On existence and uniqueness
properties for solutions of
stochastic fixed point equations
Christian Beck1,
Lukas Gonon2,3,
Martin Hutzenthaler4,
and
Arnulf Jentzen5
1
Department of Mathematics,
ETH Zurich,
Zürich,
Switzerland,
e-mail: [email protected]
2
Department of Mathematics,
ETH Zurich,
Zürich,
Switzerland,
e-mail: [email protected]
3
Faculty of Mathematics and Statistics, University of St. Gallen,
St. Gallen,
Switzerland,
e-mail: [email protected]
4
Faculty of Mathematics,
University of Duisburg-Essen,
Essen,
Germany,
e-mail: [email protected]
5
Department of Mathematics,
ETH Zurich,
Zürich,
Switzerland,
e-mail: [email protected]
Abstract
The Feynman–Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinear Kolmogorov PDE does not possess a classical solution.
Contents
-
1 Introduction
-
2 Abstract stochastic fixed point equations (SFPEs)
-
2.1 Integrability properties for certain stochastic processes
-
2.2 Continuity properties for solutions of SFPEs
-
2.3 Regularity properties for solutions of SFPEs
-
2.4 Contractivity properties for SFPEs
-
2.5 Existence and uniqueness properties for solutions of SFPEs
-
3 SFPEs associated with stochastic differential equations (SDEs)
-
3.1 A priori estimates for solutions of SDEs
-
3.2 Locality properties for solutions of SDEs
-
3.3 Continuity properties for solutions of SDEs
-
3.4 Existence and uniqueness properties for solutions of SFPEs associated with SDEs
1 Introduction
The Feynman–Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. The main result of this article, Theorem 2.9 in Section 2.5 below, shows the existence of unique solutions of certain SFPEs in an abstract setting.
As an application of Theorem 2.9 we establish in Theorem 3.8 the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinear Kolmogorov PDE does not possess a classical solution (see, for example, Hairer et al. [9]). To illustrate Theorem 3.8 in more detail we provide in the following result, Theorem 1.1 below, a special case of Theorem 3.8.
Theorem 1.1**.**
Let
d,m∈N,
L,T∈(0,∞),
let
⟨⋅,⋅⟩:Rd×Rd→R
be the standard scalar product on Rd,
let
∥⋅∥:Rd→[0,∞)
be a norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty) be a norm on Rd×m,
let
μ:Rd→Rd and
σ:Rd→Rd×m
be locally Lipschitz continuous,
let
f∈C([0,T]×Rd×R,R),
g∈C(Rd,R)
be at most polynomially growing, assume for all
t∈[0,T],
x∈Rd,
v,w∈R
that
max{⟨x,μ(x)⟩,\vvvertσ(x)\vvvert2}≤L(1+∥x∥2)
and
∣f(t,x,v)−f(t,x,w)∣≤L∣v−w∣,
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 (Ft)t∈[0,T]-Brownian motion,
and for every
t∈[0,T],
x∈Rd
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→Rd
be an (Fs)s∈[t,T]-adapted stochastic process with continuous sample paths which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
Then there exists a unique at most polynomially growing u∈C([0,T]×Rd,R) such that for all
t∈[0,T],
x∈Rd
it holds that
[TABLE]
SFPEs of the form as in (2)
have a strong connection with semilinear Kolmogorov PDEs and arise, for example, in
models from the environmental sciences as well as in pricing problems from financial engineering (cf., for example, Burgard & Kjaer [2], Crépey et al. [3], Duffie et al. [4], and Henry-Labordère [10]). SFPEs such as (2) are also important for full-history recursive multilevel Picard approximation (MLP) methods, which were recently introduced in [5, 11]; see also [1, 6, 12, 13].
In [11, 12] it has been shown that functions which satisfy SFPEs related to semilinear Kolmogorov PDEs can be approximated by MLP schemes without the curse of dimensionality. Theorem 1.1 above establishes existence of unique solutions of SFPEs related to semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities within the class of at most polynomially growing continuous functions.
Theorem 1.1 is an immediate consequence of 3.10 in Section 3.4 below. 3.10, in turn, follows from 3.9 which itself is a special case of Theorem 3.8.
Theorem 3.8 is an application of Theorem 2.9, the main result of this article. Theorem 3.8 shows the existence of unique solutions of SFPEs associated with suitable semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities within a certain class of continuous functions.
Related existence and uniqueness results can be found, e.g., in Pazy [18, Theorem 6.1.2],
Segal [20, Theorem 1], Weissler [22, Theorem 1],
and Hutzenthaler et al. [11, Corollary 3.11].
The remainder of this article is organized as follows. In Section 2 we investigate SFPEs in an abstract setting. In Theorem 2.9 in Section 2.5, the main result of this article, we obtain under suitable assumptions an abstract existence and uniqueness result for solutions of SFPEs. Its proof is based on Banach’s fixed point theorem. In Sections 2.1–2.3 we establish the well-definedness of the mapping to which Banach’s fixed point theorem is applied in the proof of Theorem 2.9. In Section 2.4 we prove a Lipschitz estimate which establishes the contractivity property of the mapping to which Banach’s fixed point theorem is applied in the proof of Theorem 2.9. In Section 3 we apply the abstract theory from Theorem 2.9 in Section 2 in the context of certain stochastic differential equations (SDEs) to obtain Theorem 3.8, the main result of Section 3. In Sections 3.1–3.3 we present several auxiliary results on certain SDEs in order to demonstrate that the hypotheses of Theorem 2.9 are satisfied in the setting of Theorem 3.8. The article is concluded by means of two simple corollaries of Theorem 3.8 (see 3.9 and 3.10 in Section 3.4 below).
2 Abstract stochastic fixed point equations (SFPEs)
In this section we study SFPEs from an abstract point of view. This section’s main result is Theorem 2.9 below. It is an application of Banach’s fixed point theorem to a suitable function.
2.7 in Section 2.3 establishes the well-definedness of this function. 2.7 is a direct consequence of Lemma 2.6 which we establish through an approximation argument building upon Lemmas 2.2 and 2.5.
The contractivity property of the function to which we apply Banach’s fixed point theorem in the proof Theorem 2.9 is established in Lemma 2.8 in Section 2.4 below.
2.1 Integrability properties for certain stochastic processes
Lemma 2.1**.**
Let
d∈N,
T∈(0,∞),
let
O⊆Rd
be a non-empty open set,
let
(Ω,F,P)
be a probability space,
for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be
(B([t,T])⊗F)/B(O)-measurable,
let
g:O→R
be B(O)/B(R)-measurable,
let
h:[0,T]×O→R
be B([0,T]×O)/B(R)-measurable,
let
V:[0,T]×O→(0,∞)
be B([0,T]×O)/B((0,∞))-measurable,
and assume for all
t∈[0,T],
s∈[t,T],
x∈O
that
E[V(s,Xst,x)]≤V(t,x)
and
supt∈[0,T]supx∈O[V(T,x)∣g(x)∣+V(t,x)∣h(t,x)∣]<∞.
Then it holds for all
t∈[0,T],
x∈O
that
[TABLE]
Throughout this proof let
c∈[0,∞)
satisfy for all
t∈[0,T],
x∈O
that
[TABLE]
Observe that the hypothesis that g:O→R is B(O)/B(R)-measurable, the hypothesis that h:[0,T]×O→R is B([0,T]×O)/B(R)-measurable, and the hypothesis that for every
t∈[0,T],
x∈O
it holds that Xt,x:[t,T]×Ω→O is (B([t,T])⊗F)/B(O)-measurable ensure that for every
t∈[0,T],
x∈O
it holds that Ω∋ω↦g(XTt,x(ω))∈R is
F/B(R)-measurable and [t,T]×Ω∋(s,ω)↦h(s,Xst,x(ω))∈R is (B([t,T])⊗F)/B(R)-measurable. The hypothesis that for all
t∈[0,T],
s∈[t,T],
x∈O
it holds that E[V(s,Xst,x)]≤V(t,x), Fubini’s theorem, and (4) hence ensure that for all
t∈[0,T],
x∈O
it holds that
[TABLE]
This demonstrates (3). The proof of Lemma 2.1 is thus completed.
∎
2.2 Continuity properties for solutions of SFPEs
In this section we establish in Lemma 2.2, Lemma 2.3, and 2.4 several elementary convergence and approximation results. The convergence result in Lemma 2.2 and the approximation result in 2.4 pave the way for Section 2.3. They will together with Lemma 2.5 be employed in the proof of Lemma 2.6 in Section 2.3. Lemma 2.6, in turn, has 2.7 as a rather direct consequence, which itself is one of the cornerstones of the proof of Theorem 2.9.
Lemma 2.2**.**
Let
d∈N,
T∈(0,∞),
let
∥⋅∥:Rd→[0,∞) be a norm on Rd,
let
O⊆Rd be a non-empty open set,
for every
r∈(0,∞) let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
(Ω,F,P) be a probability space,
for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be (B([t,T])⊗F)/B(O)-measurable,
let
V∈C([0,T]×O,(0,∞)) satisfy
for all
t∈[0,T],
s∈[t,T],
x∈O
that
E[V(s,Xst,x)]≤V(t,x),
let
gn∈C(O,R), n∈N0,
and
hn∈C([0,T]×O,R), n∈N0,
satisfy for all
n∈N
that
infr∈(0,∞)[supt∈[0,T]supx∈O∖Or(V(T,x)∣gn(x)∣+V(t,x)∣hn(t,x)∣)]=0,
and assume that
[TABLE]
Then
- (i)
it holds for every
n∈N0
that
[TABLE]
2. (ii)
it holds for every
n∈N0
that there exists a unique
un:[0,T]×O→R
which satisfies for all
t∈[0,T],
x∈O
that
[TABLE]
3. (iii)
it holds that
[TABLE]
and
4. (iv)
it holds for every
compact set K⊆O
that
[TABLE]
First, observe that for every
r∈(0,∞)
it holds that
Or
is a compact set. This and the fact that for every
n∈N0
it holds that
O∋x↦V(T,x)gn(x)∈R
and
[0,T]×O∋(t,x)↦V(t,x)hn(t,x)∈R
are continuous imply that for all
n∈N0,
r∈(0,∞)
it holds that
[TABLE]
The hypothesis that for every
n∈N
it holds that
infr∈(0,∞)[supt∈[0,T]supx∈O∖Or(V(T,x)∣gn(x)∣+V(t,x)∣hn(t,x)∣)]=0
hence ensures that for every
n∈N
it holds that
[TABLE]
Combining this with (6)
demonstrates Item (i).
Next observe that Item (i)
and Lemma 2.1 establish
Item (ii). Next note that the
hypothesis that for all
t∈[0,T],
s∈[t,T],
x∈O
it holds that
E[V(s,Xst,x)]≤V(t,x)
ensures that for all
n∈N,
t∈[0,T],
x∈O
it holds that
[TABLE]
This and (6)
establish that
[TABLE]
Furthermore, note that the hypothesis that for all
t∈[0,T],
s∈[t,T],
x∈O
it holds that
E[V(s,Xst,x)]≤V(t,x)
assures that for all
n∈N,
t∈[0,T],
x∈O
it holds that
[TABLE]
This and (6) imply that
[TABLE]
The triangle inequality, (8),
and (14) hence yield that
[TABLE]
This establishes Item (iii). Moreover, observe that Item (iii) and the fact that V:[0,T]×O→(0,∞) is continuous imply for every
compact set K⊆O
that
[TABLE]
This establishes Item (iv). The proof of Lemma 2.2 is thus completed.
∎
Lemma 2.3**.**
Let
d∈N,
T∈(0,∞),
let
∥⋅∥:Rd→[0,∞) be a norm on Rd,
let
O⊆Rd be a non-empty open set,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
and let
h∈C([0,T]×O,R)
satisfy
infr∈(0,∞)[supt∈[0,T]supx∈O∖Or∣h(t,x)∣]=0.
Then there exist compactly supported
hn∈C([0,T]×O,R), n∈N,
which satisfy that
[TABLE]
Throughout this proof let
Un⊆O, n∈N,
be the sets given by
Un={x∈O:(∃z∈On:∥z−x∥<2n1)}.
Note that for every
n∈N
it holds that On⊆O is a compact set,
it holds that Un⊆Rd is an open set which satisfies Un⊆O,
and
it holds that On⊆Un. Urysohn’s lemma (cf., for example, Rudin [19, Lemma 2.12]) hence ensures for every
n∈N
that there exists φn∈C([0,T]×O,R) which satisfies for all
t∈[0,T],
x∈O
that
\mathbbm1[0,T]×On(t,x)≤φn(t,x)≤\mathbbm1[0,T]×Un(t,x).
Observe, in particular, that this implies that the functions
φn:[0,T]×O→R,
n∈N,
have compact supports. In the next step we let
hn:[0,T]×O→R, n∈N,
satisfy for all
n∈N,
t∈[0,T],
x∈O
that
hn(t,x)=φn(t,x)h(t,x).
Note that this and the fact that for every
n∈N
it holds that φn∈C([0,T]×O,R) is compactly supported imply that for every
n∈N
it holds that
hn:[0,T]×O→R
is a compactly supported continuous function. Moreover, observe that
[TABLE]
This establishes (19). The proof of Lemma 2.3 is thus completed.
∎
Corollary 2.4**.**
Let
d∈N,
T∈(0,∞),
let
∥⋅∥:Rd→[0,∞) be a norm on Rd,
let
O⊆Rd be a non-empty open set,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
h∈C([0,T]×O,R),
V∈C([0,T]×O,(0,∞)),
and assume that infr∈(0,∞)[supt∈[0,T]supx∈O∖Or(V(t,x)∣h(t,x)∣)]=0.
Then there exist compactly supported
hn∈C([0,T]×O,R), n∈N,
which satisfy that
[TABLE]
Proof of 2.4.
Throughout this proof let
g:[0,T]×O→R
satisfy for all
t∈[0,T],
x∈O
that
g(t,x)=V(t,x)h(t,x).
Observe that the assumption that
h∈C([0,T]×O,R),
the assumption that
V∈C([0,T]×O,(0,∞)),
and the assumption that
infr∈(0,∞)[supt∈[0,T]supx∈O∖Or(V(t,x)∣h(t,x)∣)]=0
prove that
g∈C([0,T]×O,R)
and
[TABLE]
Lemma 2.3 (with h=g in the notation of Lemma 2.3) therefore ensures that there exist compactly supported
gn∈C([0,T]×O,R), n∈N,
which satisfy that
[TABLE]
Next let
hn:[0,T]×O→R, n∈N,
satisfy for all
n∈N,
t∈[0,T],
x∈O
that
hn(t,x)=gn(t,x)V(t,x).
Hence, we obtain that for all
n∈N
it holds that
hn∈C([0,T]×O,R)
and
[TABLE]
This establishes (21). The proof of 2.4 is thus completed.
∎
2.3 Regularity properties for solutions of SFPEs
In this section we establish 2.7, one of the building blocks of the proof of Theorem 2.9. 2.7 is a rather direct consequence of Lemma 2.6 which, in turn, we prove by means of an argument building upon Lemmas 2.2–2.5.
Lemma 2.5**.**
Let
d∈N,
T∈(0,∞),
let
∥⋅∥:Rd→[0,∞)
be a norm on Rd,
let
O⊆Rd
be a non-empty open set,
let
(Ω,F,P)
be a probability space,
for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be
(B([t,T])⊗F)/B(O)-measurable,
let
g∈C(O,R),
h∈C([0,T]×O,R)
be bounded,
and assume for all
ε∈(0,∞),
s∈[0,T]
and all
(tn,xn)∈[0,T]×O, n∈N0,
with
limsupn→∞[∣tn−t0∣+∥xn−x0∥]=0
that
limsupn→∞[P(∥Xmax{s,tn}tn,xn−Xmax{s,t0}t0,x0∥≥ε)]=0. Then
- (i)
it holds for all
t∈[0,T],
x∈O
that
[TABLE]
and
2. (ii)
it holds that
[TABLE]
is continuous.
Throughout this proof let
(tn,xn)∈[0,T]×O, n∈N0,
satisfy
limsupn→∞[∣tn−t0∣+∥xn−x0∥]=0.
Note that Lemma 2.1 establishes Item (i). Next we prove Item (ii). For this we intend to show that
[TABLE]
Next note that the fact that g:O→R and h:[0,T]×O→R are continuous ensures that for all
ε∈(0,∞),
s∈[0,T]
it holds that
[TABLE]
(cf., for example, Kallenberg [14, Lemma 4.3]).
Combining this and the fact that g:O→R and h:[0,T]×O→R are bounded with Vitali’s convergence theorem ensures that for all
s∈[0,T]
it holds that
[TABLE]
Lebesgue’s dominated convergence theorem
and the fact that h:[0,T]×O→R is bounded hence imply that
[TABLE]
This yields that
[TABLE]
Combining this with (29) demonstrates (27). The proof of Lemma 2.5 is thus completed.
∎
Lemma 2.6**.**
Let
d∈N,
T∈(0,∞),
let
∥⋅∥:Rd→[0,∞)
be a norm on Rd,
let
O⊆Rd
be a non-empty open set,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
(Ω,F,P)
be a probability space,
for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be (B([t,T])⊗F)/B(O)-measurable,
assume for all
ε∈(0,∞),
s∈[0,T]
and all
(tn,xn)∈[0,T]×O, n∈N0,
with
limsupn→∞[∣tn−t0∣+∥xn−x0∥]=0
that
limsupn→∞[P(∥Xmax{s,tn}tn,xn−Xmax{s,t0}t0,x0∥≥ε)]=0,
let
g∈C(O,R),
h∈C([0,T]×O,R),
V∈C([0,T]×O,(0,∞))
and
u:[0,T]×O→R
satisfy for all
t∈[0,T],
s∈[t,T],
x∈O
that
E[V(s,Xst,x)]≤V(t,x),
and assume for all
t∈[0,T],
x∈O
that infr∈(0,∞)[sups∈[0,T]supy∈O∖Or(V(T,y)∣g(y)∣+V(s,y)∣h(s,y)∣)]=0
and
[TABLE]
(cf. Item (ii) of Lemma 2.2).
Then
- (i)
it holds that u∈C([0,T]×O,R) and
2. (ii)
it holds in the case of
supr∈(0,∞)[inft∈[0,T]infx∈O∖OrV(t,x)]=∞ that
[TABLE]
Throughout this proof let
gn:O→R, n∈N,
and
hn:[0,T]×O→R, n∈N,
be compactly supported continuous functions which satisfy that
[TABLE]
(cf. 2.4)
and let
un:[0,T]×O→R, n∈N,
satisfy for all
n∈N,
t∈[0,T],
x∈O
that
[TABLE]
(cf. Lemma 2.1).
Note that Lemma 2.5 assures for every
n∈N
that un:[0,T]×O→R
is continuous.
Next observe that the fact that
gn:O→R, n∈N,
and
hn:[0,T]×O→R,
n∈N,
are compactly supported ensures that for every
n∈N
there exists
r∈(0,∞)
which satisfies that for all
t∈[0,T],
x∈O∖Or
it holds that
gn(x)=0=hn(t,x).
This implies for every
n∈N
that
[TABLE]
Item (iv) of Lemma 2.2, (34), and the fact that un:[0,T]×O→R, n∈N, are continuous therefore imply that u:[0,T]×O→R is continuous. This establishes Item (i). In the next step we prove Item (ii). For this we assume that
supr∈(0,∞)[inft∈[0,T]infx∈O∖OrV(t,x)]=∞.
Note that this entails for every
n∈N
that
[TABLE]
Combining this with Item (iii) of
Lemma 2.2 yields that
[TABLE]
This establishes Item (ii). The proof of Lemma 2.6 is thus completed.
∎
Lemma 2.6 allows to infer the next result, 2.7, which constitutes an important ingredient of the proof of Theorem 2.9.
Corollary 2.7**.**
Let
d∈N,
L,T∈(0,∞),
let
∥⋅∥:Rd→[0,∞)
be a norm on Rd,
let
O⊆Rd
be a non-empty open set,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
(Ω,F,P)
be a probability space,
for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be (B([t,T])⊗F)/B(O)-measurable,
assume for all
ε∈(0,∞),
s∈[0,T]
and all
(tn,xn)∈[0,T]×O, n∈N0,
with
limsupn→∞[∣tn−t0∣+∥xn−x0∥]=0
that
limsupn→∞[P(∥Xmax{s,tn}tn,xn−Xmax{s,t0}t0,x0∥≥ε)]=0,
let
f∈C([0,T]×O×R,R),
g∈C(O,R),
u∈C([0,T]×O,R),
V∈C([0,T]×O,(0,∞))
satisfy for all
t∈[0,T],
s∈[t,T],
x∈O
that
E[V(s,Xst,x)]≤V(t,x),
and assume for all
t∈[0,T],
x∈O,
v,w∈R
that
infr∈(0,∞)[sups∈[0,T]supy∈O∖Or(V(s,y)∣f(s,y,0)∣+∣u(s,y)∣+V(T,y)∣g(y)∣)]=0
and
∣f(t,x,v)−f(t,x,w)∣≤L∣v−w∣.
Then
- (i)
it holds for all
t∈[0,T],
x∈O
that
[TABLE]
2. (ii)
it holds that
[TABLE]
is continuous, and
3. (iii)
it holds in the case of
supr∈(0,∞)[inft∈[0,T]infx∈O∖OrV(t,x)]=∞
that
[TABLE]
Proof of 2.7.
First, observe that
[TABLE]
is a continuous function which satisfies for all
t∈[0,T],
x∈O
that
[TABLE]
The hypothesis that
infr∈(0,∞)[supt∈[0,T]supx∈O∖Or(V(t,x)∣f(t,x,0)∣+∣u(t,x)∣)]=0
therefore ensures that
[TABLE]
Lemma 2.1 and Item (i) of Lemma 2.2 hence establish Item (i). Moreover, Lemma 2.6 (with
g=g,
h=([0,T]×O∋(t,x)↦f(t,x,u(t,x))∈R),
u=([0,T]×O∋(t,x)↦E[g(XTt,x)+∫tTf(s,Xst,x,u(s,Xst,x))ds]∈R)
in the notation of Lemma 2.6) establishes Items (ii) and (iii). The proof of 2.7
is thus completed.
∎
2.4 Contractivity properties for SFPEs
In this section we establish an elementary Lipschitz estimate (see Lemma 2.8 below) which will yield the contractivity needed in the proof of Theorem 2.9.
Lemma 2.8**.**
Let
d∈N,
L,T∈(0,∞),
let
O⊆Rd
be a non-empty open set,
let
(Ω,F,P)
be a probability space,
for every
t∈[0,T],
x∈O let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be
(B([t,T])⊗F)/B(O)-measurable,
let
V:[0,T]×O→(0,∞)
be
B([0,T]×O)/B((0,∞))-measurable,
assume for all
t∈[0,T],
s∈[t,T],
x∈O
that
E[V(s,Xst,x)]≤V(t,x),
let
f:[0,T]×O×R→R
be B([0,T]×O×R)/B(R)-measurable,
assume for all
t∈[0,T],
x∈O,
v,w∈R
that
∣f(t,x,v)−f(t,x,w)∣≤L∣v−w∣,
let
v,w:[0,T]×O→R
be
B([0,T]×O)/B(R)-measurable,
and assume that
[TABLE]
Then it holds for all
λ∈(0,∞),
t∈[0,T],
x∈O
that
[TABLE]
First, note that the fact that
f:[0,T]×O×R→R
is B([0,T]×O×R)/B(R)-measurable,
the fact that
v,w:[0,T]×O→R
are B([0,T]×O)/B(R)-measurable,
and the fact that for all
t∈[0,T],
x∈O
it holds that
Xt,x:[t,T]×Ω→O
is (B([t,T])⊗F)/B(O)-measurable ensure that for all
t∈[0,T],
x∈O
it holds that
[TABLE]
is (B([t,T])⊗F)/B(R)-measurable. Next observe that the hypothesis that for all
t∈[0,T],
s∈[t,T],
x∈O
it holds that
E[V(s,Xst,x)]≤V(t,x),
the hypothesis that for all
t∈[0,T],
x∈O,
a,b∈R
it holds that
∣f(t,x,a)−f(t,x,b)∣≤L∣a−b∣,
and Fubini’s theorem ensure that for all
λ∈(0,∞),
t∈[0,T],
x∈O
it holds that
[TABLE]
This establishes (46). The proof of Lemma 2.8
is thus completed.
∎
2.5 Existence and uniqueness properties for solutions of SFPEs
Combining Banach’s fixed point theorem with 2.7 and 2.8 allows to conclude the main result of this section, Theorem 2.9 below.
Theorem 2.9**.**
Let
d∈N,
L,T∈(0,∞),
let
∥⋅∥:Rd→[0,∞) be a norm on Rd,
let
O⊆Rd
be a non-empty open set,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
(Ω,F,P)
be a probability space,
for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be (B([t,T])⊗F)/B(O)-measurable,
assume for all
ε∈(0,∞),
s∈[0,T]
and all
(tn,xn)∈[0,T]×O, n∈N0,
with
limsupn→∞[∣tn−t0∣+∥xn−x0∥]=0
that
limsupn→∞[P(∥Xmax{s,tn}tn,xn−Xmax{s,t0}t0,x0∥≥ε)]=0,
let
f∈C([0,T]×O×R,R),
g∈C(O,R),
V∈C([0,T]×O,(0,∞))
satisfy for all
t∈[0,T],
s∈[t,T],
x∈O,
v,w∈R
that
E[V(s,Xst,x)]≤V(t,x)
and
∣f(t,x,v)−f(t,x,w)∣≤L∣v−w∣,
and assume that
infr∈(0,∞)[supt∈[0,T]supx∈O∖Or(V(t,x)∣f(t,x,0)∣+V(T,x)∣g(x)∣)]=0
and
supr∈(0,∞)[inft∈[0,T]infx∈O∖OrV(t,x)]=∞.
Then there exists a unique
u∈C([0,T]×O,R)
such that
- (i)
it holds that
[TABLE]
and
2. (ii)
it holds for all
t∈[0,T],
x∈O
that
[TABLE]
Throughout this proof let
V
be the set given by
[TABLE]
let
W1 and W2
be the sets given by
[TABLE]
and
[TABLE]
let
∥⋅∥λ:V→[0,∞), λ∈R,
satisfy for every
λ∈R, v∈V
that
[TABLE]
(see Item (i) of Lemma 2.2),
and let
∥⋅∥Wi:Wi→[0,∞), i∈{1,2},
satisfy for every
i∈{1,2},
w∈Wi
that
[TABLE]
Recall that
(W1,∥⋅∥W1) is an R-Banach space.
Combining this with the fact that W2 is a closed subset of (W1,∥⋅∥W1) (see Lemma 2.2) implies that (W2,∥⋅∥W2) is an R-Banach space.
Moreover, observe that
(V,∥⋅∥λ), λ∈R, are normed R-vector spaces.
In the next step we show that
(V,∥⋅∥0)
is complete. For this let
vn∈V, n∈N,
satisfy
limsupn→∞[supm≥n∥vn−vm∥0]=0.
This implies that
Vvn:[0,T]×O→R, n∈N,
is a Cauchy sequence in (W2,∥⋅∥W2). Thus, there exists ϕ∈W2 which satisfies that
limsupn→∞[supt∈[0,T]supx∈O∣V(t,x)vn(t,x)−ϕ(t,x)∣]=0.
Hence, we obtain that
ϕV=([0,T]×O∋(t,x)↦ϕ(t,x)V(t,x)∈R)∈V
and
limsupn→∞∥vn−ϕV∥0=0.
This demonstrates that
(V,∥⋅∥0) is an R-Banach space.
Combining this with the fact that for every
ν∈R,
λ∈[ν,∞),
v∈V
it holds that
∥v∥ν≤∥v∥λ≤e(λ−ν)T∥v∥ν
shows that for every
λ∈R
it holds that
(V,∥⋅∥λ) is an R-Banach space.
Next note that 2.7 yields that there exists a unique Φ:V→V which satisfies for all
t∈[0,T],
x∈O,
v∈V
that
[TABLE]
Moreover, observe that Lemma 2.8 ensures for all
λ∈(0,∞),
v,w∈V
that
[TABLE]
Hence, we obtain for all
λ∈[2L,∞),
v,w∈V
that
[TABLE]
Banach’s fixed point theorem therefore demonstrates that there exists a unique u∈V which satisfies Φ(u)=u.
The proof of Theorem 2.9 is thus completed.
∎
3 SFPEs associated with stochastic differential equations (SDEs)
In this section we apply the abstract existence and uniqueness result which we obtained in the previous section (see Theorem 2.9 in Section 2 above) to certain SDEs (see Section 3.4 below). In Sections 3.1–3.3 we present, for the reader’s convenience and for the sake of completeness, some elementary and essentially well-known results on SDEs. These results are employed to show that the hypotheses of Theorem 2.9 are indeed satisfied in the setting of Theorem 3.8 (cf. Lemmas 3.1 and 3.7).
3.1 A priori estimates for solutions of SDEs
The following well-known result,
Lemma 3.1 below (cf., for example, Gyöngy & Krylov [8]),
can be seen as an extension of moment bounds for solutions
of SDEs in the presence
of a Lyapunov function or, in other words, a non-negative supersolution of the corresponding Kolmogorov PDE.
Lemma 3.1**.**
Let
d,m∈N,
T∈(0,∞),
let
O⊆Rd be an open set,
let
⟨⋅,⋅⟩:Rd×Rd→R be the standard scalar product on Rd,
let
μ∈C([0,T]×O,Rd),
σ∈C([0,T]×O,Rd×m),
V∈C1,2([0,T]×O,[0,∞))
satisfy for all
t∈[0,T],
x∈O
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 (Ft)t∈[0,T]-Brownian motion,
let
τ:Ω→[0,T] be an
(Ft)t∈[0,T]-stopping time,
and let
X:[0,T]×Ω→O
be an (Ft)t∈[0,T]-adapted stochastic process with continuous sample paths which satisfies that for all
t∈[0,T]
it holds P-a.s. that
[TABLE]
Then it holds that
[TABLE]
Throughout this proof let
∥⋅∥:Rd→[0,∞)
be the standard norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty)
be the Frobenius norm on Rd×m,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
Y:[0,T]×Ω→R
be an (Ft)t∈[0,T]-adapted stochastic process with continuous sample paths which satisfies that for all
t∈[0,T]
it holds P-a.s. that
[TABLE]
and let
ρn:Ω→[0,T], n∈N,
be the
(Ft)t∈[0,T]-stopping times
which satisfy for all
n∈N
that
[TABLE]
Observe that the fact that X has continuous sample paths and the fact
that [0,T] is compact ensure that for all
ω∈Ω
it holds that
{Xt(ω):t∈[0,T]}
is compact. Combining this with the fact that
Rd∋x↦∥x∥∈[0,∞)
and
Rd∋x↦inf({1}∪{∥x−y∥:y∈Rd∖O})∈[0,1]
are continuous implies that for every
ω∈Ω
there exist
ε,r∈(0,∞)
such that for all
t∈[0,T]
it holds that
{y∈Rd:∥y−Xt(ω)∥<ε}⊆O
and
supt∈[0,T]∥Xt(ω)∥≤r.
Combining this with the fact that for all
ε,r∈(0,∞)
there exists
n∈N
such that for all
k∈N
with
k≥n
it holds that
r≤k
and
\nicefrac1k≤ε
implies that for every
ω∈Ω
there exists
n∈N
such that for all
k∈N
with
k≥n
it holds that
ρk(ω)=T.
Next note that the assumption that
∇xV:[0,T]×O→Rd
and
σ:[0,T]×O→Rd×m
are continuous implies that for all
n∈N
it holds that
[TABLE]
This yields for all
n∈N
that
[TABLE]
Combining (62) and (63) hence assures for all
n∈N
that
[TABLE]
Next note that Itô’s formula ensures that for all
t∈[0,T]
it holds P-a.s. that
[TABLE]
This and the fact that X has continuous sample paths imply that for all
n∈N
it holds P-a.s. that
[TABLE]
This and (59) guarantee that for all
n∈N
it holds P-a.s. that
[TABLE]
Combining this and (66) yields for all
n∈N
that
[TABLE]
Fatou’s lemma hence ensures that
[TABLE]
The proof of Lemma 3.1 is thus completed.
∎
The next elementary result, Lemma 3.2 below, provides a way to construct from a supersolution of a suitable elliptic PDE a supersolution of a Kolmogorov PDE (cf. Lemma 3.1 above). Later we will employ Lemma 3.2 to infer 3.9 from Theorem 3.8.
Lemma 3.2**.**
Let
d,m∈N,
T∈(0,∞),
ρ∈R,
let
⟨⋅,⋅⟩:Rd×Rd→R be the standard scalar product on Rd,
let
O⊆Rd
be a non-empty open set,
let
μ∈C([0,T]×O,Rd),
σ∈C([0,T]×O,Rd×m),
V∈C2(O,(0,∞))
satisfy for all
t∈[0,T],
x∈O
that
[TABLE]
and let
V:[0,T]×O→(0,∞)
satisfy for all
t∈[0,T],
x∈O
that
[TABLE]
Then
- (i)
it holds that V∈C2([0,T]×O,(0,∞)) and
2. (ii)
it holds for all
t∈[0,T],
x∈O
that
[TABLE]
First, note that the chain rule and the fact that V∈C2(O,(0,∞)) ensure for all
t∈[0,T],
x∈O
that
- (I)
V∈C2([0,T]×O,R),
2. (II)
(∂t∂V)(t,x)=−ρe−ρtV(x)=−ρV(t,x),
3. (III)
(∇xV)(t,x)=e−ρt(∇V)(x), and
4. (IV)
(HessxV)(t,x)=e−ρt(HessV)(x).
Note that Item (I) establishes Item (i). Moreover, combining (72) with Items (II)–(IV) yields for all
t∈[0,T],
x∈O
that
[TABLE]
This establishes Item (ii). The proof of Lemma 3.2 is thus completed.
∎
The next elementary result, Lemma 3.3 below, establishes in conjunction with Lemma 3.2 above that under certain coercivity and linear growth conditions (see (76) in Lemma 3.3) Lyapunov-type functions with polynomial growth are available (cf. also Grohs et al. [7, Lemma 2.21]).
Lemma 3.3 will later on allow to infer 3.10 from 3.9.
Lemma 3.3**.**
Let
d,m∈N,
c,T,p,ρ∈(0,∞)
satisfy
ρ=2pcmax{p+1,3},
let
⟨⋅,⋅⟩:Rd×Rd→R be the standard scalar product on Rd,
let
∥⋅∥:Rd→[0,∞) be the standard norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty) be the Frobenius norm on
Rd×m,
let
O⊆Rd be a non-empty open set,
and let
μ:[0,T]×O→Rd,
σ:[0,T]×O→Rd×m,
V:O→(0,∞)
satisfy for all
t∈[0,T],
x∈O
that
[TABLE]
Then
- (i)
*it holds that
V∈C∞(O,(0,∞)) and*
2. (ii)
it holds for all
t∈[0,T],
x∈O
that
[TABLE]
Throughout this proof let
σi,j:[0,T]×O→R,
i∈{1,2,…,d},
j∈{1,2,…,m},
satisfy for all
t∈[0,T],
x∈O
that
[TABLE]
Note that the chain rule, the fact that Rd∋x↦1+∥x∥2∈(0,∞) is infinitely often differentiable, and
the fact that
(0,∞)∋s↦s2p∈(0,∞)
is infinitely often differentiable establish Item (i). It thus remains to prove Item (ii). For this, we observe for all
x∈O,
i,j∈{1,2,…,d}
that
[TABLE]
and
[TABLE]
This yields for all
t∈[0,T],
x∈O
that
[TABLE]
Next note that for all
t∈[0,T],
x∈O
it holds that
[TABLE]
Combining this with (81) shows that for all
t∈[0,T],
x∈O
it holds that
[TABLE]
This establishes Item (ii).
The proof of Lemma 3.3 is thus completed.
∎
3.2 Locality properties for solutions of SDEs
In this section we present two elementary results concerning the local
behaviour of solutions to SDEs. These results,
Lemmas 3.4 and 3.5 below,
are used in the proof of Lemma 3.7
(see Section 3.4 below). Lemma 3.4
asserts, loosely speaking, that a particle whose movements are governed
by a SDE with sufficiently regular
coefficients is almost surely at rest when it finds itself in a region
away from the supports of the coefficients.
Lemma 3.4**.**
Let
d,m∈N,
T∈(0,∞),
let
∥⋅∥:Rd→[0,∞) be the standard norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty)
be the Frobenius norm on Rd×m,
let
μ∈C([0,T]×Rd,Rd),
σ∈C([0,T]×Rd,Rd×m)
satisfy for all
r∈(0,∞)
that
[TABLE]
let
O⊆Rd be an open set which satisfies supp(μ)∪supp(σ)⊆[0,T]×O,
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 (Ft)t∈[0,T]-Brownian motion,
and let
X:[0,T]×Ω→Rd
be an (Ft)t∈[0,T]-adapted stochastic process with continuous sample paths which satisfies that for all
t∈[0,T]
it holds P-a.s. that
[TABLE]
Then
- (i)
it holds that
\Big{[}\big{(}\mathbb{P}(X_{0}\notin\mathcal{O})=1\big{)}\Rightarrow\big{(}\mathbb{P}(\forall\,t\in[0,T]\colon X_{t}=X_{0})=1\big{)}\Big{]}
and
2. (ii)
it holds that
\Big{[}\big{(}\mathbb{P}(X_{0}\in\mathcal{O})=1\big{)}\Rightarrow\big{(}\mathbb{P}(\forall\,t\in[0,T]\colon X_{t}\in\overline{\mathcal{O}})=1\big{)}\Big{]}.
We first prove Item (i).
For this we assume that P(X0∈/O)=1. Observe that this implies
\mathbb{P}(\forall\,t\in[0,T]\colon\left\|\mu(t,X_{0})\right\|+\left\vvvert\sigma(t,X_{0})\right\vvvert=0)=1.
Therefore, we obtain that
[TABLE]
is an (Ft)t∈[0,T]-adapted stochastic process with continuous sample paths which satisfies that for all t∈[0,T] it holds P-a.s. that
[TABLE]
Karatzas & Shreve [15, Theorem 5.2.5] and (84)–(86) hence assure that
[TABLE]
This establishes Item (i).
Next we prove Item (ii). For this we assume that
P(X0∈O)=1
and let
τ:Ω→[0,T]
satisfy
τ=inf({t∈[0,T]: Xt∈/O}∪{T}).
Note that τ is an (Ft)t∈[0,T]-stopping time. Let Y:[0,T]×Ω→Rd satisfy for all
t∈[0,T],
ω∈Ω
that
Yt(ω)=Xmin{t,τ(ω)}(ω).
Observe that Y:[0,T]×Ω→Rd is an (Ft)t∈[0,T]-adapted stochastic process with continuous sample paths. Moreover, note that for all
t∈[0,T]
it holds P-a.s. that
[TABLE]
Combining this with the fact that for all
t∈[0,T]
it holds that
\mathbbm1{t≤τ}Xt=\mathbbm1{t≤τ}Yt
and
\mathbbm{1}_{\{\tau<t\}}[\left\|\mu(t,Y_{t})\right\|+\left\vvvert\sigma(t,Y_{t})\right\vvvert]=0
we obtain that for all
t∈[0,T]
it holds P-a.s. that
[TABLE]
Karatzas & Shreve [15, Theorem 5.2.5],
(84),
and
(85)
hence demonstrate that
[TABLE]
This establishes Item (ii). The proof of Lemma 3.4 is thus completed.
∎
The next result, Lemma 3.5 below,
basically asserts that the solutions of SDEs
coincide as long as the trajectories stay in a domain in which the drift
and diffusion coefficients are the same.
Lemma 3.5**.**
Let
d,m∈N,
T∈(0,∞),
let
∥⋅∥:Rd→[0,∞)
be the standard norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty)
be the Frobenius norm on Rd×m,
let
O⊆Rd
be an open set,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
C⊆[0,T]×Rd
be a closed set which satisfies
C⊆[0,T]×O,
let
μ1,μ2∈C([0,T]×O,Rd),
σ1,σ2∈C([0,T]×O,Rd×m)
satisfy for all
r∈(0,∞)
that
μ1∣C=μ2∣C,
σ1∣C=σ2∣C,
and
[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 (Ft)t∈[0,T]-Brownian motion,
let
X(i)=(Xt(i))t∈[0,T]:[0,T]×Ω→O,
i∈{1,2},
be (Ft)t∈[0,T]-adapted stochastic processes with continuous
sample paths which satisfy that for every
i∈{1,2},
t∈[0,T]
it holds P-a.s. that
[TABLE]
assume that
X0(1)=X0(2),
and let
τ:Ω→[0,T]
satisfy
τ=inf({t∈[0,T]:(t,Xt(1))∈/C or (t,Xt(2))∈/C}∪{T}).
Then it holds that
[TABLE]
Throughout this proof let
ρn:Ω→[0,T],
n∈N,
satisfy for all
n∈N
that
\rho_{n}=\inf\!\big{(}\{t\in[0,T]\colon X^{(1)}_{t}\in\mathcal{O}\setminus O_{n}~{}\text{or}~{}X^{(2)}_{t}\in\mathcal{O}\setminus O_{n}\}\cup\{T\}\big{)}
and let
Ln∈[0,∞), n∈N,
be real numbers which satisfy for all
t∈[0,T],
x,y∈On
that
[TABLE]
Observe that for all
n∈N
it holds that τ and ρn are (Ft)t∈[0,T]-stopping times.
Moreover, note that for every K⊆O compact there exists n∈N such that K⊆On. This and the fact that X(1) and X(2) have continuous sample paths ensure that for all
ω∈Ω
there exists
n∈N
such that for all
k∈N
with
k≥n
it holds that
ρk(ω)=T.
Next note that the assumption that
X(1)
and
X(2)
have continuous sample paths and the fact that
On, n∈N,
are compact imply that for all
n∈N,
ω∈{ρn>0},
i∈{1,2}
it holds that
Xρn(ω)(i)(ω)∈On.
Combining this with the assumption that
X0(1)=X0(2)
assures that for all
n∈N,
t∈[0,T]
it holds that
[TABLE]
This ensures for every
n∈N
that
[TABLE]
Next note that the fact that for all
s∈(0,T]
it holds that
\mathbbm{1}_{\{s\leq\tau\}}\big{[}\|\mu_{1}(s,X^{(2)}_{s})-\mu_{2}(s,X^{(2)}_{s})\|+\vvvert\sigma_{1}(s,X^{(2)}_{s})-\sigma_{2}(s,X^{(2)}_{s})\vvvert\big{]}=0,
the assumption that
X0(1)=X0(2),
and
(93)
ensure that for all
n∈N,
t∈[0,T]
it holds P-a.s. that
[TABLE]
This implies that for all
n∈N,
t∈[0,T]
it holds P-a.s. that
[TABLE]
Minkowski’s inequality, Itô’s isometry, and (95)–(97) hence yield for all
n∈N,
t∈[0,T]
that
[TABLE]
The fact that for all
a,b∈[0,∞)
it holds that
(a+b)2≤2a2+2b2
and Hölder’s inequality hence demonstrate for all
n∈N,
t∈[0,T]
that
[TABLE]
Combining this with Gronwall’s inequality and (97) implies for all
n∈N,
t∈[0,T]
that
[TABLE]
The fact that X(1) and X(2) have continuous sample paths hence ensures for all
n∈N
that
[TABLE]
Therefore we obtain that
[TABLE]
This implies that
[TABLE]
This establishes (94). The proof of Lemma 3.5 is thus completed.
∎
3.3 Continuity properties for solutions of SDEs
The well-known Lemma 3.6 below (cf. also Stroock [21, Theorem I.2.2]) estimates the difference between two solutions to the same SDE that start at different times and different places. Lemma 3.6 is a crucial ingredient in the proof of Lemma 3.7, where it is used to show that the solution to an auxiliary SDE evaluated at a certain time is stochastically continuous as a function of the initial values.
Lemma 3.6**.**
Let
d,m∈N,
L,T∈(0,∞),
let
∥⋅∥:Rd→[0,∞)
be the standard norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty)
be the Frobenius norm on
Rd×m,
let
O⊆Rd be a non-empty open set,
let
μ∈C([0,T]×O,Rd),
σ∈C([0,T]×O,Rd×m)
be compactly supported functions which satisfy for all
t∈[0,T],
x,y∈O
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 (Ft)t∈[0,T]-Brownian motion,
and for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be an (Fs)s∈[t,T]-adapted stochastic process with continuous sample paths which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
Then it holds for all
t∈[0,T],
t∈[t,T],
s∈[t,T],
x,x∈O
that
[TABLE]
Proof of
Lemma 3.6.
Throughout this proof let
m:[0,T]×Rd→Rd
and
s:[0,T]×Rd→Rd×m
satisfy for all
t∈[0,T],
x∈Rd
that
[TABLE]
Observe that (106)
ensures that
m:[0,T]×Rd→Rd
and
s:[0,T]×Rd→Rd×m
are compactly supported continuous functions which satisfy for all
t∈[0,T],
x,y∈Rd
that
[TABLE]
Karatzas & Shreve [15, Theorem
5.2.9] hence guarantees
for every
t∈[0,T],
x∈O
that there exists an (Fs)s∈[t,T]-adapted stochastic process
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→Rd
with continuous sample paths
- (A)
which satisfies that
sups∈[t,T]E[∥Xst,x∥2]<∞
and
2. (B)
which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
This, Karatzas & Shreve [15, Theorem
5.2.5], and
(107) ensure that for all
t∈[0,T],
x∈O
it holds that
[TABLE]
Combining this with the fact that for all
t∈[0,T],
x∈O
it holds that sups∈[t,T]E[∥Xst,x∥2]<∞
implies that for all
t∈[0,T],
x∈O
it holds that
[TABLE]
Next note that (107) ensures that for all
t∈[0,T],
t∈[t,T],
s∈[t,T],
x,x∈O
it holds P-a.s. that
[TABLE]
Minkowski’s inequality hence yields that for all
t∈[0,T],
t∈[t,T],
s∈[t,T],
x,x∈O
it holds that
[TABLE]
Itô’s isometry and (106) therefore ensure that for all
t∈[0,T],
t∈[t,T],
s∈[t,T],
x,x∈O
it holds that
[TABLE]
This and
(106) imply
for all
t∈[0,T],
t∈[t,T],
s∈[t,T],
x,x∈O
that
[TABLE]
The fact that for all
a,b,c∈[0,∞)
it holds that
(a+b+c)2≤3(a2+b2+c2)
and Hölder’s inequality therefore ensure that for all
t∈[0,T],
t∈[t,T],
s∈[t,T],
x,x∈O
it holds that
[TABLE]
Gronwall’s inequality and (113) hence imply for all
t∈[0,T],
t∈[t,T],
s∈[t,T],
x,x∈O
that
[TABLE]
In the next step we observe that (107) guarantees that for all
t∈[0,T],
t∈[t,T],
x,x∈O
it holds P-a.s. that
[TABLE]
Minkowski’s inequality hence demonstrates that for all
t∈[0,T],
t∈[t,T],
x,x∈O
it holds that
[TABLE]
Itô’s isometry therefore implies that for all
t∈[0,T],
t∈[t,T],
x,x∈O
it holds that
[TABLE]
This, Minkowski’s inequality, and (106) yield that for all
t∈[0,T],
t∈[t,T],
x,x∈O
it holds that
[TABLE]
The fact that for all
a,b,c∈[0,∞)
it holds that
(a+b+c)2≤3(a2+b2+c2)
hence implies for all
t∈[0,T],
t∈[t,T],
x,x∈O
that
[TABLE]
This demonstrates for all
t∈[0,T],
t∈[t,T],
x,x∈O
that
[TABLE]
Hence, we obtain for all
t∈[0,T],
t∈[t,T],
x,x∈O
that
[TABLE]
Gronwall’s inequality and
(113) hence ensure that for all
t∈[0,T],
t∈[t,T],
x,x∈O
it holds that
[TABLE]
Combining this with (119) demonstrates (108). The proof of Lemma 3.6 is thus completed.
∎
3.4 Existence and uniqueness properties for solutions of SFPEs associated with SDEs
In this section we provide the announced application of Theorem 2.9 (see Theorem 3.8 below). The next essentially well-known result, Lemma 3.7 below (cf., for example, Liu & Röckner [17, Proposition 3.2.1]), ascertains that the stochastic continuity hypothesis of Theorem 2.9 is satisfied in the setting of Theorem 3.8.
Lemma 3.7**.**
Let
d,m∈N,
T∈(0,∞),
let
⟨⋅,⋅⟩:Rd×Rd→R
be the standard scalar product on Rd,
let
∥⋅∥:Rd→[0,∞)
be the standard norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty) be the Frobenius norm on Rd×m,
let
O⊆Rd be a non-empty open set,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
μ∈C([0,T]×O,Rd),
σ∈C([0,T]×O,Rd×m)
satisfy for all
r∈(0,∞)
that
[TABLE]
let
V∈C1,2([0,T]×O,(0,∞))
satisfy for all
t∈[0,T],
x∈O
that
[TABLE]
assume that
supr∈(0,∞)[inft∈[0,T]infx∈O∖OrV(t,x)]=∞, 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
(Ft)t∈[0,T]-Brownian motion,
and for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be an (Fs)s∈[t,T]-adapted stochastic process with continuous sample paths which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
Then it holds for all
ε∈(0,∞),
s∈[0,T]
and all
(tn,xn)∈[0,T]×O, n∈N0,
with
limsupn→∞[∣tn−t0∣+∥xn−x0∥]=0
that
[TABLE]
Throughout this proof let
(tn,xn)∈[0,T]×O, n∈N0,
satisfy
limsupn→∞[∣tn−t0∣+∥xn−x0∥]=0.
Note that for the proof of Lemma 3.7 it is
sufficient to demonstrate that for all
ε∈(0,∞),
s∈[0,T]
it holds that
[TABLE]
Next note that the assumption that it holds that
supr∈(0,∞)inft∈[0,T]infx∈O∖OrV(t,x)=∞
ensures that for every
n∈N
there exists
r∈(0,∞)
such that
inft∈[0,T]infx∈O∖OrV(t,x)>n.
This yields that for every
n∈N
there exists
r∈(0,∞)
such that
{V≤n}⊆[0,T]×Or.
Hence, we obtain for every
n∈N
that
{V≤n} is a bounded set.
Combining this with the fact that
V:[0,T]×O→(0,∞)
is continuous demonstrates that for every
n∈N
it holds that
{V≤n}
is a compact set.
Lang [16, Theorem II.3.7]
therefore ensures that there exist
φn∈Cc∞([0,T]×O,R), n∈N,
which satisfy for all
n∈N,
t∈[0,T],
x∈O
that
[TABLE]
Next let
mn:[0,T]×Rd→Rd, n∈N,
and
sn:[0,T]×Rd→Rd×m, n∈N,
satisfy for all
n∈N,
t∈[0,T],
x∈Rd
that
[TABLE]
This, (128), and (133) assure that
mn:[0,T]×Rd→Rd, n∈N,
and
sn:[0,T]×Rd→Rd×m, n∈N,
are compactly supported continuous functions which satisfy that
- (A)
for all
n∈N
it holds that
[TABLE]
2. (B)
for all
n∈N,
t∈[0,T],
x∈O
it holds that
[TABLE]
and
3. (C)
for all
n∈N,
t∈[0,T],
x∈O
it holds that
[TABLE]
Note that Karatzas & Shreve [15, Theorem 5.2.9] (cf. also Gyöngy & Krylov [8, Corollary 2.6] and Liu & Röckner [17, Theorem 3.1.1]) and Item (A) yield that for every
n∈N,
t∈[0,T],
x∈O
there exists an (Fs)s∈[t,T]-adapted stochastic process with continuous sample paths Xn,t,x=(Xsn,t,x)s∈[t,T]:[t,T]×Ω→Rd which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
Moreover, note that Item (C) ensures for all
n∈N
that
supp(mn)∪supp(sn)⊆{V≤n+1}.
The fact that for every
n∈N
there exists
r∈(0,∞)
such that
{V≤n}⊆[0,T]×Or
hence implies that for every
n∈N
there exists
r∈(0,∞)
such that
supp(mn)∪supp(sn)⊆[0,T]×Or.
Furthermore, observe that Item (i) of Lemma 3.4 ensures that for all
n∈N,
r∈(0,∞),
m∈N∩(r,∞),
t∈[0,T],
x∈O∖{y∈O:(∃z∈Or:∥y−z∥<\nicefrac1m)}
with
supp(mn)∪supp(sn)⊆[0,T]×Or
it holds that
P(∀s∈[t,T]:Xsn,t,x=x)=1.
Combining this with the fact that for all
r∈(0,∞)
it holds that
Or=∩m∈N∩(r,∞){y∈Rd:(∃x∈Or:∥x−y∥<\nicefrac1m)} implies that for all
n∈N,
r∈(0,∞),
t∈[0,T],
x∈O∖Or
with
supp(mn)∪supp(sn)⊆[0,T]×Or
it holds that
[TABLE]
Next we observe that Item (ii) of Lemma 3.4 ensures that for all
n∈N,
r∈(0,∞),
m∈N∩(r,∞),
t∈[0,T],
x∈{y∈O:(∃z∈Or:∥y−z∥<\nicefrac1m)}
with
supp(mn)∪supp(sn)⊆[0,T]×Or
it holds that
P(∀s∈[t,T]:(∃y∈Or:∥Xsn,t,x−y∥≤\nicefrac1m))=1.
This yields that for all
n∈N,
r∈(0,∞),
t∈[0,T],
x∈Or
with
supp(mn)∪supp(sn)⊆[0,T]×Or
it holds that
[TABLE]
The fact that for every
n∈N
there exists
r∈(0,∞)
such that
supp(mn)∪supp(sn)⊆[0,T]×Or
and
(139) hence demonstrate that for every
n∈N
there exists
r∈(0,∞)
such that
- (I)
it holds for all
t∈[0,T],
x∈Or
that
P(∀s∈[t,T]:Xsn,t,x∈Or)=1
and
2. (II)
it holds for all
t∈[0,T],
x∈O∖Or
that
P(∀s∈[t,T]:Xsn,t,x=x)=1.
Therefore, we obtain that for every
n∈N,
t∈[0,T],
x∈O
there exists an (Fs)s∈[t,T]-adapted stochastic process with continuous sample paths
Xn,t,x=(Xsn,t,x)s∈[t,T]:[t,T]×Ω→O
which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
In the next step let
τn,t,x:Ω→[t,T],
n∈N,
t∈[0,T],
x∈O,
satisfy for every
n∈N,
t∈[0,T],
x∈O,
ω∈Ω
that
τn,t,x(ω)=inf({s∈[t,T]:max{V(s,Xsn,t,x(ω)),V(s,Xst,x(ω))}>n}∪{T}).
Note that for every
n∈N,
t∈[0,T],
x∈O
it holds that τn,t,x:Ω→[t,T] is an (Fs)s∈[t,T]-stopping time. Next observe that the fact that for every
n∈N
it holds that
{V≤n}
is a compact set, Item (B), and Lemma 3.5 (with
T=T−t,
C={(s,y)∈[0,T−t]×O:V(t+s,y)≤n},
μ1=([0,T−t]×O∋(s,y)↦μ(t+s,y)∈Rd),
μ2=([0,T−t]×O∋(s,y)↦mn(t+s,y)∈Rd),
σ1=([0,T−t]×O∋(s,y)↦σ(t+s,y)∈Rd×m),
σ2=([0,T−t]×O∋(s,y)↦sn(t+s,y)∈Rd×m),
F=(Ft+s)s∈[0,T−t],
W=([0,T−t]×Ω∋(s,ω)↦Wt+s(ω)−Wt(ω)∈Rm),
X(1)=([0,T−t]×Ω∋(s,ω)↦Xt+st,x(ω)∈O),
X(2)=([0,T−t]×Ω∋(s,ω)↦Xt+sn,t,x(ω)∈O),
τ=τn,t,x−t
for
n∈N,
t∈[0,T],
x∈O
in the notation of Lemma 3.5)
ensure for all
n∈N,
t∈[0,T],
x∈O
that
[TABLE]
This, Markov’s inequality, and Lemma 3.1
(with
T=T−t,
μ=([0,T−t]×O∋(s,y)↦μ(t+s,y)∈Rd),
σ=([0,T−t]×O∋(s,y)↦σ(t+s,y)∈Rd×m),
V=([0,T−t]×O∋(s,y)↦V(t+s,y)∈[0,∞)),
F=(Ft+s)s∈[0,T−t],
W=([0,T−t]×Ω∋(s,ω)↦Wt+s(ω)−Wt(ω)∈Rm),
X=([0,T−t]×Ω∋(s,ω)↦Xt+st,x(ω)∈O),
τ=τk,t,x−t
for
k∈N,
t∈[0,T],
x∈O
in the notation of Lemma 3.1) imply for all
ε∈(0,∞),
k∈N,
t∈[0,T],
x∈O,
s∈[t,T]
that
[TABLE]
Furthermore, observe that Lemma 3.6 ensures that there exist real numbers
ck∈[0,∞), k∈N,
which satisfy that for every k,n∈N,
s∈[t0,T]
it holds that
[TABLE]
Moreover, observe that (141) ensures that for all
k,n∈N,
s∈[t0,T]
it holds P-a.s. that
[TABLE]
Minkowski’s inequality, the fact that
mk:[0,T]×Rd→Rd, k∈N,
and
sk:[0,T]×Rd→Rd×m, k∈N,
are compactly supported continuous functions, and Itô’s isometry hence imply that for all
k,n∈N,
s∈[t0,T]
it holds that
[TABLE]
This, the fact that for all
a,b∈R
it holds that (a+b)2≤2a2+2b2,
and (144)
ensure that there exist real numbers
ck∈[0,∞), k∈N,
which satisfy for every
k,n∈N,
s∈[t0,T]
that
[TABLE]
In addition, observe that (141) ensures that for all
k,n∈N,
s∈[0,t0]
it holds P-a.s. that
[TABLE]
Minkowski’s inequality, the fact that
mk:[0,T]×Rd→Rd, k∈N,
and
sk:[0,T]×Rd→Rd×m, k∈N,
are compactly supported continuous functions, and Itô’s isometry hence imply that for all
k,n∈N,
s∈[0,t0]
it holds that
[TABLE]
This and (147) ensure that there exist real numbers
ck∈[0,∞), k∈N,
which satisfy for every
k,n∈N,
s∈[0,T]
that
[TABLE]
Combining this with Markov’s inequality and (143) demonstrates that for all
ε∈(0,∞),
s∈[0,T]
it holds that
[TABLE]
This demonstrates (132).
The proof of Lemma 3.7 is thus completed.
∎
The next result, Theorem 3.8 below, is the main result of this article. It is an application of Theorem 2.9. Lemmas 3.1 and 3.7 above ensure that the crucial hypotheses of Theorem 2.9 are satisfied in the setting of Theorem 3.8.
Theorem 3.8**.**
Let
d,m∈N,
L,T∈(0,∞),
let
⟨⋅,⋅⟩:Rd×Rd→R
be the standard scalar product on Rd,
let
∥⋅∥:Rd→[0,∞)
be the standard norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty)
be the Frobenius norm on
Rd×m,
let
O⊆Rd be a non-empty open set,
for every
r∈(0,∞)
let
Or⊆O
satisfy
Or={x∈O:∥x∥≤r and {y∈Rd:∥y−x∥<\nicefrac1r}⊆O},
let
μ∈C([0,T]×O,Rd),
σ∈C([0,T]×O,Rd×m)
satisfy for all
r∈(0,∞)
that
[TABLE]
let
f∈C([0,T]×O×R,R),
g∈C(O,R),
V∈C1,2([0,T]×O,(0,∞)),
assume for all
t∈[0,T],
x∈O,
v,w∈R
that
∣f(t,x,v)−f(t,x,w)∣≤L∣v−w∣
and
[TABLE]
assume that
supr∈(0,∞)[inft∈[0,T]infx∈O∖OrV(t,x)]=∞
and
infr∈(0,∞)[supt∈[0,T]supx∈O∖Or(V(t,x)∣f(t,x,0)∣+V(T,x)∣g(x)∣)]=0,
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
(Ft)t∈[0,T]-Brownian motion,
and for every
t∈[0,T],
x∈O
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→O
be an
(Fs)s∈[t,T]-adapted stochastic process with continuous sample paths which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
Then there exists a unique u∈C([0,T]×O,R) such that
- (i)
it holds that
[TABLE]
and
2. (ii)
it holds for all
t∈[0,T],
x∈O
that
[TABLE]
Proof of
Theorem 3.8.
First, note that Lemma 3.1
(with
T=T−t,
μ=([0,T−t]×O∋(s,y)↦μ(t+s,y)∈Rd),
σ=([0,T−t]×O∋(s,y)↦σ(t+s,y)∈Rd×m),
V=([0,T−t]×O∋(s,y)↦V(t+s,y)∈[0,∞)),
F=(Ft+s)s∈[0,T−t],
W=([0,T−t]×Ω∋(s,ω)↦Wt+s(ω)−Wt(ω)∈Rm),
X=([0,T−t]×Ω∋(s,ω)↦Xt+st,x(ω)∈O) for
t∈[0,T],
x∈O
in the notation of Lemma 3.1)
ensures that for all
t∈[0,T],
s∈[t,T],
x∈O
it holds that
[TABLE]
Next observe that Lemma 3.7 ensures that for all
ε∈(0,∞),
s∈[0,T]
and all
(tn,xn)∈[0,T]×O, n∈N0,
with
limsupn→∞[∣tn−t0∣+∥xn−x0∥]=0
it holds that
limsupn→∞[P(∥Xmax{s,tn}tn,xn−Xmax{s,t0}t0,x0∥≥ε)]=0.
Combining this with (157) and Theorem 2.9 demonstrates that there exists a unique u∈C([0,T]×O,R) which satisfies that for all
t∈[0,T],
x∈O
it holds that
limsupr→∞[sups∈[0,T]supy∈O∖Or(V(s,y)∣u(s,y)∣)]=0
and
[TABLE]
This establishes Items (i) and (ii). The proof of Theorem 3.8 is thus completed.
∎
Lemma 3.2 implies the following corollary of Theorem 3.8 in the situation in which the drift and diffusion coefficients μ:[0,T]×O→Rd and σ:[0,T]×O→Rd×m depend only on the spatial variable x∈O and are independent of the time variable t∈[0,T]. For the sake of simplicity we take the spatial domain O to be Rd in 3.9 below.
Corollary 3.9**.**
Let d,m∈N,
L,T∈(0,∞),
ρ∈R,
let
⟨⋅,⋅⟩:Rd×Rd→R be the standard scalar product on Rd,
let
∥⋅∥:Rd→[0,∞)
be the standard norm on Rd,
let
μ:Rd→Rd
and
σ:Rd→Rd×m
be locally Lipschitz continuous, let
f∈C([0,T]×Rd×R,R),
g∈C(Rd,R),
V∈C2(Rd,(0,∞)),
assume for all
t∈[0,T],
x∈Rd,
v,w∈R
that
∣f(t,x,v)−f(t,x,w)∣≤L∣v−w∣
and
[TABLE]
assume that
supr∈(0,∞)[infx∈Rd,∥x∥>rV(x)]=∞
and infr∈(0,∞)[supt∈[0,T]supx∈Rd,∥x∥>r(V(x)∣f(t,x,0)∣+∣g(x)∣)]=0, 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 (Ft)t∈[0,T]-Brownian motion,
and for every
t∈[0,T],
x∈Rd
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→Rd
be an (Fs)s∈[t,T]-adapted stochastic process with continuous sample paths which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
Then there exists a unique u∈C([0,T]×Rd,R) such that
- (i)
it holds that
[TABLE]
and
2. (ii)
for all
t∈[0,T],
x∈Rd
it holds that
[TABLE]
Proof of 3.9.
Throughout this proof let
V:[0,T]×Rd→(0,∞)
satisfy for all
t∈[0,T],
x∈Rd
that
V(t,x)=e−ρtV(x).
Observe that Lemma 3.2 (with
O=Rd,
μ=([0,T]×Rd∋(t,x)↦μ(x)∈Rd),
σ=([0,T]×Rd∋(t,x)↦σ(x)∈Rd×m)
in the notation of Lemma 3.2)
ensures that for all
t∈[0,T],
x∈Rd
it holds that
V∈C1,2([0,T]×Rd,(0,∞))
and
[TABLE]
Next, observe that the hypothesis that
supr∈(0,∞)[infx∈Rd,∥x∥>rV(x)]=∞
implies that
[TABLE]
Furthermore, observe that the hypothesis that
infr∈(0,∞)[supt∈[0,T]supx∈Rd,∥x∥>r(V(x)∣f(t,x,0)∣+∣g(x)∣)]=0
demonstrates that
[TABLE]
Theorem 3.8 (with O=Rd, μ=([0,T]×Rd∋(t,x)↦μ(x)∈Rd), σ=([0,T]×Rd∋(t,x)↦σ(x)∈Rd×m), V=V in the notation of Theorem 3.8) and (164) hence ensure that there exists a unique u∈C([0,T]×Rd,R) which satisfies that
- (I)
it holds that
[TABLE]
and
2. (II)
it holds for all
t∈[0,T],
x∈Rd
that
[TABLE]
Next note that Item (I) implies that
[TABLE]
This establishes Item (i). Moreover, note that Item (II) establishes Item (ii). The proof of 3.9 is thus completed. ∎
Finally, in 3.10 below, we specialize in the setting of 3.9 to the situation in which the drift and diffusion coefficients μ:Rd→Rd and σ:Rd→Rd×m satisfy a coercivity condition and the nonlinearity f:[0,T]×Rd×R→R as well as the terminal condition g:Rd→R are at most polynomially growing with respect to the spatial variable x∈Rd. Suitable choices for the Lyapunov-type function V are provided by Lemma 3.3.
Corollary 3.10** (Existence and uniqueness of at most polynomially growing solutions of SFPEs).**
Let
d,m∈N,
L,T∈(0,∞),
let
⟨⋅,⋅⟩:Rd×Rd→R
be the standard scalar product on Rd,
let
∥⋅∥:Rd→[0,∞)
be the standard norm on Rd,
let
\left\vvvert\cdot\right\vvvert\colon\mathbb{R}^{d\times m}\to[0,\infty) be the Frobenius norm on Rd×m,
let
μ:Rd→Rd and
σ:Rd→Rd×m
be locally Lipschitz continuous,
let
f∈C([0,T]×Rd×R,R),
g∈C(Rd,R)
be at most polynomially growing,
assume for all
t∈[0,T],
x∈Rd,
v,w∈R
that
max{⟨x,μ(x)⟩,\vvvertσ(x)\vvvert2}≤L(1+∥x∥2)
and
∣f(t,x,v)−f(t,x,w)∣≤L∣v−w∣,
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 (Ft)t∈[0,T]-Brownian motion,
and for every
t∈[0,T],
x∈Rd
let
Xt,x=(Xst,x)s∈[t,T]:[t,T]×Ω→Rd
be an (Fs)s∈[t,T]-adapted stochastic process with continuous sample paths which satisfies that for all
s∈[t,T]
it holds P-a.s. that
[TABLE]
Then there exists a unique u∈C([0,T]×Rd,R) such that
- (i)
it holds that u is at most polynomially growing and
2. (ii)
it holds for all
t∈[0,T],
x∈Rd
that
[TABLE]
Proof of 3.10.
Throughout this proof let
ρq∈(0,∞),
q∈(0,∞),
satisfy for every
q∈(0,∞)
that
ρq=2qLmax{q+1,3},
let
p∈(0,∞)
satisfy that
supt∈[0,T]supy∈Rd[1+∥y∥p∣f(t,y,0)∣+∣g(y)∣]<∞,
and let
Vq:Rd→R,
q∈(0,∞),
satisfy for all
q∈(0,∞),
x∈Rd
that
Vq(x)=[1+∥x∥2]\nicefracq2.
Note that the fact that
supt∈[0,T]supx∈Rd[1+∥x∥p∣f(t,x,0)∣+∣g(x)∣]<∞
implies that for all
q∈(p,∞)
it holds that
[TABLE]
Moreover, observe that for all
q∈(p,∞)
it holds that
[TABLE]
Next note that Lemma 3.3 ensures for all
q∈(0,∞),
x∈Rd
that
[TABLE]
Combining this with (171), (172), and 3.9 (with V=V2p in the notation of 3.9) yields that there exists a unique u∈C([0,T]×Rd,R) which satisfies for all
t∈[0,T],
x∈Rd
that
limsupr→∞[sups∈[0,T]supy∈Rd,∥y∥>r(V2p(y)∣u(s,y)∣)]=0
and
u(t,x)=\mathbb{E}[g(X^{t,x}_{T})+\int_{t}^{T}f\big{(}s,X^{t,x}_{s},u(s,X^{t,x}_{s})\big{)}\,ds].
In particular, this ensures that u:[0,T]×Rd→R is at most polynomially growing. This establishes that u∈C([0,T]×Rd,R) satisfies Items (i) and (ii).
It remains to prove that u:[0,T]×Rd→R is the only continuous function which satisfies Items (i) and (ii). For this, let
v∈C([0,T]×Rd,R)
be an at most polynomially growing function which satisfies for all
t∈[0,T],
x∈Rd
that
v(t,x)=E[g(XTt,x)+∫tTf(s,Xst,x,v(s,Xst,x))ds].
The fact that v:[0,T]×Rd→R is at most polynomially growing ensures that there exists
q∈(0,∞)
which satisfies that
supt∈[0,T]supx∈Rd[1+∥x∥q∣v(t,x)∣]<∞.
This implies that u,v∈C([0,T]×Rd,R) satisfy for all
t∈[0,T],
x∈Rd
that
limsupr→∞[sups∈[0,T]supy∈Rd,∥y∥>r(Vmax{2q,2p}(y)∣u(s,y)∣+∣v(s,y)∣)]=0,
u(t,x)=\mathbb{E}[g(X^{t,x}_{T})+\int_{t}^{T}f\big{(}s,X^{t,x}_{s},u(s,X^{t,x}_{s})\big{)}\,ds],
and
[TABLE]
3.9 (with V=Vmax{2q,2p} in the notation of 3.9) hence ensures that u=v.
This establishes that u:[0,T]×Rd→R is the unique continuous function which satisfies Items (i) and (ii). The proof of 3.10 is thus completed.
∎
