Lipschitz continuous function,
and
(100) therefore imply that
for all closed and non-empty sets A⊆E it holds that
[TABLE]
The fact that
B(E)=S({A⊆E:A is closed}),
the fact that {A⊆E:A is closed} is closed under intersections,
and the uniqueness theorem for measures (see, e.g., Klenke [66, Lemma 1.42])
hence assure that
for all B∈B(E) it holds that
This subsection collects elementary and well-known results about random variables originating from evaluations of random fields at random indices.
Lemma 2.14**.**
Let (Ω,F), (S,S), (E,E) be measurable spaces,
let
U=(U(s))s∈S=(U(s,ω))s∈S,ω∈Ω:S×Ω→E
be an
(S⊗F)/E-measurable function,
and let X:Ω→S be an F/S-measurable function.
Then it holds that the function
U(X)=(U(X(ω),ω))ω∈Ω:Ω→E
is F/E-measurable.
Throughout this proof let X:Ω→S×Ω be the function which satisfies
for all ω∈Ω that
[TABLE]
Observe that the hypothesis that X:Ω→S is an F/S-measurable function assures that
X:Ω→S×Ω is an F/(S⊗F)-measurable function.
Combining this with the fact that U:S×Ω→E is an (S⊗F)/E-measurable function demonstrates that
[TABLE]
is an F/E-measurable function.
The proof of Lemma 2.14 is thus completed.
∎
A proof for the next two elementary and well-known results (see Lemma 2.15 and Lemma 2.16 below) can, e.g., be found in [59, Lemma 2.3 and Lemma 2.4].
Lemma 2.15**.**
Let (Ω,F,P) be a probability space,
let (S,δ) be a separable metric space,
let U=(U(s))s∈S:S×Ω→[0,∞)
be a continuous random field,
let X:Ω→S be a random variable,
and
assume that U and X are independent.
Then
it holds that
[TABLE]
Lemma 2.16**.**
Let (Ω,F,P) be a probability space,
let (S,δ) be a separable metric space,
let U=(U(s))s∈S:S×Ω→R
be a continuous random field,
let X:Ω→S be a random variable,
assume that U and X are independent,
and assume that
∫SE[∣U(s)∣](X(P)B(S))(ds)<∞.
Then
it holds that
(X(P)B(S))({s∈S:E[∣U(s)∣]=∞})=0, E[∣U(X)∣]<∞, and
[TABLE]
2.7 Brownian motions and right-continuous filtrations
The next result, Lemma 2.17 below, states that a Brownian motion with respect to a filtration is also a Brownian motion with respect to the smallest right-continuous filtration containing the original filtration (cf. (117)).
Lemma 2.17 and its proof are very similar to Prévôt & Röckner [86, Proposition 2.1.13].
Lemma 2.17**.**
Let
m∈N,
T∈(0,∞),
let
(Ω,F,P,(Ft)t∈[0,T])
be a filtered probability space,
let
W:[0,T]×Ω→Rm
be a standard
(Ω,F,P,(Ft)t∈[0,T])-Brownian motion,
and let Ht⊆F, t∈[0,T], satisfy
for all t∈[0,T] that
[TABLE]
Then it holds that W is a standard
(Ω,F,P,(Ht)t∈[0,T])-Brownian motion.
Throughout this proof
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
for every n∈N
let hn:[0,∞)→[0,1] be the function which satisfies
for all r∈[0,∞) that
[TABLE]
for every closed and non-empty set A⊆Rd let
DA:Rd→[0,∞)
be the function which satisfies
for all x∈Rd that
[TABLE]
and for every n∈N and every closed and non-empty set A⊆Rd let
fA,n:Rd→[0,1]
be the function which satisfies
for all x∈Rd that
[TABLE]
Observe that the fact that W has continuous sample paths,
the fact that for all t∈[0,T), s∈(t,T], k∈N it holds that
Ws−Wmin{t+\nicefrac1k,s} and Ht are independent,
Klenke [66, Theorem 5.4],
and the theorem of dominated convergence
assure that
for all t∈[0,T), s∈(t,T], B∈Ht and all globally bounded and continuous functions g:Rd→R it holds that
[TABLE]
Next note that
the fact that closed and non-empty sets A⊆Rd and all x∈Rd it holds that
DA(x)=0⇔x∈A assures that
for all closed and non-empty sets A⊆Rd and all x∈Rd it holds that
[TABLE]
Moreover, note that the fact that
for every n∈N
it holds that hn:[0,∞)→[0,1] is a continuous function and
the fact that
for every closed and non-empty set A⊆Rd
it holds that
DA:Rd→[0,∞) is a continuous function
assure that
for every n∈N and every closed and non-empty set A⊆Rd it holds that
fA,n:Rd→[0,1] is a continuous function.
Combining this, (121), (122), and the theorem of dominated convergence
shows that
for all t∈[0,T), s∈(t,T], B∈Ht and all closed and non-empty sets A⊆Rd it holds that
[TABLE]
This proves that
for all t∈[0,T), s∈(t,T], B∈Ht it holds that
(\mathbbm1B)−1({},{0},{1},{0,1})
and
(Ws−Wt)−1({A⊆Rd:A is a closed set})
are independent.
The fact that {A⊆Rd:A is a closed set} is closed under intersections,
the fact that S({A⊆Rd:A is a closed set})=B(Rd),
and Klenke [66, Theorem 2.16] hence assure that
for all t∈[0,T), s∈(t,T], B∈Ht it holds that
Ws−Wt and B are independent.
This implies that
for all t∈[0,T], s∈[t,T] it holds that
Ws−Wt and Ht are independent.
Combining this with the hypothesis that W is a Brownian motion, and the fact that W:[0,T]×Ω→Rm is an (Ht)t∈[0,T]/B(Rm)-adapted stochastic processes establishes that
W:[0,T]×Ω→Rm
is a standard
(Ω,F,P,(Ht)t∈[0,T])-Brownian motion.
The proof of Lemma 2.17 is thus completed.
∎
2.8 On a distributional flow property for solutions of SDEs
In this subsection we prove a distributional flow property for solutions of SDEs in Lemma 2.19 below.
The idea for the proof of Lemma 2.19 is based on the observation that if we replace solution processes of SDEs by Euler-Maruyama approximations the flow-type condition trivially holds (cf. the argument below (150) in the proof of Lemma 2.19 below).
To prove Lemma 2.19 below we also need, besides several auxiliary results of the previous subsections, the following well-known statement (see Lemma 2.18 below).
Lemma 2.18**.**
Let d,m∈N, T∈(0,∞), t∈[0,T], s∈[t,T],
let (Ω,F,P,(Ft)t∈[0,T]) be a filtered probability space which satisfies the usual conditions,
let W:[0,T]×Ω→Rm be a standard (Ω,F,P,(Fr)r∈[0,T])-Brownian motion,
let μ:[0,T]×Rd→Rd and σ:[0,T]×Rd→Rd×m be globally Lipschitz continuous functions,
let
X=(Xr(x))r∈[t,s],x∈Rd:[t,s]×Rd×Ω→Rd
be a continuous random field
which satisfies
for every x∈Rd that
(Xr(x))r∈[t,s]:[t,s]×Ω→Rd is an (Fr)r∈[t,s]/B(Rd)-adapted stochastic process and
which satisfies that for all r∈[t,s], x∈Rd
it holds P-a.s. that
[TABLE]
and let ξ:Ω→Rd be an Ft/B(Rd)-measurable function with E[∥ξ∥2]<∞.
Then
for all r∈[t,s] it holds P-a.s. that
Throughout this proof assume w.l.o.g. that s>t,
let (unN,r)n∈{0,1,2,…,N},N∈N,r∈(t,s]⊆[t,s] satisfy
for all N∈N, n∈{0,1,2,…,N}, r∈(t,s] that
unN,r=t+Nn(r−t),
for every N∈N, r∈(t,s]
let XN,r=(XnN,r(x))n∈{0,1,2,…,N},x∈Rd:{0,1,2,…,N}×Rd×Ω→Rd
be the continuous random field which satisfies
for all n∈{1,2,…,N}, x∈Rd that
X0N,r(x)=x and
[TABLE]
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let ∣∣∣⋅∣∣∣:Rd×m→[0,∞) be the Frobenius norm on Rd×m,
and let L∈[0,∞) satisfy
for all r,h∈[0,T], x,y∈Rd
that
[TABLE]
Note that (124), (126), (127),
Corollary 2.12
(with
d=d,
m=m,
N=N,
T=T,
t=t,
s=r,
L=L,
(Ω,F,P,(Fh)h∈[0,T])=(Ω,F,P,(Fh)h∈[0,T]),
W=W,
ζ=x,
μ=μ,
σ=σ,
(Xh)h∈[t,s]=(Xh)h∈[t,r],
(rn)n∈{0,1,…,N}=(unN,r)n∈{0,1,…,N},
(Xn)n∈{0,1,…,N}=(XnN,r(x))n∈{0,1,…,N}
for N∈N, x∈Rd, r∈(t,s]
in the notation of Corollary 2.12),
and
Lemma 2.10
(with
d=d,
m=m,
T=r−t,
ξ=x,
L=L,
(Ω,F,P,(Fh)h∈[0,T])=(Ω,F,P,(Ft+r)r∈[0,r−t]),
(Wh)h∈[0,T]=(Wt+h−Wt)h∈[0,r−t],
(μ(h,x))h∈[0,T],x∈Rd=(μ(t+h,x))h∈[0,r−t],x∈Rd,
(σ(h,x))h∈[0,T],x∈Rd=(σ(t+h,x))h∈[0,r−t],x∈Rd,
(Xh)h∈[0,T]=(Xt+h)h∈[0,r−t]
for x∈Rd, r∈(t,s]
in the notation of Lemma 2.10)
assure that
for all x∈Rd, N∈N, r∈(t,s] it holds that
[TABLE]
This ensures that
for all r∈[t,s], x∈Rd it holds that
limsupN→∞E[∥Xr(x)−XNN,r(x)∥2]=0.
This and the fact that
for all r∈[t,s], x∈Rd, N∈N
it holds that XNN,r(x):Ω→Rd is
S(Wh−Wt:h∈[t,r])/B(Rd)-measurable
imply that
for all r∈[t,s], x∈Rd it holds that Xr(x):Ω→Rd is
S(S(Wh−Wt:h∈[t,r])∪{A∈F:P(A)=0})/B(Rd)-measurable.
Combining this with the fact that ξ:Ω→Rd is an Ft/B(Rd)-measurable function and the fact that W:[0,T]×Ω→Rm is a standard (Ω,F,P,(Fr)r∈[0,T])-Brownian motion demonstrates
for all r∈[t,s], N∈N
it holds that (Xr(x)−XNN,r(x))x∈Rd and ξ are independent.
Lemma 2.15 and (128) hence assure that
for all N∈N, r∈(t,s] it holds that
[TABLE]
Next observe that
the hypothesis that μ and σ are globally Lipschitz continuous functions,
the hypothesis that E[∥ξ∥2]<∞,
and the existence theorem for the solutions of SDEs (see, e.g., Karatzas & Shreve [64, Proposition 5.2.9])
prove that
there exists an
(Fr)r∈[t,s]/B(Rd)-adapted stochastic process Y:[t,s]×Ω→Rd with continuous sample paths
which satisfies that
for all r∈[t,s]
it holds P-a.s. that
[TABLE]
Moreover, observe that (126) ensures that
for all N∈N, n∈{1,2,…,N}, r∈(t,s] and all functions ζ:Ω→Rd it holds that
X0N,r(ζ)=ζ and
[TABLE]
Combining this, (127), the fact that E[∥Yt∥2]=E[∥ξ∥2]<∞, and (130) with Corollary 2.12
(with
d=d,
m=m,
N=N,
T=T,
t=t,
s=r,
L=L,
(Ω,F,P,(Fh)h∈[0,T])=(Ω,F,P,(Fh)h∈[0,T]),
W=W,
ζ=ξ,
μ=μ,
σ=σ,
(Xh)h∈[t,s]=(Yh)h∈[t,r],
(rn)n∈{0,1,…,N}=(unN,r)n∈{0,1,…,N},
(Xn)n∈{0,1,…,N}=(XnN,r(ξ))n∈{0,1,…,N}
for N∈N, r∈(t,s]
in the notation of Corollary 2.12)
demonstrates
that
for all N∈N, r∈(t,s] it holds that
[TABLE]
The triangle inequality and (129) hence show that
for all r∈(t,s] it holds that
[TABLE]
Combining this with the fact that (Xr(ξ))r∈[t,s] and (Yr)r∈[t,s] are continuous random fields demonstrates that
[TABLE]
This and (130) prove that
for all r∈[t,s]
it holds P-a.s. that
Let d,m∈N, T∈(0,∞),
let μ:[0,T]×Rd→Rd and σ:[0,T]×Rd→Rd×m be globally Lipschitz continuous functions,
let (Ω,F,P) be a complete probability space,
let (Ft1)t∈[0,T] and (Ft2)t∈[0,T] be filtrations on (Ω,F,P) which satisfy the usual conditions,
assume that FT1 and FT2 are independent,
for every i∈{1,2}
let Wi:[0,T]×Ω→Rm
be a standard (Ω,F,P,(Fti)t∈[0,T])-Brownian motion,
and
for every i∈{1,2} let
Xi=(Xt,si(x))s∈[t,T],t∈[0,T],x∈Rd:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
be a continuous random field
which satisfies for every t∈[0,T], x∈Rd that
(Xt,si(x))s∈[t,T]:[t,T]×Ω→Rd is an (Fsi)s∈[t,T]/B(Rd)-adapted stochastic process and
which satisfies that
for all t∈[0,T], s∈[t,T], x∈Rd
it holds P-a.s. that
[TABLE]
Then it holds
for all r,s,t∈[0,T], x∈Rd, B∈B(Rd) with t≤s≤r that
P(Xt,t1(x)=x)=1
and
Throughout this proof let r,s,t∈[0,T], x∈Rd satisfy that t≤s≤r,
let (unN)n∈{0,1,2,…,N},N∈N⊆[t,s], (vnN)n∈{0,1,2,…,N},N∈N⊆[s,r] satisfy
for all N∈N, n∈{0,1,2,…,N} that
unN=t+Nn(s−t)
and
vnN=s+Nn(r−s),
for every N∈N
let XN:{0,1,2,…,2N}×Ω→Rd and
YN,ZN:{0,1,2,…,N}×Ω→Rd
be the stochastic processes which satisfy
for all n∈{1,2,…,N} that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
let Gh⊆F, h∈[0,T], and Hh⊆F, h∈[0,T], be the sigma-algebras which satisfy
for all h∈[0,T] that
[TABLE]
let ⟨⋅,⋅⟩:Rd×Rd→R be the Euclidean scalar product on Rd,
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
and
let ∣∣∣⋅∣∣∣:Rd×m→[0,∞) be the Frobenius norm on Rd×m.
Note that the hypothesis that (Ft1)t∈[0,T] and (Ft2)t∈[0,T] are filtrations on (Ω,F,P) which satisfy the usual conditions and (142) imply that
(Ht)t∈[0,T] is a filtration on (Ω,F,P) which satisfies the usual conditions.
Moreover, observe that (136) assures that
[TABLE]
Furthermore, note that
the hypothesis that μ and σ are globally Lipschitz continuous,
(136),
(138),
(139),
(140),
and Corollary 2.12
demonstrate that
there exists a real number C∈(0,∞) which satisfies that
for all N∈N it holds that
[TABLE]
This implies that
[TABLE]
Moreover, observe that the hypothesis that μ and σ are globally Lipschitz continuous implies that
Next note that the fact that
for all h∈[0,T], l∈[h,T] it holds that
Wl1−Wh1, Fh1, and Fh2 are independent
assures that
for all h∈[0,T], l∈[h,T] it holds that
Wl1−Wh1 and Gh are independent.
This, the fact that W1:[0,T]×Ω→Rd is a Brownian motion, and
the fact that W1:[0,T]×Ω→Rd is an (Gh)h∈[0,T]/B(Rd)-adapted stochastic process
imply that
W1:[0,T]×Ω→Rd is a standard (Ω,F,P,(Gh)h∈[0,T])-Brownian motion.
Lemma 2.17 and (142) hence
ensure that W1:[0,T]×Ω→Rd is a standard (Ω,F,P,(Hh)h∈[0,T])-Brownian motion.
Combining this,
the fact that (Ω,F,P,(Hh)h∈[0,T]) is a filtered probability space which satisfies the usual conditions,
the fact that
for all y∈Rd it holds that (Xs,h1(y))h∈[s,r]:[s,r]×Ω→Rd is an (Hh)h∈[s,r]/B(Rd)-adapted stochastic process,
(136),
the fact that
Xt,s2(x):Ω→Rd is Hs/B(Rd)-measurable,
and
(147)
with Lemma 2.18
(with
d=d,
m=m,
T=T,
t=s,
s=r,
(Ω,F,P,(Fh)h∈[0,T])=(Ω,F,P,(Hh)h∈[0,T]),
W=W1,
μ=μ,
σ=σ,
(Xh(y))h∈[t,s],y∈Rd=(Xs,h1(y))h∈[s,r],y∈Rd,
ξ=Xt,s2(x)
in the notation of Lemma 2.18)
proves that
for all h∈[s,r] it holds P-a.s. that
[TABLE]
The fact that (Ω,F,P,(Hh)h∈[0,T]) is a filtered probability space which satisfies the usual conditions,
the fact that W1:[0,T]×Ω→Rd is a standard (Ω,F,P,(Hh)h∈[0,T])-Brownian motion,
the fact that YNN:Ω→Rd is Hs/B(Rd)-measurable,
the hypothesis that μ and σ are globally Lipschitz continuous functions,
the fact that (Xs,h1(Xt,s2(x)))h∈[s,r]:[s,r]×Ω→Rd is an (Hh)h∈[s,r]/B(Rd)-adapted stochastic process with continuous sample paths,
(147),
(141),
and Corollary 2.12
(with
d=d,
m=m,
N=N,
T=T,
t=s,
s=r,
L=suph,l∈[0,T],y,z∈Rd:(h,y)=(l,z)∣h−l∣+∥y−z∥∥μ(h,y)−μ(l,z)∥+∣∣∣σ(h,y)−σ(l,z)∣∣∣,
(Ω,F,P,(Fh)h∈[0,T])=(Ω,F,P,(Hh)h∈[0,T]),
W=W1,
ζ=YNN,
μ=μ,
σ=σ,
(Xh)h∈[t,s]=(Xs,h1(Xt,s2(x)))h∈[s,r],
(rn)n∈{0,1,…,N}=(vnN)n∈{0,1,…,N},
(Xn)n∈{0,1,…,N}=(ZnN)n∈{0,1,…,N}
for N∈N
in the notation of Corollary 2.12)
hence demonstrate that there exists a real number K∈(0,∞) which satisfies that
for all N∈N it holds that
Furthermore, observe that (138)–(141)
assure that
for all N∈N it holds that
X2NN and ZNN have the same distribution.
This, (145), and (150) imply that
for all globally bounded and Lipschitz continuous functions g:Rd→R it holds that
[TABLE]
Lemma 2.13 hence assures that Xs,r1(Xt,s2(x)) and Xt,r1(x) are identically distributed.
Combining this with (143) completes the proof of Lemma 2.19.
∎
3 Full history recursive multilevel Picard (MLP) approximation algorithms
In this section we present the proposed MLP scheme and perform a rigorous complexity analysis.
First, we introduce our MLP scheme (cf. (156) in Subsection 3.1 below) as an approximation algorithm for a solution (cf. u in Setting 3.1 in Subsection 3.1 below) of certain type of stochastic fixed point equation (cf. (155) in Subsection 3.1 below) in Subsection 3.1.
Subsequently, the goal of Subsections 3.2–3.4 is to obtain an estimate for the L2-error between the MLP scheme and the solution of the stochastic fixed point equation.
This results in Proposition 3.15 and Corollary 3.16 in Subsection 3.4 below.
In Subsection 3.5 we estimate the computational effort needed to simulate realizations of the MLP scheme and combine this with the L2-error estimate in Corollary 3.16 to obtain a computational complexity analysis for the MLP algorithm in Proposition 3.18.
Finally, in Subsection 3.6, we exploit a connection between stochastic fixed point equations and viscosity solutions of semilinear Kolmogorov PDEs to carry over the complexity analysis of Subsection 3.5 to semilinear Kolmogorov PDEs (cf. (300) in Theorem 3.20 below) demonstrating that our proposed MLP algorithm overcomes the curse of dimensionality in the approximation of semilinear Kolmogorov PDEs in Theorem 3.20, the main result of this paper.
3.1 Stochastic fixed point equations and MLP approximations
Setting 3.1**.**
Let d∈N, T∈(0,∞), L∈[0,∞), Θ=∪n=1∞Zn,
u∈C([0,T]×Rd,R),
g∈C(Rd,R),
f∈C([0,T]×Rd×R,R) satisfy
for all t∈[0,T], x∈Rd, v,w∈R that
[TABLE]
let (Ω,F,P) be a probability space,
let
Rθ:Ω→[0,1],
θ∈Θ,
be independent U[0,1]-distributed random variables,
let
Rθ=(Rtθ)t∈[0,T]:[0,T]×Ω→[0,T],
θ∈Θ,
be the stochastic processes which satisfy
for all t∈[0,T], θ∈Θ that
[TABLE]
let
Xθ=(Xt,sθ(x))s∈[t,T],t∈[0,T],x∈Rd:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd,
θ∈Θ,
be independent continuous random fields
which satisfy
for all r,s,t∈[0,T], x∈Rd, θ,ϑ∈Θ, B∈B(Rd) with t≤s≤r and θ=ϑ that
P(Xt,tθ(x)=x)=1
and
[TABLE]
assume that
(Xθ)θ∈Θ
and
(Rθ)θ∈Θ
are independent,
assume
for all t∈[0,T], x∈Rd that
\mathbb{E}\big{[}|g(X^{0}_{t,T}(x))|+\int_{t}^{T}|f(r,X^{0}_{t,r}(x),u(r,X^{0}_{t,r}(x)))|\,dr\big{]}<\infty
and
[TABLE]
and let
VM,nθ:[0,T]×Rd×Ω→R, M,n∈Z, θ∈Θ,
be functions which satisfy
for all M,n∈N, θ∈Θ, t∈[0,T], x∈Rd that
VM,−1θ(t,x)=VM,0θ(t,x)=0
and
[TABLE]
3.2 A priori bounds for solutions of stochastic fixed point equations
In our L2-error analysis (see Subsection 3.4 below) of the MLP scheme introduced in Setting 3.1 we need to estimate expectations involving the solution of the stochastic fixed point equation. This estimate is carried out in Lemma 3.3 below.
In order to prove Lemma 3.3 we need the elementary and well known time reversed Gronwall inequality in Lemma 3.2.
Let T,α,β∈[0,∞) and let ϵ:[0,T]→[0,∞] be a B([0,T])/B([0,∞])-measurable function which satisfies
for all t∈[0,T] that
∫0Tϵ(r)dr<∞
and
ϵ(t)≤α+β∫tTϵ(r)dr.
Then
(i)
it holds
for all t∈[0,T] that
ϵ(t)≤αexp(β(T−t))
and
2. (ii)
Throughout this proof
let
Φ:[0,T]→[0,T]
and
ε:[0,T]→[0,∞]
be the functions which satisfy
for all t∈[0,T] that
[TABLE]
Observe that the integral transformation theorem (see, e.g., Klenke [66, Theorem 4.10]) implies that
for all t∈[0,T] it holds that
[TABLE]
Hence, we obtain that
[TABLE]
Moreover, observe that (157), (158), and the hypothesis that
for all t∈[0,T] it holds that
ϵ(t)≤α+β∫tTϵ(r)dr
assure that
for all t∈[0,T] it holds that
[TABLE]
Combining this and (159) with Gronwall’s integral inequality (cf, e.g., Grohs et al. [48, Lemma 2.11]) demonstrates that
for all t∈[0,T] it holds that
[TABLE]
Hence, we obtain that
for all t∈[0,T] it holds that
[TABLE]
This establishes items (i)–(ii).
The proof of Lemma 3.2 is thus completed.
∎
Lemma 3.3**.**
Assume Setting 3.1, let ξ∈Rd, C∈[0,∞] satisfy
that
[TABLE]
and assume that
∫0T(E[∣u(t,X0,t0(ξ))∣2])\nicefrac12dt<∞.
Then
(i)
it holds
for all t∈[0,T] that
(E[∣u(t,X0,t0(ξ))∣2])\nicefrac12≤Cexp(L(T−t))
and
2. (ii)
it holds that
supt∈[0,T](E[∣u(t,X0,t0(ξ))∣2])\nicefrac12≤Cexp(LT).
Throughout this proof assume w.l.o.g. that C<∞
and
let μt:B(Rd)→[0,1], t∈[0,T], be the probability measures which satisfy
for all t∈[0,T], B∈B(Rd) that
[TABLE]
(cf. item (iv) in Lemma 3.6).
Note that (155) and the triangle inequality ensure that
for all t∈[0,T] it holds that
[TABLE]
Jensen’s inequality hence assures that
for all t∈[0,T] it holds that
[TABLE]
Furthermore, observe that (164), the fact that X0 and X1 are independent and continuous random fields, (154), and Lemma 2.15 demonstrate that
for all t∈[0,T] it holds that
[TABLE]
In addition, note that
Minkowski’s integral inequality (cf., e.g., Jentzen & Kloeden [61, Proposition 8 in Appendix A.1]),
(164),
the fact that X0 and X1 are independent and continuous random fields,
(154),
and Lemma 2.15
imply that
for all t∈[0,T] it holds that
[TABLE]
Moreover, observe that (152) ensures that
for all t∈[0,T], x∈Rd, v∈R it holds that
[TABLE]
This, (168), and the triangle inequality imply that
for all t∈[0,T] it holds that
[TABLE]
Furthermore, note that Lemma 2.9 assures that
for all t∈[0,T] it holds that
[TABLE]
Combining this with (163), (166), (167), and (170) implies that
for all t∈[0,T] it holds that
[TABLE]
The hypothesis that
∫0T(E[∣u(t,X0,t0(ξ))∣2])\nicefrac12dt<∞
and Lemma 3.2
(with
T=T,
α=C,
β=L,
(\epsilon(t))_{t\in[0,T]}=\big{(}(\mathbb{E}[|u(t,X^{0}_{0,t}(\xi))|^{2}])^{\nicefrac{{1}}{{2}}}\big{)}_{t\in[0,T]}
in the notation of Lemma 3.2)
hence establish items (i)–(ii).
The proof of Lemma 3.3 is thus completed.
∎
3.3 Properties of MLP approximations
In this subsection we establish in Lemma 3.6 below some elementary properties of the MLP approximations (cf. (156) in Setting 3.1 above) introduced in Setting 3.1 above.
For this we need two elementary and well known results on identically distributed random variables (see Lemma 3.4 and Lemma 3.5 below).
Lemma 3.4**.**
Let d,N∈N,
let (Ω,F,P) be a probability space,
let Xk:Ω→Rd, k∈{1,2,…,N}, be independent random variables,
let Yk:Ω→Rd, k∈{1,2,…,N}, be independent random variables,
and assume for every k∈{1,2,…,N} that Xk and Yk are identically distributed.
Then it holds that
\big{(}\sum_{k=1}^{N}X_{k}\big{)}\colon\Omega\to\mathbb{R}^{d}
and
\big{(}\sum_{k=1}^{N}Y_{k}\big{)}\colon\Omega\to\mathbb{R}^{d}
are identically distributed random variables.
Throughout this proof
let X,Y:Ω→RNd be the random variables which satisfy that
[TABLE]
and
let f∈C(RNd,Rd) be the function which satisfies
for all v1,v2,…,vN∈Rd that
f(v1,v2,…,vN)=∑k=1Nvk.
Observe that
the hypothesis that (Xk)k∈{1,2,…,N} are independent,
the hypothesis that (Yk)k∈{1,2,…,N} are independent,
and
the hypothesis that for every k∈{1,2,…,N} it holds that Xk and Yk are identically distributed random variables
assure that
for all (Bk)k∈{1,2,…,N}⊆B(Rd) it holds that
[TABLE]
This, the fact that
[TABLE]
and the uniqueness theorem for measures (see, e.g., Klenke [66, Lemma 1.42])
imply that
it holds for all B∈B(RNd) that
[TABLE]
Hence, we obtain that
for all B∈B(Rd) it holds that
[TABLE]
This shows that
\big{(}\sum_{k=1}^{N}X_{k}\big{)}\colon\Omega\to\mathbb{R}^{d}
and
\big{(}\sum_{k=1}^{N}Y_{k}\big{)}\colon\Omega\to\mathbb{R}^{d}
are identically distributed random variables.
The proof of Lemma 3.4 is thus completed.
∎
Lemma 3.5**.**
Let (Ω,F,P) be a probability space,
let (S,δ) be a separable metric space,
let (E,δ) be a metric space,
let U,V:S×Ω→E
be continuous random fields,
let X,Y:Ω→S be random variables,
assume that U and X are independent,
assume that V and Y are independent,
assume
for all s∈S that U(s) and V(s) are identically distributed, and
assume that X and Y are identically distributed.
Then
it holds that
U(X)=(U(X(ω),ω))ω∈Ω:Ω→E and
V(Y)=(V(Y(ω),ω))ω∈Ω:Ω→E
are identically distributed random variables.
First, note that Grohs et al. [3, Lemma 2.4], the fact that U and V are continuous random fields, and
Lemma 2.14 ensure that U(X) and V(Y) are random variables.
Next observe
the hypothesis that U and X are independent,
the hypothesis that V and Y are independent,
the hypothesis that
for all s∈S it holds that
U(s) and V(s) are identically distributed,
the hypothesis that X and Y are identically distributed and Lemma 2.16
demonstrate that
for all globally bounded and Lipschitz continuous functions g:E→R it holds that
[TABLE]
Combining this with Lemma 2.13 assures that U(X) and V(Y) are identically distributed.
The proof of Lemma 3.5 is thus completed.
∎
We first prove item (i) by induction on n∈N0.
For the base case n=0 observe that the hypothesis that
for all θ∈Θ it holds that
VM,0θ=0
demonstrates that
for all θ∈Θ it holds that
VM,0θ:[0,T]×Rd×Ω→Rd
is a continuous random field.
This establishes item (i) in the base case n=0.
For the induction step N0∋(n−1)→n∈N
let n∈N and assume that
for every k∈N0∩[0,n), θ∈Θ it holds that
VM,kθ:[0,T]×Rd×Ω→Rd
is a continuous random field.
Combining this, the hypothesis that g and f are continuous functions, and
the fact that
for all θ∈Θ
it holds that
Rθ:[0,T]×Ω→[0,T]
and
Xθ:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
are continuous random fields
with (156),
Grohs et al. [3, Lemma 2.4], and Lemma 2.14
proves that for all θ∈Θ it holds that
VM,nθ:[0,T]×Rd×Ω→Rd
is a continuous random field.
Induction thus establishes item (i).
Next we prove item (ii) by induction on n∈N0.
For the base case n=0 observe that the hypothesis that
for all θ∈Θ it holds that
VM,0θ=0
demonstrates that
for all θ∈Θ it holds that
VM,0θ:[0,T]×Rd×Ω→R is
(B([0,T]×Rd)⊗S((R(θ,ϑ))ϑ∈Θ,(X(θ,ϑ))ϑ∈Θ))/B(R)-measurable.
This implies item (ii) in the base case n=0.
For the induction step N0∋(n−1)→n∈N
let n∈N and assume that
for all k∈N0∩[0,n), θ∈Θ it holds that
VM,kθ is
(B([0,T]×Rd)⊗S((R(θ,ϑ))ϑ∈Θ,(X(θ,ϑ))ϑ∈Θ))/B(R)-measurable.
Combining this, the fact that f and g are Borel measurable, and
the fact that
for all θ∈Θ
it holds that
Xθ:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
is a continuous random field
with (156) and Lemma 2.14
proves that for all θ∈Θ, t∈[0,T], x∈Rd it holds that
[TABLE]
Moreover, observe that item (i) and Grohs et al. [3, Lemma 2.4] ensure that for all θ∈Θ it holds that VM,nθ is
(B([0,T]×Rd)⊗S(VM,nθ))/B(R)-measurable.
Combining this with (180) demonstrates that
for all θ∈Θ it holds that
VM,nθ is
(B([0,T]×Rd)⊗S((R(θ,ϑ))ϑ∈Θ,(X(θ,ϑ))ϑ∈Θ))/B(R)-measurable.
Induction thus establishes item (ii).
Furthermore, observe that item (ii),
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent,
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
and Lemma 2.14
prove item (iii).
Next observe that (154), the hypothesis that (Xθ)θ∈Θ are independent, Lemma 2.16
(with
S=Rd,
U=g(Xs,sθ(⋅)),
X=Xt,sϑ(x)
for g∈C(Rd,R), t∈[0,T], s∈[t,T], x∈Rd, θ,ϑ∈Θ in the notation of Lemma 2.16),
and the fact that
for all t∈[0,T], x∈Rd, θ∈Θ it holds that
P(Xt,tθ(x)=x)=1
assure that
for all t∈[0,T], s∈[t,T], x∈Rd, θ,ϑ∈Θ with θ=ϑ and
all globally bounded and continuous functions g:Rd→R it holds that
[TABLE]
Combining this with Lemma 2.13 demonstrates that
for all t∈[0,T], s∈[t,T], x∈Rd, θ,ϑ∈Θ it holds that
Xt,sθ(x):Ω→Rd and Xt,sϑ(x):Ω→Rd are identically distributed random variables.
This establishes item (iv).
Next we prove item (v) by induction on n∈N0.
For the base case n=0 observe that the hypothesis that
for all θ∈Θ it holds that
VM,0θ=0
demonstrates that
for all t∈[0,T], x∈Rd it holds that
VM,0θ(t,x):Ω→Rd, θ∈Θ,
are identically distributed random variables.
This establishes item (v) in the base case n=0.
For the induction step N0∋(n−1)→n∈N
let n∈N and assume that
for all k∈N0∩[0,n), t∈[0,T], x∈Rd it holds that
VM,kθ(t,x):Ω→Rd, θ∈Θ,
are identically distributed random variables.
This,
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent,
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
item (ii),
Lemma 3.4,
and
Lemma 3.5
(with
S=[0,T]×Rd,
E=R,
U=\big{(}f(s,y,V^{(\theta,k,m)}_{M,k}(s,y))-\mathbbm{1}_{\mathbb{N}}(k)f(s,y,V^{(\theta,k,-m)}_{M,k-1}(s,y))\big{)}_{(s,y)\in[0,T]\times\mathbb{R}^{d}},
V=\big{(}f(s,y,V^{(\vartheta,k,m)}_{M,k}(s,y))-\mathbbm{1}_{\mathbb{N}}(k)f(s,y,V^{(\vartheta,k,-m)}_{M,k-1}(s,y))\big{)}_{(s,y)\in[0,T]\times\mathbb{R}^{d}},
X=(Rt(θ,k,m),Xt,Rt(θ,k,m)(θ,k,m)(x)),
Y=(Rt(ϑ,k,m),Xt,Rt(ϑ,k,m)(ϑ,k,m)(x))
for θ,ϑ∈Θ, t∈[0,T], x∈Rd, k∈N0∩[0,n), m∈N with θ=ϑ in the notation of Lemma 3.5)
assure that
for all t∈[0,T], x∈Rd, k∈N0∩[0,n), m∈N it holds that
[TABLE]
are identically distributed random variables.
Items (iii)–(iv),
(156),
and
Lemma 3.4
therefore
ensure that
for all t∈[0,T], x∈Rd it holds that
VM,nθ(t,x):Ω→Rd, θ∈Θ,
are identically distributed random variables.
Induction thus establishes item (v).
The proof of Lemma 3.6 is thus completed.
∎
3.4 Analysis of approximation errors of MLP approximations
Proposition 3.15, resp. Corollary 3.16, in Subsection 3.4.5 below presents estimates for the L2-approximation error of the MLP scheme (cf. (156) in Setting 3.1 above) introduced in Setting 3.1 with respect to the solution of the stochastic fixed point equation (cf. (155) in Setting 3.1 above)
for every iteration (cf. n∈N in (156) in Subsection 3.1 above) and every Monte Carlo accuracy (cf. M∈N in (156) in Subsection 3.1 above) of the MLP scheme.
The essential idea for the proof of those statements is to decompose the L2-approximation error into a bias and a variance part and to analyze them separately (see Subsections 3.4.1–3.4.3). This approach leads to a recursive inequality (cf. (240) in the proof of Proposition 3.15 below) which can be treated using an elementary Gronwall inequality, proven in Subsection 3.4.4 (see Lemma 3.12).
For the proofs of the statements in this subsection we need some elementary and well-known results
(see Lemma 3.7, Lemma 3.10, and Lemma 3.14)
which we state and prove where they are used.
3.4.1 Expectations of MLP approximations
Lemma 3.7**.**
Assume Setting 3.1,
let θ∈Θ, t∈[0,T],
let U1:[t,T]×Ω→[0,∞] and U2:[t,T]×Ω→R be continuous random fields which satisfy
for all i∈{1,2} that Ui and Rθ are independent and
∫tTE[∣U2(r)∣]dr<∞.
Then
it holds
for all i∈{1,2} that
Borel[t,T]({r∈[t,T]:E[∣U2(r)∣]=∞})=0,
E[∣U2(Rtθ)∣]<∞,
and
Throughout this proof assume w.l.o.g. that t<T.
Observe that (153) implies that Rtθ is U[t,T]-distributed.
Combining this with the fact that U1 is continuous, the fact that U1 and Rtθ are independent,
and Lemma 2.15 assures that
[TABLE]
In addition, note that
the fact that Rtθ is U[t,T]-distributed,
the fact that U2 is continuous,
the fact that U2 and Rtθ are independent,
the hypothesis that
∫tTE[∣U2(r)∣]dr<∞,
and Lemma 2.16 ensure that
Borel[t,T]({r∈[t,T]:E[∣U2(r)∣]=∞})=0,
E[∣U2(Rtθ)∣]<∞,
and
[TABLE]
Combining this with (184) establishes (183).
The proof of Lemma 3.7 is thus completed.
∎
Lemma 3.8** (Expectations of MLP approximations).**
Assume Setting 3.1 and assume
for all t∈[0,T], x∈Rd that
∫tTE[∣f(r,Xt,r0(x),0)∣]dr<∞.
Then
(i)
for all M∈N, n∈N0, t∈[0,T], s∈[t,T], x∈Rd it holds that
Throughout this proof let M∈N, x∈Rd.
Observe that
Lemma 3.7,
items (i)–(ii) in Lemma 3.6,
and the fact that
for all n∈N it holds that VM,n0, X0, and R0 are independent
demonstrate that
for all n∈N0, t∈[0,T] it holds that
[TABLE]
Next we claim that
for all n∈N0, t∈[0,T], s∈[t,T] it holds that
[TABLE]
We now prove (189) by induction on n∈N0.
For the base case n=0 observe that
the hypothesis that VM,00=0
and
the hypothesis that
for all t∈[0,T] it holds that
∫tTE[∣f(r,Xt,r0(x),0)∣]dr<∞
imply that
for all t∈[0,T], s∈[t,T] it holds that
[TABLE]
This establishes (189) in the case n=0.
For the induction step N0∋(n−1)→n∈N
let n∈N and assume that
for all k∈N0∩[0,n), t∈[0,T], s∈[t,T] it holds that
[TABLE]
Note that (156) and the triangle inequality ensure that
for all t∈[0,T], s∈[t,T] it holds that
[TABLE]
Furthermore, observe that
(154),
(155),
and
item (iv) in Lemma 3.6
assure that
for all m∈Z, t∈[0,T], s∈[t,T] it holds that
[TABLE]
Moreover, note that
Lemma 3.7,
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent,
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
items (i)–(ii) & (iv)–(v) in Lemma 3.6,
(154),
and
Lemma 2.15
demonstrate that
for all i,j,l,m∈Z, k∈N0, t∈[0,T], s∈[t,T] it holds that
[TABLE]
Combining this with (191), (192), and (193) establishes that
for all t∈[0,T], s∈[t,T] it holds that
[TABLE]
Hence, we obtain that
for all t∈[0,T] it holds that
[TABLE]
The hypothesis that
for all t∈[0,T] it holds that
∫tTE[∣f(r,Xt,r0(x),0)∣]dr<∞
and
the fact that
for all t∈[0,T], x∈Rd, v∈R it holds that
∣f(t,x,v)∣≤∣f(t,x,0)∣+L∣v∣
therefore assure that
for all t∈[0,T] it holds that
[TABLE]
This, (195), and (196) establish that
for all t∈[0,T], s∈[t,T] it holds that
[TABLE]
Induction thus proves (189).
Combining (188) and (189) establishes item (i).
Next observe that
(156),
(189),
items (i)–(ii) & (iv)–(v) in Lemma 3.6,
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent,
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
and
Lemma 3.5
ensure that
for all n∈N, t∈[0,T] it holds that
[TABLE]
Lemma 3.7,
items (i)–(ii) in Lemma 3.6,
the fact that
for all n∈N0 it holds that VM,n0, X0, and R0 are independent,
(189),
and
Fubini’s theorem
therefore imply that
for all n∈N, t∈[0,T] it holds that
[TABLE]
This establishes item (ii).
The proof of Lemma 3.8 is thus completed.
∎
3.4.2 Biases of MLP approximations
Lemma 3.9** (Biases of MLP approximations).**
Assume Setting 3.1
and assume
for all t∈[0,T], x∈Rd that
∫tTE[∣f(r,Xt,r0(x),0)∣]dr<∞.
Then
it holds
for all M,n∈N, t∈[0,T], x∈Rd that
Note that Lemma 3.8, the hypothesis that
for all t∈[0,T], x∈Rd it holds that
∫tTE[∣f(r,Xt,r0(x),0)∣]dr<∞,
(152), (155),
and Tonelli’s theorem
demonstrate that
for all M,n∈N, t∈[0,T], x∈Rd it holds that
[TABLE]
Lemma 2.9 and Jensen’s inequality hence show that
for all M,n∈N, t∈[0,T], x∈Rd it holds that
3.4.3 Estimates for the variances of MLP approximations
Lemma 3.10**.**
Let n∈N,
let (Ω,F,P) be a probability space, and
let X1,X2,…,Xn:Ω→R be independent random variables which satisfy
for all i∈{1,2,…,n} that
E[∣Xi∣]<∞.
Then
it holds that
Note that the fact that
for all independent random variables Y,Z:Ω→R with E[∣Y∣+∣Z∣]<∞ it holds that
E[∣YZ∣]<∞
and
E[YZ]=E[Y]E[Z]
(cf., e.g., Klenke [66, Theorem 5.4])
and the hypothesis that
Xi:Ω→R, i∈{1,2,…,n},
are independent random variables
assure that
Throughout this proof
let M,n∈N, t∈[0,T], x∈Rd.
Observe that Lemma 3.10,
item (i) in Lemma 3.8,
the fact that
for all θ∈Θ it holds that
E[∣g(Xt,T0(x))∣]<∞,
item (iii) in Lemma 3.6,
and
(156)
imply that
[TABLE]
Moreover, note that item (iv) in Lemma 3.6
and the fact that
for all Z∈L1(P,R) it holds that Var(Z)≤E[∣Z∣2] ensure that
[TABLE]
In addition, note that
items (i)–(ii) & (iv)–(v) in Lemma 3.6,
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent,
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
the fact that
for all Z∈L1(P,R) it holds that Var(Z)≤E[∣Z∣2],
and
Lemma 3.5
show that
for all k∈N0∩[0,n) it holds that
[TABLE]
Lemma 3.7,
the fact that X0 and R0 are independent,
and the hypothesis that
for all θ∈Θ it holds that VM,0θ=0
therefore demonstrate that
[TABLE]
In addition, observe that
(152),
(209),
the fact that
for all x,y∈[0,∞) it holds that
∣x+y∣2≤2(∣x∣2+∣y∣2),
items (i)–(ii) & (v) in Lemma 3.6,
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent,
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
and
Lemma 3.5
assure that
for all k∈N∩[1,n) it holds that
[TABLE]
Lemma 3.7, items (i)–(ii) in Lemma 3.6,
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent, and
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent
hence ensure that
for all k∈N∩[1,n) it holds that
[TABLE]
Combining this with (207), (208), and (210) establishes that
Throughout this proof assume w.l.o.g. that β>0.
We prove (215) by induction on n∈N0.
For the base case n=0 observe that (214) assures that
for all q∈N0 it holds that
[TABLE]
This proves (215) in the base case n=0.
For the induction step N0∋(n−1)→n∈N observe that (214) ensures that
for all n∈N, q∈N0 with
∀k∈N0∩[0,n),p∈N0:ϵk,p≤αMk+p(1+β)k
it holds that
[TABLE]
Induction hence establishes (215).
The proof of Lemma 3.12 is thus completed.
∎
3.4.5 Error estimates for MLP approximations
Corollary 3.13**.**
Assume Setting 3.1
and assume
for all t∈[0,T], x∈Rd that
∫tTE[∣f(r,Xt,r0(x),0)∣]dr<∞.
Then
it holds
for all M,n∈N, t∈[0,T], x∈Rd that
Throughout this proof
let M,n∈N, t∈[0,T], x∈Rd, C∈[0,∞], (ek)k∈N0∩[0,n)⊆[0,∞] satisfy that
for all k∈N0∩[0,n) that
[TABLE]
and
[TABLE]
Note that
item (i) in Lemma 3.8,
the bias variance decomposition of the mean square error (cf., e.g., Jentzen & von Wurstemberger [62, Lemma 2.2]),
the hypothesis that
for all s∈[0,T], z∈Rd it holds that
∫sTE[∣f(r,Xs,r0(z),0)∣]dr<∞,
Lemma 3.9,
and
Lemma 3.11 demonstrate that
Throughout this proof
assume w.l.o.g. that C<∞,
let M∈N,
let
ϵn,q∈[0,∞], n,q∈N0, be the extended real numbers which satisfy
for all n,q∈N0 that
[TABLE]
[TABLE]
and let μt:B(Rd)→[0,1], t∈[0,T], be the probability measures which satisfy
for all t∈[0,T], B∈B(Rd) that
[TABLE]
(cf. item (iv) in Lemma 3.6).
Note that the fact that
for all x,y∈[0,∞) it holds that
(x+y)2≥x2+y2
assures that
[TABLE]
Next observe that
items (i)–(ii) in Lemma 3.6,
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent,
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
Tonelli’s theorem,
Corollary 3.13,
and
Lemma 2.15 ensure that
for all n∈N, t∈[0,T] it holds that
[TABLE]
Moreover, observe that
(228),
(229),
the fact that
X0
and
X1
are independent and continuous random fields, (154),
and
Lemma 2.15 imply that
for all t∈[0,T] it holds that
[TABLE]
In addition, note that
(228),
items (i)–(ii) in Lemma 3.6,
the hypothesis that (Xθ)θ∈Θ are independent,
the hypothesis that (Rθ)θ∈Θ are independent, and
the hypothesis that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
(154),
Lemma 2.15,
and
Lemma 3.5
assure that
for all n∈N0, t∈[0,T], r∈[t,T] it holds that
[TABLE]
Combining this with (230) and (231) ensures that
for all n∈N, t∈[0,T] it holds that
[TABLE]
The fact that
P(X0,00(ξ)=ξ)=1,
the fact that
for all n∈N it holds that VM,n0, X0, and R0 are independent,
Lemma 3.5,
and
(226)
hence imply that
for all n∈N it holds that
[TABLE]
Moreover, observe that Lemma 3.14
(with
T=T,
q=q,
(U(r))r∈[0,T]=(E[∣u(r,X0,r0(ξ))−VM,n0(r,X0,r0(ξ))∣2])r∈[0,T]
for n∈N0, q∈N
in the notation of Lemma 3.14)
demonstrates that
for all n∈N0, q∈N it holds that
[TABLE]
This and (233) imply that
for all n,q∈N it holds that
[TABLE]
Furthermore, note the fact that
∫0T(E[∣u(r,X0,r0(ξ))∣2])\nicefrac12dr<∞
and
Lemma 3.3 prove that
[TABLE]
The fact that
P(X0,00(ξ)=ξ)=1
and
the fact that VM,00=0
hence
assure that
[TABLE]
Moreover, observe that
(237)
and
the fact that VM,00=0
ensure that
for all q∈N it holds that
[TABLE]
Combining this, (234), (236), and (238) demonstrates that
for all n,q∈N0 it holds that
[TABLE]
Lemma 3.12
(with
α=C2exp(M),
β=4L2T2,
M=M,
(ϵn,q)n,q∈N0=(ϵn,q)n,q∈N0
in the notation of Lemma 3.12)
therefore
proves that
for all n,q∈N0 it holds that
[TABLE]
This implies that
for all n∈N0 it hold that
[TABLE]
The fact that
for all x,y∈[0,∞) it holds that
x+y≤x+y
hence demonstrates that
for all n∈N0 it holds that
[TABLE]
The proof of Proposition 3.15 is thus completed.
∎
Corollary 3.16**.**
Assume Setting 3.1, let ξ∈Rd, C∈[0,∞] satisfy
that
[TABLE]
and assume
for all t∈[0,T], x∈Rd that
∫0T(E[∣u(r,X0,r0(ξ))∣2])\nicefrac12dr+∫tTE[∣f(r,Xt,r0(x),0)∣]dr<∞.
Then
it holds
for all N∈N that
Proposition 3.15 establishes Corollary 3.16.
The proof of Corollary 3.16 is thus completed.
∎
3.5 Complexity analysis for MLP approximation algorithms
In this subsection we consider the computational effort of the MLP scheme (cf. (156) in Setting 3.1 above) introduced in Setting 3.1 and combine it with the L2-error estimate in Corollary 3.16 to obtain a complexity analysis for the MLP scheme in Proposition 3.18 below.
In Lemma 3.17 we think for all M,n∈N of CM,n as the number of realizations of 1-dimensional random variables needed to simulate one realization of VM,nθ(t,x) for any θ∈Θ, t∈[0,T], x∈Rd.
The recursive inequality in (246) in Lemma 3.17 is based on (156) and the assumption that the number of
realizations of 1-dimensional random variables needed to simulate Xt,rθ(x) for any θ∈Θ, t∈[0,T], r∈[t,T], x∈Rd is bounded by αd.
Lemma 3.17**.**
Let d∈N, α∈[1,∞),
(CM,n)M,n∈Z⊆[0,∞) satisfy
for all n,M∈N that
CM,0=0
and
First, observe that (246) and the hypothesis that
for all M∈N it holds that
CM,0=0 imply that
for all n∈N, M∈N∩[2,∞) it holds
that
[TABLE]
The discrete Gronwall inequality in Corollary 2.2
(with
N=∞,
α=3αd+2,
β=(1+M1),
(ϵn)n∈N0=(M−(n+1)CM,(n+1))n∈N0
in the notation of Corollary 2.2)
hence ensures that
for all n∈N0, M∈N∩[2,∞) it holds that
[TABLE]
This establishes that
for all n∈N, M∈N∩[2,∞) it holds that
[TABLE]
Moreover, observe that the fact that C1,0=0 and (246) demonstrate that
for all n∈N it holds that
[TABLE]
Hence, we obtain
for all n∈N, k∈N∩(0,n] that
[TABLE]
Combining this with the discrete Gronwall inequality in Corollary 2.2
(with
N=n−1,
α=αd+n(αd+1),
β=2,
(ϵk)k∈N0∩[0,N]=(C1,k+1)k∈N0∩[0,n)
in the notation of Corollary 2.2)
proves that
for all n∈N, k∈N0∩[0,n) it holds that
[TABLE]
The fact that
for all n∈N it holds that
(1+2n)3n−1≤5n
hence
shows that
for all n∈N it holds that
[TABLE]
Combining this with (249) completes the proof of Lemma 3.17.
∎
Proposition 3.18**.**
Assume 3.1,
let ξ∈Rd, C∈[0,∞), α∈[1,∞),
(CM,n)M,n∈Z⊆N0 satisfy
for all n,M∈N that
[TABLE]
[TABLE]
and assume
for all t∈[0,T], x∈Rd that
∫0T(E[∣u(r,X0,r0(ξ))∣2])\nicefrac12dr+∫tTE[∣f(r,Xt,r0(x),0)∣]dr<∞.
Then
there exists a function
N:(0,∞)→N
such that
for all ε,δ∈(0,∞) it holds that
let
N:(0,∞)→N
be the function which satisfies
for all ε∈(0,∞) that
[TABLE]
and let δ∈(0,∞).
Note that (259) and Corollary 3.16 assure that
for all ε∈(0,∞) it holds that
[TABLE]
Moreover, observe that (259) ensures that
for all ε∈(0,∞) with Nε≥2 it holds that
[TABLE]
Lemma 3.17 and (254) hence show that
for all ε∈(0,∞) with Nε≥2 it holds that
[TABLE]
Next note that
for all n∈N∩[2,∞) it holds that
[TABLE]
Furthermore, observe that the fact that κ≥e and the fact that 5e≤4 imply that
for all n∈N∩[2,∞) it holds that
[TABLE]
Combining this, (263), and
the fact that
for all n∈N it holds that
n≤(4+8LT)n
demonstrates that
[TABLE]
In addition, observe that
[TABLE]
This, (262), and (265) prove that
for all ε∈(0,∞) with Nε≥2 it holds that
[TABLE]
Next note that
the hypothesis that C1,0=0, (254), and
the fact that
3≤supn∈N[n\nicefrac(nδ)2(4+8LT)(n+1)(3+δ)]<∞
assure that
for all ε∈(0,∞) with Nε=1 it holds that
[TABLE]
This and (267) demonstrate that
for all ε∈(0,∞) it holds that
[TABLE]
Combining this with (260) completes the proof of Proposition 3.18.
∎
3.6 MLP approximations for semilinear partial differential equations (PDEs)
Thanks to an equivalence between semilinear Kolmogorov PDEs and stochastic fixed points equations
we can carry over the complexity analysis of Subsection 3.5 for the approximation of solutions of stochastic fixed points equations to our proposed MLP scheme for the approximation of solutions of semilinear Kolmogorov PDEs (cf. (275) in Subsection 3.6.1 below) resulting in Proposition 3.19.
Considering this complexity analysis over variable dimensions shows that our proposed MLP algorithm overcomes the curse of dimensionality in the approximation of solutions of certain semilinear Kolmogorov PDEs (see Theorem 3.20 in Subsection 3.6.2 below, the main result of this paper, for details).
3.6.1 MLP approximations in fixed space dimensions
Proposition 3.19**.**
Let d,m∈N, T∈(0,∞), L,K,p,C1,C2,C∈[0,∞), α∈[1,∞), ξ∈Rd, Θ=∪n=1∞Zn,
let ⟨⋅,⋅⟩:Rd×Rd→R be the Euclidean scalar product on Rd,
let ∥⋅∥:Rd→[0,∞) be the Euclidean norm on Rd,
let ∣∣∣⋅∣∣∣:Rd×m→[0,∞) be the Frobenius norm on Rd×m,
assume that
[TABLE]
let g∈C(Rd,R), f∈C([0,T]×Rd×R,R) satisfy
for all t∈[0,T], x∈Rd, v,w∈R that
[TABLE]
let μ:[0,T]×Rd→Rd and σ:[0,T]×Rd→Rd×m be globally Lipschitz continuous functions
which satisfy
for all t∈[0,T], x∈Rd that
[TABLE]
let (Ω,F,P) be a complete probability space,
let
Rθ:Ω→[0,1],
θ∈Θ,
be independent U[0,1]-distributed random variables,
let
Rθ=(Rtθ)t∈[0,T]:[0,T]×Ω→[0,T],
θ∈Θ,
be the stochastic processes which satisfy
for all t∈[0,T], θ∈Θ that
[TABLE]
let (Ftθ)t∈[0,T], θ∈Θ, be filtrations on (Ω,F,P) which satisfy the usual conditions,
assume that (FTθ)θ∈Θ is an independent family of sigma-algebras,
assume that
(FTθ)θ∈Θ
and
(Rθ)θ∈Θ
are independent,
for every θ∈Θ
let Wθ:[0,T]×Ω→Rm
be a standard (Ω,F,P,(Ftθ)t∈[0,T])-Brownian motion,
for every θ∈Θ let
Xθ=(Xt,sθ(x))s∈[t,T],t∈[0,T],x∈Rd:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
be a continuous random field
which satisfies for every t∈[0,T], x∈Rd that
(Xt,sθ(x))s∈[t,T]:[t,T]×Ω→Rd is an (Fsθ)s∈[t,T]/B(Rd)-adapted stochastic process and
which satisfies that
for all t∈[0,T], s∈[t,T], x∈Rd
it holds P-a.s. that
[TABLE]
let
VM,nθ:[0,T]×Rd×Ω→R, M,n∈Z, θ∈Θ,
be functions which satisfy
for all M,n∈N, θ∈Θ, t∈[0,T], x∈Rd that
VM,−1θ(t,x)=VM,0θ(t,x)=0
and
[TABLE]
and let (CM,n)M,n∈Z⊆N0 satisfy
for all n,M∈N that
CM,0=0 and
[TABLE]
Then
(i)
there exists a unique at most polynomially growing function u∈C([0,T]×Rd,R) which satisfies that
u∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
u(T,x)=g(x),
2. (ii)
it holds
for all M∈N, n∈N0 that
[TABLE]
and
3. (iii)
there exists a function
N:(0,∞)→N
such that
for all ε,δ∈(0,∞) it holds that
Throughout this proof let (ρ1(q))q∈[0,∞),(ρ2(q))q∈[0,∞)⊆(0,∞), C∈[0,∞] satisfy
for all q∈[0,∞) that
[TABLE]
[TABLE]
Observe that the fact that μ and σ are globally Lipschitz continuous functions
and
(271)
assure that
there exists a unique at most polynomially growing function u∈C([0,T]×Rd,R) which satisfies that
u∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
u(T,x)=g(x)
(cf., e.g., Hairer et al. [50, Section 4]).
This proves item (i).
In addition, note that
the fact that μ and σ are globally Lipschitz continuous functions,
(271),
(274),
and
the Feynman-Kac formula
assure that
for all t∈[0,T], x∈Rd it holds that
[TABLE]
(cf., e.g., Hairer et al. [50, Section 4]).
Moreover, observe that
the hypothesis that μ and σ are globally Lipschitz continuous functions,
the fact that
for all θ,ϑ∈Θ with θ=ϑ it holds that FTθ and FTϑ are independent,
(274), and
Lemma 2.19 assure that
for all θ,ϑ∈Θ, r,s,t∈[0,T], x∈Rd, B∈B(Rd) with t≤s≤r and θ=ϑ
it holds
that
P(Xt,tθ(x)=x)=1
and
[TABLE]
Next note that
the hypothesis that μ and σ are globally Lipschitz continuous functions,
(272),
(274),
and Lemma 2.6
(with
d=d,
m=m,
T=T−t,
C1=C1,
C2=C2,
ξ=x,
(μ(r,y))r∈[0,T],y∈Rd=(μ(t+r,y))r∈[0,T−t],y∈Rd,
(σ(r,y))r∈[0,T],y∈Rd=(σ(t+r,y))r∈[0,T−t],y∈Rd, (Ω,F,P,(Fr)r∈[0,T])=(Ω,F,P,(Ft+r)r∈[0,T−t]),
(Wr)r∈[0,T]=(Wt+r0−Wt0)r∈[0,T−t],
(Xr)r∈[0,T]=(Xt,t+r0(x))r∈[0,T−t]
for t∈[0,T], x∈Rd
in the notation of Lemma 2.6)
assure that
for all x∈Rd, t∈[0,T], s∈[t,T], q∈[0,∞) it holds that
[TABLE]
For the next step let K,p∈[0,∞) satisfy
for all t∈[0,T], x∈Rd that
[TABLE]
This, Tonelli’s theorem, and (271) assure that
for all t∈[0,T], x∈Rd it holds that
[TABLE]
Moreover, observe that (271), (286), (287), and the triangle inequality demonstrate that
for all t∈[0,T], x∈Rd it holds that
[TABLE]
Combining this,
(271),
(275),
(284),
(285),
(288),
the fact that (Xθ)θ∈Θ are independent,
and
the fact that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent
with Proposition 3.15
(with
d=d,
T=T,
L=L,
u=u,
g=g,
f=f,
Rθ=Rθ,
Xθ=Xθ,
VM,nθ=VM,nθ,
ξ=ξ,
C=C
for M,n∈Z, θ∈Θ
in the notation of Proposition 3.15)
proves that
for all M∈N, n∈N0 it holds that
[TABLE]
Next observe that (271), (286), and the triangle inequality imply that
[TABLE]
In addition note that (271), (286), and the triangle inequality imply that
[TABLE]
Combining this and (291) with (281) and (282) demonstrates that
[TABLE]
This and (290) establish item (ii).
In addition, observe that
(271),
(275),
(276),
(284),
(285),
(288)
(289),
the fact that (Xθ)θ∈Θ are independent,
the fact that (Xθ)θ∈Θ and (Rθ)θ∈Θ are independent,
(293),
and
Proposition 3.18
(with
d=d,
T=T,
L=L,
u=u,
g=g,
f=f,
Xθ=Xθ,
VM,nθ=VM,nθ,
ξ=ξ,
C=C,
α=α,
CM,n=CM,n
for M,n∈Z, θ∈Θ
in the notation of Proposition 3.15)
prove that
there exists a function
N:(0,∞)→N
such that
for all ε,δ∈(0,∞) it holds that
[TABLE]
[TABLE]
This establishes item (iii).
The proof of Proposition 3.19 is thus completed.
∎
3.6.2 MLP approximations in variable space dimensions
Theorem 3.20**.**
Let T∈(0,∞), α,c,K∈[1,∞), L,p,P,P,q,C1,C2∈[0,∞),
for every d∈N
let ∥⋅∥Rd:Rd→[0,∞) be the Euclidean norm on Rd,
let ⟨⋅,⋅⟩Rd:Rd×Rd→R be the Euclidean scalar product on Rd, and
let ∣∣∣⋅∣∣∣d:Rd×d→[0,∞) be the Frobenius norm on Rd×d,
for every d∈N let ξd∈Rd satisfy that ∥ξd∥Rd≤cdq,
for every d∈N let gd∈C(Rd,R), fd∈C([0,T]×Rd×R,R) satisfy
for all t∈[0,T], x∈Rd, v,w∈R that
[TABLE]
for every d∈N let μd:[0,T]×Rd→Rd and σd:[0,T]×Rd→Rd×d be globally Lipschitz continuous functions which satisfy
for all t∈[0,T], x∈Rd that
[TABLE]
let (Ω,F,P) be a complete probability space,
let
Rθ:Ω→[0,1],
θ∈Θ,
be independent U[0,1]-distributed random variables,
let
Rθ=(Rtθ)t∈[0,T]:[0,T]×Ω→[0,T],
θ∈Θ,
be the stochastic processes which satisfy
for all t∈[0,T], θ∈Θ that
[TABLE]
let (Ftd,θ)t∈[0,T], d∈N, θ∈Θ, be filtrations on (Ω,F,P) which satisfy the usual conditions,
assume for every d∈N that (FTd,θ)θ∈Θ is an independent family of sigma-algebras,
assume that
(FTd,θ)d∈N,θ∈Θ
and
(Rθ)θ∈Θ
are independent,
for every d∈N, θ∈Θ
let Wd,θ:[0,T]×Ω→Rd
be a standard (Ω,F,P,(Ftd,θ)t∈[0,T])-Brownian motion,
for every d∈N, θ∈Θ
let Xd,θ=(Xt,sd,θ(x))s∈[t,T],t∈[0,T],x∈Rd:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
be a continuous random field
which satisfies for every t∈[0,T], x∈Rd that
(Xt,sd,θ(x))s∈[t,T]:[t,T]×Ω→Rd is an (Fsd,θ)s∈[t,T]/B(Rd)-adapted stochastic process and
which satisfies that
for all t∈[0,T], s∈[t,T], x∈Rd
it holds P-a.s. that
[TABLE]
let
VM,nd,θ:[0,T]×Rd×Ω→R, M,n∈Z, θ∈Θ, d∈N,
be functions which satisfy
for all d,M,n∈N, θ∈Θ, t∈[0,T], x∈Rd that
VM,−1d,θ(t,x)=VM,0d,θ(t,x)=0
and
[TABLE]
and let (Cd,M,n)M,n∈Z,d∈N⊆N0 satisfy
for all d,n,M∈N that
Cd,M,0=0
and
[TABLE]
Then
(i)
for every d∈N there exists a unique at most polynomially growing function ud∈C([0,T]×Rd,R) which satisfies that
ud∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
ud(T,x)=gd(x)
and
2. (ii)
there exists a function
N=(Nd,ε)d∈N,ε∈(0,∞):N×(0,∞)→N
such that
for all d∈N, ε,δ∈(0,∞) it holds that
Throughout this proof let (βδ)δ∈(0,∞)⊆(0,∞), (Cd)d∈N⊆[0,∞) satisfy
for all δ∈(0,∞), d∈N that
βδ=[supn∈Nn(nδ/2)(4+8LT)(3+δ)(n+1)] and
[TABLE]
Observe that Proposition 3.19
(with
d=d,
m=d,
T=T,
L=L,
K=KdP,
p=p,
C1=C1dP,
C2=C2,
α=α,
ξ=ξd,
g=gd,
f=fd,
μ=μd,
σ=σd,
Rθ=Rθ,
Fθ=Fd,θ,
Wθ=Wd,θ,
Xθ=Xd,θ,
VM,nθ=VM,nd,θ,
CM,n=Cd,M,n
for d∈N, M,n∈Z, θ∈Θ
in the notation of Proposition 3.19)
proves that for every d∈N
(I)
there exists a unique at most polynomially growing function ud∈C([0,T]×Rd,R) which satisfies that
ud∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
ud(T,x)=gd(x) and
2. (II)
there exists a function
Nd=(Nd,ε)ε∈(0,∞):(0,∞)→N
such that
for all ε,δ∈(0,∞) it holds that
[TABLE]
[TABLE]
Observe that item (I) proves item (i).
Moreover, note that the hypothesis that
for all d∈N it holds that ∥ξd∥Rd≤cdq
and the fact that
(2p+1)≤4p+1
imply that
for all d∈N it holds that
[TABLE]
This and (308) demonstrate that
for all d∈N, δ,ε∈(0,∞) it holds that
[TABLE]
Combining this and (307) establishes item (ii).
The proof of Theorem 3.20 is thus completed.
∎
4 MLP approximations for PDE models
The MLP scheme for semilinear Kolmogorov PDEs (cf. (300) in Theorem 3.20 above) proposed in Subsection 3.6 can only be implemented for semilinear Kolmogorov PDEs for which an explicit solution of the corresponding SDE is known.
In this section, we consider the MLP algorithm for two examples of such semilinear Kolmogorov PDEs, semilinear heat equations (see Subsection 4.1 below) and semilinear Black-Scholes equations (see Subsections 4.2–4.3 below).
Apart from specifying the linear part of the PDE we also choose a particular nonlinearity (cf. (357) in Corollary 4.5 below) in Subsection 4.3 to obtain a PDE, which is used in the pricing of financial derivatives with default risk (cf., e.g., Han et al. [51, (10)] and Duffie et al. [33]).
4.1 MLP approximations for semilinear heat equations
Theorem 4.1**.**
Let T∈(0,∞), κ,p,P,q∈[0,∞), Θ=∪n=1∞Zn,
for every d∈N let ∥⋅∥Rd:Rd→[0,∞) be the Euclidean norm on Rd,
for every d∈N let ξd∈Rd satisfy that ∥ξd∥Rd≤κdq,
for every d∈N let gd∈C(Rd,R), fd∈C([0,T]×Rd×R,R) satisfy
for all t∈[0,T], x∈Rd, v,w∈R that
[TABLE]
let (Ω,F,P) be a probability space,
let
Rθ:Ω→[0,1],
θ∈Θ,
be independent U[0,1]-distributed random variables,
let
Rθ=(Rtθ)t∈[0,T]:[0,T]×Ω→[0,T],
θ∈Θ,
be the stochastic processes which satisfy
for all t∈[0,T], θ∈Θ that
[TABLE]
for every d∈N let Wd,θ:[0,T]×Ω→Rd,
θ∈Θ,
be independent standard Brownian motions,
assume that
(Wd,θ)d∈N,θ∈Θ
and
(Rθ)θ∈Θ
are independent,
for every d∈N, θ∈Θ
let Xd,θ=(Xt,sd,θ(x))s∈[t,T],t∈[0,T],x∈Rd:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
be the function which satisfies
for all t∈[0,T], s∈[t,T], x∈Rd that
[TABLE]
let
VM,nd,θ:[0,T]×Rd×Ω→R, M,n∈Z, θ∈Θ, d∈N,
be functions which satisfy
for all d,M,n∈N, θ∈Θ, t∈[0,T], x∈Rd that
VM,−1d,θ(t,x)=VM,0d,θ(t,x)=0
and
[TABLE]
and let (Cd,M,n)M,n∈Z,d∈N⊆N0 satisfy
for all d,n,M∈N that
Cd,M,0=0
and
[TABLE]
Then
(i)
for every d∈N there exists a unique at most polynomially growing function ud∈C([0,T]×Rd,R) which satisfies that
ud∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
ud(T,x)=gd(x)
and
2. (ii)
there exist functions
N=(Nd,ε)d∈N,ε∈(0,∞):N×(0,∞)→N
and
C=(Cδ)δ∈(0,∞):(0,∞)→(0,∞)
such that
for all d∈N, ε,δ∈(0,∞) it holds that
Throughout this proof
assume w.l.o.g. that κ≥1,
assume w.l.o.g. that (Ω,F,P) is a complete probability space,
for every d∈N
let ⟨⋅,⋅⟩Rd:Rd×Rd→R be the Euclidean scalar product on Rd and
let ∣∣∣⋅∣∣∣d:Rd×d→[0,∞) be the Frobenius norm on Rd×d,
let μd∈C([0,T]×Rd,Rd), d∈N, and σd∈C([0,T]×Rd,Rd×d), d∈N, satisfy
for all d∈N, t∈[0,T], x∈Rd that
[TABLE]
and
for every d∈N, θ∈Θ, t∈[0,T] let Ftd,θ⊆F
be the sigma-algebra which satisfies that
[TABLE]
Note that (320) implies that
for every d∈N, θ∈Θ it holds that
(Ftd,θ)t∈[0,T] is a filtration on (Ω,F,P) which satisfies the usual conditions.
Moreover, observe that (320) and Lemma 2.17 demonstrate that
for every d∈N, θ∈Θ it holds that
Wd,θ:[0,T]×Ω→Rd is a standard (Ω,F,P,(Ftd,θ)t∈[0,T])-Brownian motion.
Next note that (313) and (319) assure that
for every d∈N, θ∈Θ it holds that
Xd,θ is continuous random field which satisfies
for every t∈[0,T], x∈Rd that
(Xt,sd,θ(x))s∈[t,T]:[t,T]×Ω→Rd is an (Fsd,θ)s∈[t,T]/B(Rd)-adapted stochastic process
and which satisfies
that for all t∈[0,T], s∈[t,T], x∈Rd
it holds P-a.s. that
[TABLE]
In addition, note that
for all d∈N, t∈[0,T], x∈Rd it holds that
[TABLE]
This,
(311),
(312),
(314),
(315),
(319),
(321),
and Theorem 3.20
(with
T=T,
α=1,
c=κ,
K=κ,
L=κ,
p=p,
P=1,
P=P,
q=q,
C1=1,
C2=0,
ξd=ξd,
gd=gd,
fd=fd,
μd=μd,
σd=σd,
Rθ=Rθ,
Fd,θ=Fd,θ,
Wd,θ=Wd,θ,
Xd,θ=Xd,θ,
VM,nd,θ=VM,nd,θ,
Cd,M,n=Cd,M,n
for d∈N, M,n∈Z, θ∈Θ
in the notation of Theorem 3.20)
establish that
(I)
for every d∈N there exists a unique at most polynomially growing function ud∈C([0,T]×Rd,R) which satisfies that
ud∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
ud(T,x)=gd(x)
and
2. (II)
there exists a function
N=(Nd,ε)d∈N,ε∈(0,∞):N×(0,∞)→N
such that
for all d∈N, ε,δ∈(0,∞) it holds that
[TABLE]
[TABLE]
Note that item (I) establishes item (i).
Moreover, observe that item (II) establishes item (ii).
The proof of Theorem 4.1 is thus completed.
∎
4.2 MLP approximations for semilinear Black-Scholes equations
Lemma 4.2**.**
Let d∈N, T∈(0,∞), (αi)i∈{1,2,…,d}, (βi)i∈{1,2,…,d}⊆R,
let ⟨⋅,⋅⟩:Rd×Rd→R be the Euclidean scalar product on Rd,
let Σ=(ζ1,…,ζd)∈Rd×d satisfy
for all i∈{1,2,…,d} that
⟨ζi,ζi⟩=1,
let (Ω,F,P) be a complete probability space,
let W:[0,T]×Ω→Rd be a d-dimensional standard Brownian motion,
and
let
X=(Xt,s(i)(x))s∈[t,T],t∈[0,T],x∈Rd,i∈{1,2,…,d}:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
be the function which satisfies
for all
i∈{1,2,…,d}, t∈[0,T], s∈[t,T], x=(x1,x2,…,xd)∈Rd that
[TABLE]
Then
it holds that X is a continuous random field which satisfies
that for all t∈[0,T], s∈[t,T], x∈Rd it holds
P-a.s. that
Throughout this proof let t∈[0,T], s∈(0,T], x=(x1,x2,…,xd)∈Rd,
let
fi:[0,T]×Rd→R, i∈{1,2,…,d},
be the functions which satisfy
for all i∈{1,2,…,d}, r∈[0,T], w∈Rd that
[TABLE]
let B=(B(i))i∈{1,2,…,d}:[0,s−t]×Ω→Rd satisfy
for all r∈[0,s−t] that
Br=Wt+r−Wt,
and let ζi(j)∈R, i,j∈{1,2,…,d}, satisfy
for all i∈{1,2,…,d} that
ζi=(ζi(j))j∈{1,2,…,d}.
Observe that Itô’s formula (cf., e.g., Karatzas & Shreve [64, Theorem 3.3.6]) assures that
for all i∈{1,2,…,d}
it holds P-a.s. that
[TABLE]
The fact that
for all i∈{1,2,…,d} it holds that \sum_{j=1}^{d}\big{|}\zeta_{i}^{(j)}\big{|}^{2}=\left<\zeta_{i},\zeta_{i}\right>=1
and the fact that
for all i∈{1,2,…,d}, r∈[0,s−t] it holds that
fi(r,Br)=Xt,t+r(i)(x)
hence assure that
for all i∈{1,2,…,d}
it holds P-a.s. that
[TABLE]
This implies (327).
The proof of Lemma 4.2 is thus completed.
∎
Theorem 4.3**.**
Let T∈(0,∞), κ,p,P,q∈[0,∞), (αd,i)i∈{1,2,…,d},d∈N, (βd,i)i∈{1,2,…,d},d∈N⊆R, Θ=∪n=1∞Zn satisfy that
supd∈N,i∈{1,2,…,d}max{∣αd,i∣,∣βd,i∣2}≤κ,
for every d∈N
let ⟨⋅,⋅⟩Rd:Rd×Rd→R be the Euclidean scalar product on Rd and
let ∥⋅∥Rd:Rd→[0,∞) be the Euclidean norm on Rd,
for every d∈N let ξd∈Rd, Σd=(ζd,1,…,ζd,d)∈Rd×d satisfy
for all i∈{1,2,…,d} that
∥ξd∥Rd≤κdq and
∥ζd,i∥Rd=1,
for every d∈N let μd:[0,T]×Rd→Rd and σd:[0,T]×Rd→Rd×d be the functions which satisfy
for all t∈[0,T], x=(x1,x2,…,xd)∈Rd that
[TABLE]
for every d∈N let gd∈C(Rd,R), fd∈C([0,T]×Rd×R,R) satisfy
for all t∈[0,T], x∈Rd, v,w∈R that
[TABLE]
let (Ω,F,P) be a probability space,
let
Rθ:Ω→[0,1],
θ∈Θ,
be independent U[0,1]-distributed random variables,
let
Rθ=(Rtθ)t∈[0,T]:[0,T]×Ω→[0,T],
θ∈Θ,
be the stochastic processes which satisfy
for all t∈[0,T], θ∈Θ that
[TABLE]
for every d∈N
let
Wd,θ:[0,T]×Ω→Rd,
θ∈Θ,
be independent standard Brownian motions,
assume that
(Wd,θ)d∈N,θ∈Θ
and
(Rθ)θ∈Θ
are independent,
for every d∈N, θ∈Θ
let
Xd,θ=(Xt,sd,θ,i(x))s∈[t,T],t∈[0,T],x∈Rd,i∈{1,2,…,d}:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
be the function which satisfies
for all
t∈[0,T], s∈[t,T], x=(x1,x2,…,xd)∈Rd, i∈{1,2,…,d} that
[TABLE]
let
VM,nd,θ:[0,T]×Rd×Ω→R, M,n∈Z, θ∈Θ, d∈N,
be functions which satisfy
for all d,M,n∈N, θ∈Θ, t∈[0,T], x∈Rd that
VM,−1d,θ(t,x)=VM,0d,θ(t,x)=0
and
[TABLE]
and let (Cd,M,n)M,n∈Z,d∈N⊆N0 satisfy
for all d,n,M∈N that
Cd,M,0=0
and
[TABLE]
Then
(i)
for every d∈N there exists a unique at most polynomially growing function ud∈C([0,T]×Rd,R) which satisfies that
ud∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
ud(T,x)=gd(x)
and
2. (ii)
there exist
functions
N=(Nd,ε)d∈N,ε∈(0,∞):N×(0,∞)→N
and
C=(Cδ)δ∈(0,∞):(0,∞)→(0,∞)
such that
for all d∈N, ε,δ∈(0,∞) it holds that
Throughout this proof
assume w.l.o.g. that κ≥1,
assume w.l.o.g. that (Ω,F,P) is a complete probability space,
for every d∈N
let ∣∣∣⋅∣∣∣d:Rd×d→[0,∞) be the Frobenius norm on Rd×d,
for every d∈N, i∈{1,2,…,d} let ζd,i(j)∈R, j∈{1,2,…,d}, satisfy
that
ζd,i=(ζd,i(j))j∈{1,2,…,d},
and
for every d∈N, θ∈Θ, t∈[0,T] let Ftd,θ⊆F
be the sigma-algebra which satisfies that
[TABLE]
Note that (340) implies that
for every d∈N, θ∈Θ it holds that
(Ftd,θ)t∈[0,T] is a filtration on (Ω,F,P) which satisfies the usual conditions.
Moreover, observe that (340) and Lemma 2.17 demonstrate that
for every d∈N, θ∈Θ it holds that
Wd,θ:[0,T]×Ω→Rd is a standard (Ω,F,P,(Ftd,θ)t∈[0,T])-Brownian motion.
In addition, note that
(331)
and
the fact that
supd∈N,i∈{1,2,…,d}∣αd,i∣≤κ
imply that
for all d∈N, t∈[0,T], x=(x1,x2,…,xd)∈Rd it holds that
[TABLE]
Furthermore, observe that
(331),
the fact that
supd∈N,i∈{1,2,…,d}∣βd,i∣2≤κ,
and
the hypothesis that
for all d∈N, i∈{1,2,…,d} it holds that
∥ζd,i∥Rd=1
assure that
for all d∈N, t∈[0,T], x=(x1,x2,…,xd)∈Rd it holds that
[TABLE]
This and (341) ensure that
for all d∈N, t∈[0,T], x∈Rd it holds that
[TABLE]
Next note that (331), (334), and Lemma 4.2
(with
d=d,
T=T,
(αi)i∈{1,…,d}=(αd,i)i∈{1,…,d},
(βi)i∈{1,…,d}=(βd,i)i∈{1,…,d},
Σ=Σd,
W=Wd,θ,
X=Xd,θ
for θ∈Θ, d∈N
in the notation of Lemma 4.2)
demonstrate that
for all d∈N, θ∈Θ it holds that
Xd,θ is continuous random field which satisfies
for every t∈[0,T], x∈Rd that
(Xt,sd,θ(x))s∈[t,T]:[t,T]×Ω→Rd is an (Fsd,θ)s∈[t,T]/B(Rd)-adapted stochastic process
and which satisfies
that for all t∈[0,T], s∈[t,T], x∈Rd
it holds P-a.s. that
[TABLE]
Combining
this,
(332),
the fact that for all d∈N it holds that μd and σd are globally Lipschitz continuous functions,
and
(343)
with Theorem 3.20
(with
T=T,
α=1,
c=κ,
K=κ,
L=κ,
p=p,
P=0,
P=P,
q=q,
C1=0,
C2=κ,
ξd=ξd,
gd=gd,
fd=fd,
μd=μd,
σd=σd,
Rθ=Rθ,
Fd,θ=Fd,θ,
Wd,θ=Wd,θ,
Xd,θ=Xd,θ,
VM,nd,θ=VM,nd,θ,
Cd,M,n=Cd,M,n
for d∈N, θ∈Θ, M,n∈Z,
in the notation of Theorem 3.20)
establishes that
(I)
for every d∈N there exists a unique at most polynomially growing function ud∈C([0,T]×Rd,R) which satisfies that
ud∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
ud(T,x)=gd(x)
and
2. (II)
there exists a function
N=(Nd,ε)ε∈(0,∞):N×(0,∞)→N
such that
for all ε,δ∈(0,∞) it holds that
[TABLE]
[TABLE]
Observe that item (I) proves item (i).
Furthermore, note that item (II) establishes item (ii).
The proof of Theorem 4.3 is thus completed.
∎
Theorem 4.4**.**
Let T∈(0,∞), κ,p,P,q∈[0,∞), (αd,i)i∈{1,2,…,d},d∈N, (βd,i)i∈{1,2,…,d},d∈N⊆R, Θ=∪n=1∞Zn satisfy that
supd∈N,i∈{1,2,…,d}max{∣αd,i∣,∣βd,i∣2}≤κ,
for every d∈N
let ⟨⋅,⋅⟩Rd:Rd×Rd→R be the Euclidean scalar product on Rd and
let ∥⋅∥Rd:Rd→[0,∞) be the Euclidean norm on Rd,
for every d∈N let ξd∈Rd, Σd=(ζd,1,…,ζd,d)∈Rd×d satisfy
for all i∈{1,2,…,d} that
∥ξd∥Rd≤κdq and
∥ζd,i∥Rd=1,
for every d∈N let μd:[0,T]×Rd→Rd and σd:[0,T]×Rd→Rd×d be the functions which satisfy
for all t∈[0,T], x=(x1,x2,…,xd)∈Rd that
[TABLE]
let f:R→R be a Lipschitz continuous function,
for every d∈N let gd∈C(Rd,R) satisfy
for all t∈[0,T], x∈Rd that
[TABLE]
let (Ω,F,P) be a probability space,
let
Rθ:Ω→[0,1],
θ∈Θ,
be independent U[0,1]-distributed random variables,
let
Rθ=(Rtθ)t∈[0,T]:[0,T]×Ω→[0,T],
θ∈Θ,
be the stochastic processes which satisfy
for all t∈[0,T], θ∈Θ that
[TABLE]
for every d∈N
let
Wd,θ:[0,T]×Ω→Rd,
θ∈Θ,
be independent standard Brownian motions,
assume that
(Wd,θ)d∈N,θ∈Θ
and
(Rθ)θ∈Θ
are independent,
for every d∈N, θ∈Θ
let
Xd,θ=(Xt,sd,θ,i(x))s∈[t,T],t∈[0,T],x∈Rd,i∈{1,2,…,d}:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
be the function which satisfies
for all
t∈[0,T], s∈[t,T], x=(x1,x2,…,xd)∈Rd, i∈{1,2,…,d} that
[TABLE]
let
VM,nd,θ:[0,T]×Rd×Ω→R, M,n∈Z, θ∈Θ, d∈N,
be functions which satisfy
for all d,M,n∈N, θ∈Θ, t∈[0,T], x∈Rd that
VM,−1d,θ(t,x)=VM,0d,θ(t,x)=0
and
[TABLE]
and let (Cd,M,n)M,n∈Z,d∈N⊆N0 satisfy
for all d,n,M∈N that
Cd,M,0=0
and
[TABLE]
Then
(i)
for every d∈N there exists a unique at most polynomially growing function ud∈C([0,T]×Rd,R) which satisfies that
ud∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)∈(0,T)×Rd and
which satisfies
for all x∈Rd that
ud(T,x)=gd(x)
and
2. (ii)
there exist
functions
N=(Nd,ε)d∈N,ε∈(0,∞):N×(0,∞)→N
and
C=(Cδ)δ∈(0,∞):(0,∞)→(0,∞)
such that
for all d∈N, ε,δ∈(0,∞) it holds that
[TABLE]
[TABLE]
4.3 MLP approximations for the pricing of financial derivatives with default risks
Corollary 4.5**.**
Let
T,R,γl,γh,vl,vh∈(0,∞), p,q∈[0,∞), ϵ∈[0,1), α,β∈R,
f∈C(R,R),
Θ=∪n=1∞Zn
satisfy
for all u∈R that
γl<γh, vl>vh, and
[TABLE]
let ξd∈Rd, d∈N satisfy that
supd∈Ndq∥ξd∥Rd<∞,
let gd∈C(Rd,R), d∈N, satisfy that
supd∈N,x∈Rd1+∥x∥Rdp∣gd(x)∣∣<∞,
let (Ω,F,P) be a probability space,
let
Rθ:Ω→[0,1],
θ∈Θ,
be independent U[0,1]-distributed random variables,
let
Rθ=(Rtθ)t∈[0,T]:[0,T]×Ω→[0,T],
θ∈Θ,
be the stochastic processes which satisfy
for all t∈[0,T], θ∈Θ that
Rtθ=t+(T−t)Rθ,
for every d∈N let
Wd,θ=(Wd,θ,i)i∈{1,2,…,d}:[0,T]×Ω→Rd,
θ∈Θ,
be independent standard Brownian motions,
assume that
(Wd,θ)d∈N,θ∈Θ
and
(Rθ)θ∈Θ
are independent,
for every d∈N, θ∈Θ
let
Xd,θ=(Xt,sd,θ,i(x))s∈[t,T],t∈[0,T],x∈Rd,i∈{1,2,…,d}:{(t,s)∈[0,T]2:t≤s}×Rd×Ω→Rd
be the function which satisfies
for all
i∈{1,2,…,d}, t∈[0,T], s∈[t,T], x=(x1,x2,…,xd)∈Rd that
[TABLE]
let
VM,nd,θ:[0,T]×Rd×Ω→R, M,n∈Z, θ∈Θ, d∈N,
be functions which satisfy
for all d,M,n∈N, θ∈Θ, t∈[0,T], x∈Rd that
VM,−1d,θ(t,x)=VM,0d,θ(t,x)=0
and
[TABLE]
and let (Cd,M,n)M,n∈Z,d∈N⊆N0 satisfy
for all d,n,M∈N that
Cd,M,0=0
and
[TABLE]
*Then
*
(i)
for every d∈N there exists a unique at most polynomially growing function ud∈C([0,T]×Rd,R) which satisfies that
ud∣(0,T)×Rd:(0,T)×Rd→R is a viscosity solution of
[TABLE]
for (t,x)=(t,(x1,x2,…,xd))∈(0,T)×Rd and
which satisfies
for all x∈Rd that
ud(T,x)=gd(x)
and
2. (ii)
there exist functions
N=(Nd,ε)d∈N,ε∈(0,1]:N×(0,1]→N
and
C=(Cδ)δ∈(0,∞):(0,∞)→(0,∞)
such that
for all d∈N, ε∈(0,1], δ∈(0,∞) it holds that
Cd,Nd,ε,Nd,ε≤Cδd1+qp(2+δ)ε−(2+δ)
and
[TABLE]
Acknowledgments
This project has been partially supported through the SNSF-Research project 200020_175699 “Higher order numerical approximation methods for stochastic partial differential equations”.
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