Numerical scheme for stochastic differential equations driven by fractional Brownian motion with 1/4 < H < 1/2

TL;DR

This paper develops a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter between 1/4 and 1/2, achieving a convergence rate of n^(2H+rho).

Contribution

It introduces a novel numerical approximation method using Doss-Sussmann representation and Taylor expansion for fractional Brownian motion with H in (1/4, 1/2).

Findings

Convergence rate of n^(2H+rho) for the scheme.

Application of Doss-Sussmann representation in numerical approximation.

Effective handling of fractional Brownian motion with H in (1/4, 1/2).

Abstract

In this article, we study a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter H in (1/4; 1/2). Towards this end, we apply Doss-Sussmann representation of the solution and an approximation of this representation using a first order Taylor expansion. The obtained rate of convergence is n^(2H+rho), for rho small enough.

Equations

$$X_t=x+{\int }_0^{t}b(X_s)ds+{\int }_0^t\sigma (X_s)\circ d{B}_s,t\in [0,T],$$

$$\mathbb{E}\left(B_t{B}_s\right)=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s{|}^{2H}\right),t\in [0,T].$$

$$X_t=x+{\int }_0^{t}b(X_s)ds+{\int }_0^t\sigma (X_s)\circ d{B}_s,t\in [0,T],$$

$$X_t=\varphi \left(Y_t,B_t\right).$$

$$\frac{\partial \varphi }{\partial \beta }(\alpha ,\beta )$$

$$\varphi (\alpha ,0)$$

$$Y_t=x+{\int }_0^t{\left(\frac{\partial \varphi }{\partial \alpha }(Y_s,B_s)\right)}^{-1}b\left(\varphi (Y_s,B_s)\right)ds,t\in [0,T].$$

$$\frac{\partial \varphi }{\partial \alpha }(\alpha ,\beta )=exp\left({\int }_0^{\beta }{\sigma }^{\prime }(\varphi (\alpha ,s))ds\right),$$

$$Y_t=x+{\int }_0^t\exp \left(-{\int }_0^{B_s}{\sigma }^{\prime }(\varphi (Y_s,u))du\right)b\left(\varphi (Y_s,B_s)\right)ds.$$

$$\varphi (x,u)=x+{\int }_0^u\sigma (\varphi (x,s))ds.$$

$$u_{i+1}^l=u_i^l+\frac{\Vert B{\Vert }_{\infty }}{l}=\frac{(i+1)\Vert B{\Vert }_{\infty }}{l},u_{-(i+1)}^l=u_{-i}^l-\frac{\Vert B{\Vert }_{\infty }}{l}=-\frac{(i+1)\Vert B{\Vert }_{\infty }}{l}.$$

$$M:=|x|+T\left(M_1\exp (M_2\Vert B{\Vert }_{\infty })+\Vert B{\Vert }_{H-\rho }{C}_3{T}^{H-\rho }\right),$$

$$C_3=M_1{M}_2\exp \left(M_2\Vert B{\Vert }_{\infty }\right)+M_4\exp \left(M_2\Vert B{\Vert }_{\infty }\right)M_5\Vert B{\Vert }_{\infty }\left(1+M_2\right)$$

$${\varphi }^l(z,u)=0\text{mbox}if(z,u)\notin [-M,M]\times [-\Vert B{\Vert }_{\infty },\Vert B{\Vert }_{\infty }];$$

$${\varphi }^l(z,u)={\varphi }^l(z,u_{k-1}^l)+{\int }_{u_{k-1}^l}^u\sigma \left({\varphi }^l(z,u_{k-1}^l)+(s-u_{k-1}^l)\sigma \left({\varphi }^l(z,u_{k-1}^l)\right)\right)ds,$$

$${\varphi }^l(z,u_0^l)=z,ifz\in [-M,M].$$

$${\varphi }^l(z,u)={\varphi }^l(z,u_k^l)-{\int }_u^{u_k^l}\sigma \left({\varphi }^l(z,u_k^l)+(s-u_k^l)\sigma \left({\varphi }^l(z,u_k^l)\right)\right)ds,$$

$${\Psi }^l(z,u)=0\text{mbox}if(z,u)\notin [-M,M]\times [-\Vert B{\Vert }_{\infty },\Vert B{\Vert }_{\infty }],$$

$${\Psi }^l(z,u)$$

$${\Psi }^l(z,u_0^l)=z,ifz\in [-M,M].$$

$${\Psi }^l(z,u)={\Psi }^l(z,u_k^l)-{\int }_u^{u_k^l}\left(\sigma \left({\Psi }^l(z,u_k^l)\right)+{\sigma }^{\prime }\sigma \left({\Psi }^l(z,u_k^l)\right)(s-u_k^l)\right)ds,$$

$$Y_t^l$$

$$g^l\left(B_s,Y_s^l\right)=\exp \left(-{\int }_0^{B_s}{\sigma }^{\prime }({\Psi }^l(Y_s^l,u))du\right)b\left({\Psi }^l(Y_s^l,B_s)\right).$$

$$Y_0^{l,m}$$

$$Y_t^{l,m}$$

$$\frac{\partial g^l(u,z)}{\partial u}=-g^l(u,z){\sigma }^{\prime }({\Psi }^l(z,u))+\exp \left(-{\int }_0^u{\sigma }^{\prime }({\Psi }^l(z,r))dr\right)b^{\prime }({\Psi }^l(z,u))\frac{\partial {\Psi }^l(z,u)}{\partial u}.$$

$$\left|g^l(u,z)\right|\le M_1\exp \left(M_2|u|\right).$$

$$|h_1^l(B_{t_k^m},Y_{t_k^m}^{l,m})|\le C_3,$$

$$\underset{t\in [0,T]}{\sup }|Y_t^{n,n}|\le |x|+T\left(M_1\exp (M_2\Vert B{\Vert }_{\infty })+T^{H-\rho }\Vert B{\Vert }_{H-\rho }{C}_3\right)=M.$$

$$X_t^n={\Psi }^n\left(Y_t^{n,n},B_t\right),$$

$$\left|X_t-X_t^n\right|\le C{n}^{-2(H-\rho )},$$

Full text

Key words: Doss-Sussmann representation, fractional Brownian motion, stochastic differential equation, Taylor expansion.

1 Introduction

In this article we are interested in a pathwise approximation of the solution to the stochastic differential equation

[TABLE]

where $x\in\mathbb{R}$ and $b,\sigma:\mathbb{R}\rightarrow\mathbb{R}$ are measurable functions. The stochastic integral in (1) is understood in the sense of Stratonovich, (see Alòs et.al. [1] for details) and $B=\{B_{t},t\in[0,T]\}$ is a fractional Brownian motion (fBm) with Hurst parameter $H\in(1/4,1/2)$. $B$ is a centered Gaussian process with a covariance structure given by

[TABLE]

In [1], the existence and uniqueness for the solution of equation (1) have been established under suitable conditions, which follows from our assumption (see hypothesis (H) in Section 2.1).

Equation (1) has been analyzed by several authors, for different interpretations of stochastic integrals, because of the properties of fractional Brownian motion $B$ . Among these properties, we can mention self-similarity, stationary increments, $\rho$-Hölder continuity, for any $\rho\in(0,H)$, and the covariance of its increments on intervals decays asymptotically as a negative power of the distance between the intervals. Therefore, equation (1) becomes quite useful in applications in different areas such as physics, biology, finance, etc (see, e.g., [2, 9, 11]). Hence, it is important to provide approximations to the solution of (1).

For $H=1/2$ (i.e., B is a Brownian motion), a large number of numerical schemes to approximate the unique solution of (1) has been considered in the literature. The reader can consult Kloeden and Platen [10] (and the references therein), for a complete exposition of this topic. In particular, Talay [16] introduces the Doss-Sussmann transformation [6, 15] in the study of numerical methods to the solution of stochastic differential equations (see Section 2.1 for the definition of this transformation).

For $H>1/2$, numerical schemes for equation (1) have been analyzed by several authors. For instance, we can mention [4, 8, 13] and [14], where the stochastic integrals is interpreted as the extension of the Young integral given in [17] and the forward integral, respectively. It is well-known that these integrals agree with the Stratonovich one under suitable conditions (see Alòs and Nualart [3]).

In this paper we are interested in the case $H<1/2$, because numerical schemes for the solution to (1) have been studied only in some particular situations. Namely, Garzón et. al. [7] use the Doss-Sussmann transformation in order to prove the convergence for the Euler scheme associated to (1) by means of an approximation of fBm via fractional transport processes. In [14], the authors also take advantage of the Doss-Sussmann transformation in order to discuss the Crank-Nicholson method, for $H\in\left(1/6,1/2\right)$ and $b\equiv 0$. Here, they show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable. Specifically, the authors state that the rate of convergence of the scheme is of order $n^{1/2-3H}$. In [12] the authors consider the so-called modified Euler scheme for multidimensional stochastic differential equations driven by fBm with $H\in\left(1/3,1/2\right)$. They utilize rough paths techniques in order to obtain the convergence rate of order $n^{1/2-2H}$. Also, they prove that this rate is sharp. In [5] a numerical scheme for stochastic differential equations driven by a multidimensional fBm with Hurst parameter greater than $1/3$ is introduced. The method is based on a second-order Taylor expansion, where the Lévy area terms are replaced by products of increments of the driving fBm. Here, the order of convergence is $n^{-(H-\rho)}$, with $\rho\in\left(1/3,H\right)$. In order to get this rate of convergence, the authors use a combination of rough paths techniques and error bounds for the discretization of the Lévy area terms.

In this work we propose an approximation scheme for the solution to (1) with $H\in\left(1/4,1/2\right)$. To do so, we use a first order Taylor expansion in the Doss-Sussmann representation of the solution. We consider the case $H\in\left(1/4,1/2\right)$ because it is showed in [3] that the solution of (1) is given by this transformation. However, even in the case $\left(0,1/4\right)$, our scheme tends to the mentioned transformation. The rate of convergence in this paper is $n^{-2H+\rho}$, where $\rho<2H$ small enough, improving the ones given in [14], [16], [5] and [12]. Also our rate is better than the one obtained in [7] when the fBm is not approximated by means of fractional transport process. We observe that our method only establishes this rate of convergence for $H<1/2$ because we could only see that the auxiliary inequality (39) below is satisfied in this case. However, the same construction holds for $H>1/2$ (see [14], Proposition 1). In this case, the rate of convergence for the scheme is not the same as the case $1/4<H<1/2$. In fact, for $H>1/2$, we only get that the rate of convergence is $n^{-1+\rho}$ for $\rho$ small enough.

The paper is organized as follows: In Section 2 we introduce the notations needed in this article. In particular, we explain the Doss-Sussmann-type transformation related to the unique solution to (1). Also, in this section, the scheme is presented and the main result is stated (Theorem 2.2 below). In Section 3, we establish the auxiliary lemmas, which are needed to show, in Section 4, that the main result is true. The proof of the auxiliary lemmas are presented in Section 5. Finally, in the Appendix (Section 6), other auxiliary result is also studied because it is a general result concerning the Taylor expansion for some continuous functions.

2 Preliminaries and main result

In this section, we introduce the basic notions and the framework that we use in this paper. That is, we first describe the Doss-Sussmann transformation given in Doss [6] and Sussmann [15], which is the link between the stochastic and ordinary differential equations (see Alòs et al. [1], or Nourdin and Neuenkirch [14], for fractional Brownian motion case). Then, we provide a numerical method and its rate of convergence for the unique solution of (1). These are the main result of this article (see Theorem 2.2).

2.1 Doss-Sussmann transformation

Henceforth, we consider the stochastic differential equation

[TABLE]

where $B=\{B_{t}:t\in[0,T]\}$ is a fractional Brownian motion with Hurst parameter $1/4<H<1/2$, $x\in\mathbb{R}$ and the stochastic integral in (3) is understood in the sense of Stratonovich, which is introduced in [1]. Remember that $B$ is defined in (2). The coefficients $b,\sigma:\mathbb{R}\rightarrow\mathbb{R}$ are measurable functions such that

• (H)

$b\in C^{2}_{b}(\mathbb{R})$ and $\sigma\in C^{2}_{b}(\mathbb{R})$.

Remark 2.1*.*

By assumption(H), we have, for $z\in\mathbb{R}$,

• •

$|b(z)|\leq M_{1}$, $|b^{\prime}(z)|\leq M_{4}$ and $|b^{\prime\prime}(z)|\leq M_{6}$.

• •

$|\sigma(z)|\leq M_{5}$, $|\sigma^{\prime}(z)|\leq M_{2}$ and $|\sigma^{\prime\prime}(z)|\leq M_{3}$.

We explicitly give these constants so that it will be clear where we use them in our analysis.

Now, we explain the relation between (3) and ordinary differential equations: the so call Doss-Sussmann transformation.

In Alòs et al. (Proposition 6) is proven that the equation (3) has a unique solution of the form

[TABLE]

The function $\phi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is the solution of the ordinary differential equation

[TABLE]

and the process $Y$ is the pathwise solution to the equation

[TABLE]

By Doss [6], we have

[TABLE]

which implies

[TABLE]

2.2 Numerical Method

In this section, we describe our numerical scheme associated to the unique solution of (3). Towards this end, in Section 2.2.1, we first propose an approximation to the function $\phi$ given in (6), and then, in Section 2.2.2 we approximate the process $Y$. In both sections we suppose that (H) holds.

2.2.1 Approximation of $\phi$

Note that, for $x\in\mathbb{R}$, equation (2.1) has the form

[TABLE]

For each $l\in\mathbb{N}$, we take the partition $\left\{u_{i}^{l},i\in\{-l,\ldots,l\}\right\}$ of the interval $[-\|B\|_{\infty},\|B\|_{\infty}]$ given by $-\|B\|_{\infty}=u^{l}_{-l}<\ldots<u^{l}_{-1}<u^{l}_{0}=0<u^{l}_{1}<\ldots<u^{l}_{l}=\|B\|_{\infty}$. Here, $\|B\|_{\infty}=\sup_{t\in[0,T]}|B_{t}|$,

[TABLE]

Let $x\in\mathbb{R}$ be given in (7). Set

[TABLE]

where $\rho\in(0,H)$, $\|B\|_{H-\rho}$ is the $(H-\rho$)-Hölder norm of $B$ on $[0,T]$,

[TABLE]

and $M_{i}$, $i\in\{1,\ldots,6\}$ are defined in Remark 2.1.

Now, we define the function $\phi^{l}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by

[TABLE]

and, for $k=1,\ldots,l$,

[TABLE]

if $z\in[-M,M]$ and $u\in(u^{l}_{k-1},u^{l}_{k}]$, with

[TABLE]

The definition of $\phi^{l}$ for the case $k=-l,\ldots,0$ is similar. That is,

[TABLE]

if $z\in[-M,M]$ and $u\in[u^{l}_{k-1},u^{l}_{k})$.

Also, we consider the function $\Psi^{l}:\mathbb{R}^{2}\rightarrow\mathbb{R}$, which is equal to

[TABLE]

and, for $k=1,\ldots,l$,

[TABLE]

if $z\in[-M,M]$ and $u\in(u^{l}_{k-1},u^{l}_{k}]$, with

[TABLE]

For $k=-l,\ldots,0$, $\Psi^{l}$ is introduced as

[TABLE]

If $z\in[-M,M]$ and $u\in[u^{l}_{k-1},u^{l}_{k}]$. From equation (LABEL:psin) and last equality, it can be seen that $\Psi^{l}(z,\cdot)$ is continuous on $[-\|B\|_{\infty},\|B\|_{\infty}]$.

We remark that the function $\phi^{l}$ given in (11) and (13) is an auxiliary tool that allows us to use Taylor’s theorem in the analysis of the numerical scheme proposed in this paper (i.e., Theorem 2.2). Indeed, the Taylor’s theorem is utilized in Lemma 3.2.

2.2.2 Approximation of $Y$

Here, we approximate the solution of equation (7).

For $l\in\mathbb{N}$, we define the process $Y^{l}$ as the solution of the following ordinary differential equation, where the existence and uniqueness is guaranteed since the coefficient $g^{l}:\mathbb{R}^{2}\to\mathbb{R}$ satisfies Lipschitz and linear growth conditions in the second variable (see Remark 2.1 and Lemma 3.3).

[TABLE]

where

[TABLE]

Now, for $m\in\mathbb{N}$, we set the partition $0=t_{0}<\ldots<t_{m}=T$ of $[0,T]$ with $t_{i+1}=t_{i}+{T\over m}$ and we define the process $Y^{l,m}$ as:

[TABLE]

for $t_{k}^{m}\leq t<t_{k+1}^{m}$, where $h_{1}^{l}(u,z)={\partial g^{l}(u,z)\over\partial u}$ and $g^{l}$ is given by (17). So

[TABLE]

By Remark 2.1, we can see

[TABLE]

Also we have

[TABLE]

where $C_{3}$ is given in (9). Moreover, assuming that (20) and (21) are satisfied, it is not hard to prove by induction that

[TABLE]

Finally, in a similar way to Garzón et al. [7], for $n\in\mathbb{N}$, we define the approximation $X^{n}$ of $X$ by:

[TABLE]

where $\Psi^{n}$ and $Y_{t}^{n,n}$ are given by (LABEL:psin) and (18), respectively.

Now, we are in position to state our main result

Theorem 2.2*.*

Let (H) be satisfied and $1/4<H<1/2$, then

[TABLE]

where $\rho>0$ is small enough and $C$ is a constant that does not depend on $n$.

Remark 2.3*.*

The constant $C$ has the form

[TABLE]

with

[TABLE]

Remark 2.4*.*

We choose the constant $M$ because the processes given in (16) and (18), as well as the solution to (7), are bounded by $M$, as it is pointed out in this section.

3 Preliminary lemmas

In this section, we stated the auxiliary tools that we need in order to prove our main result Theorem 2.2. The first four lemmas are related to the apriori estimates of $\phi$. We recall you that the constants $M_{i},i\in\{1,\ldots,6\}$ are introduced in Remark 2.1.

Lemma 3.1**.**

Let $\phi$ and $\phi^{l}$ be given by (8) and (11), respectively. Then, Hypothesis (H) implies that , for $(z,u)\in[-M,M]\times[-\|B\|_{\infty},\|B\|_{\infty}]$, we have

[TABLE]

Lemma 3.2**.**

Let $\phi^{l}$ and $\Psi^{l}$ be given by (11) and (LABEL:psin), respectively. Then, Hypothesis (H) implies

[TABLE]

for $(z,u)\in[-M,M]\times[-\|B\|_{\infty},\|B\|_{\infty}]$.

Lemma 3.3**.**

Let $\Psi^{l}$ be introduced in (LABEL:psin) and Hypothesis (H) hold. Then, for $(z_{1},z_{2},u)\in[-M,M]^{2}\times[-\|B\|_{\infty},\|B\|_{\infty}]$,

[TABLE]

Lemma 3.4**.**

Let $\phi^{l}$ be given in (11). Then, under Hypothesis (H),

[TABLE]

for $(z_{1},z_{2},u)\in[-M,M]^{2}\times[-\|B\|_{\infty},\|B\|_{\infty}]$.

Now we proceed to state the lemmas referred to the estimates on $Y^{n}-Y$.

Lemma 3.5**.**

Assume that Hypothesis (H) is satisfied. Let $Y$ and $Y^{n}$ be given in (7) and (16), respectively. Then, for $t\in[0,T]$,

[TABLE]

where

[TABLE]

and

[TABLE]

Lemma 3.6**.**

Let $Y^{n,n}$ be defined in (18). Then Hypothesis (H) implies, for $s\in(t_{k}^{n},t_{k+1}^{n}]$,

[TABLE]

where $C_{4}=M_{1}\exp\left(M_{2}\|B\|_{\infty}\right)+C_{3}T^{H-\rho}\|B\|_{H-\rho}$ and

[TABLE]

Lemma 3.7**.**

Suppose that Hypothesis (H) holds. Let $Y^{n}$ and $Y^{n,n}$ be given in (16) and (18), respectively. Then,

[TABLE]

where $0<\rho<H$,

[TABLE]

[TABLE]

with $C_{4}$ given in Lemma 3.6, and

[TABLE]

[TABLE]

4 Convergence of the Scheme: Proof of Theorem 2.2

We are now ready to prove the main result of this article, which gives a theoretical bound on the speed of convergence for $X^{n}$ defined in (22). Remember that the constants $M_{i},i\in\{1,\ldots,6\}$, are given in Remark 2.1.

Proof.

By (4) and (22), we have, for $t\in[0,T]$,

[TABLE]

where

[TABLE]

Now we proceed to obtain estimates of $H_{1}$, $H_{2}$ and $H_{3}$. By property (6), we get

[TABLE]

Therefore, by Lemmas 3.1 and 3.5

[TABLE]

Also Lemmas 3.4 and 3.7, yield

[TABLE]

For $H_{3}$, we use Lemma 3.2. So

[TABLE]

Finally, from (26) to (28), we have

[TABLE]

which shows that Theorem 2.2 holds. ∎

5 Proofs of preliminary lemmas

Here, we provide the proofs of Lemmas 3.1 to 3.7. First, we will prove, by induction, that the statements of Lemmas 3.1 to 3.4 hold for all $k=1,2,\ldots l$ and $u\in(u_{k-1}^{l},u^{l}_{k}]$. We will consider for simplicity the case $u>0$, the other case can be treated similarly.

Proof of Lemma 3.1

Proof.

Let $z\in[-M,M]$. We will prove by induction that, for all $k\in\{1,\ldots,l\}$ and $u\in(u_{k-1}^{l},u^{l}_{k}]$, we have

[TABLE]

where $\tilde{C}_{k}=\exp\left(M_{2}(u^{l}_{k}-u^{l}_{0})\right)+\ldots+\exp\left(M_{2}(u^{l}_{k}-u^{l}_{k-1})\right).$ As a consequence we obtain the global bound

[TABLE]

where $M_{2}$ and $M_{5}$ are constants independent of $k$ and they are given in Remark 2.1.

First for $k=1$, let $0=u_{0}^{l}<u\leq u^{l}_{1}$, then (8), (11), the Lipschitz condition on $\sigma$ (with constant $M_{2}$) and the fact that $\phi(z,u_{0}^{l})=\phi^{l}(z,u_{0}^{l})=z$ imply

[TABLE]

Next, we bound the term $\mathbb{I}_{0}^{l}$,

[TABLE]

From (11), the Lipschitz condition and the bound on $\sigma$, we get

[TABLE]

Therefore by (31), (32) and the Gronwall lemma we obtain

[TABLE]

Now, consider an index $k\in\{2,\ldots,l\}$. Our induction assumption is that (29) is true for $u\in(u_{k-1}^{l},u^{l}_{k}]$. We shall now propagate the induction, that is prove that the inequality is also true for its successor, $k+1$. We will thus study (29) for $u\in(u_{k}^{l},u^{l}_{k+1}]$. Following (8), (11) and our induction hypothesis we establish

[TABLE]

From Lipschitz condition on $\sigma$,

[TABLE]

where $\tilde{C}_{k}=\exp\left(M_{2}(u^{l}_{k}-u^{l}_{0})\right)+\ldots+\exp\left(M_{2}(u^{l}_{k}-u^{l}_{k-1})\right).$

Now, we analyze the term ${\mathbb{I}}^{l}_{k}$, given in equation (33). From (11), the Lipschitz condition and the bound on $\sigma$ we obtain

[TABLE]

Therefore inequalities (33) and (34) yield

[TABLE]

Thus, the Gronwall lemma allows us to establish

[TABLE]

which shows that (29) is satisfied for $k+1$.

Now, we prove that (30) is true. For all $(z,u)\in[-M,M]\times[-\|B\|_{\infty},\|B\|_{\infty}]$, there is some $k\in\{1,\ldots,l\}$ such that $u^{l}_{k-1}<u\leq u^{l}_{k}$ and by the previous calculations

[TABLE]

proving the lemma. ∎

Proof of Lemma 3.2

Proof.

As in the proof of Lemma 3.1, we will prove by induction that, for all $k\in\{1,\ldots,l\}$ and $u\in(u_{k-1}^{l},u^{l}_{k}]$, we have

[TABLE]

with $A_{l}=1+{M_{2}\|B\|_{\infty}\over l}+{1\over 2}({M_{2}\|B\|_{\infty}\over l})^{2}$. Hence,

[TABLE]

where $M_{2}$, $M_{3}$ and $M_{5}$ are constants independent on $k$ and are given in Remark 2.1.

We first assume that $k=1$. If $0=u_{0}^{l}<u\leq u^{l}_{1}$ and from equalities (11) to (LABEL:psin) we obtain that

[TABLE]

By Taylor’s theorem there exists a point $\theta\in\left(\inf\{z,z+(s-u^{l}_{0})\sigma(z)\},\sup\{z,z+(s-u^{l}_{0})\sigma(z)\}\right)$ such that

[TABLE]

Now,let us consider $k\in\{2,\ldots,l\}$. Our induction assumption is that (35) is true for $u\in(u_{k-1}^{l},u^{l}_{k}]$. We will thus study (35) for $u\in(u_{k}^{l},u^{l}_{k+1}]$. Following equations (11) to (LABEL:psin) and our induction hypothesis, we get

[TABLE]

From the Lipschitz condition on $\sigma$, and our induction hypothesis

[TABLE]

By Taylor’s theorem there exists a point

[TABLE]

such that

[TABLE]

Therefore

[TABLE]

Since $(u-u^{l}_{k})\leq{\|B\|_{\infty}\over l}$ we obtain

[TABLE]

Since $1<\left(1+A_{l}\right)^{k+1}$, then

[TABLE]

Thus (35) holds for any $k\in\{1,\ldots,l\}$.

Finally, we see that (36) is satisfied. For all $(z,u)\in[-M,M]\times[-\|B\|_{\infty},\|B\|_{\infty}]$, there is some $k\in\{1,\ldots,l\}$ such that $u^{l}_{k-1}<u\leq u^{l}_{k}$ and by (35),

[TABLE]

Thus, the proof is complete. ∎

Proof of Lemma 3.3

Proof.

We will prove by induction that, for all $k\in\{1,2,\ldots,l\}$ and $u\in(u_{k-1}^{l},u^{l}_{k}]$, we have

[TABLE]

Furthermore, for all $k\in\mathbb{N}$ we have obtained a global bound

[TABLE]

In a similar way as in previous lemmas, if $0=u_{0}^{l}<u\leq u^{l}_{1}$, then by equation (LABEL:psin) and the fact that $\Psi^{l}(z,u_{0}^{l})=z$, we have

[TABLE]

Then for $k=1$ (37) is satisfied. Now, consider that (37) is true for $k$. Then, we will prove that the inequality is true for its successor, $k+1$. For that, we will study (37) for $u\in(u_{k}^{l},u^{l}_{k+1}]$.

Following (LABEL:psin), Lipschitz condition and hypothesis on the second derivative of $\sigma$, we have

[TABLE]

Consequently, from our induction hypothesis, we get

[TABLE]

which implies that (37) is satisfied.

Now, for all $(z,u)\in[-M,M]\times[-\|B\|_{\infty},\|B\|_{\infty}]$, there is some $k\in\{1,\ldots,l\}$ such that $u^{l}_{k-1}<u\leq u^{l}_{k}$ and from (37)

[TABLE]

where the last inequality is due by the fact that for $l$ large enough ${M_{2}\|B\|_{\infty}+M_{3}M_{5}\|B\|_{\infty}/M_{2}\over 2l}<1$ and $\left(1+{2M_{2}\|B\|_{\infty}\over l}\right)^{l}<\exp(2M_{2}\|B\|_{\infty})$. Thus the proof is complete.

∎

Proof of Lemma 3.4

Proof.

We will prove by induction that, for all $k\in\{1,\ldots,l\}$ and $u\in(u_{k-1}^{l},u^{l}_{k}]$, we have

[TABLE]

As a consequence, for all $k\in\mathbb{N}$,

[TABLE]

is true.

If $0=u_{0}^{l}<u\leq u^{l}_{1}$, then by equations (10) and (11) and the fact that $\phi^{l}(z,u_{0}^{l})=z$, we have

[TABLE]

Therefore for $k=1$ (38) is satisfied. Now, let (38) be true until $k$. Therefore, it remains to prove that this inequality is true for its successor, $k+1$. For that, we choose $u\in(u_{k}^{l},u^{l}_{k+1}]$.

Using (11) and Lipschitz condition on $\sigma$ again, we can write

[TABLE]

Our induction hypothesis leads us to

[TABLE]

Therefore, (38) for any $k\in\{1,\ldots,l\}$.

Now, for all $u\in[-\|B\|_{\infty},\|B\|_{\infty}]$, there is some $k\in\{1,\ldots,l\}$ such that $u^{l}_{k-1}<u\leq u^{l}_{k}$ and, by (38),

[TABLE]

where the last inequality is due by the fact that for $l$ large enough ${M_{2}\over l}<1$ and $\left(1+{2M_{2}\|B\|_{\infty}\over l}\right)^{l}<\exp(2M_{2}\|B\|_{\infty})$. Therefore (23) is satisfied and the proof is complete. ∎

Proof of Lemma 3.5

Proof.

By equations (7) and (16), we have, for $t\in[0,T]$,

[TABLE]

with

[TABLE]

and

[TABLE]

Therefore by (6), the Lipschitz properties on $b$ and $\sigma$, and Lemmas 3.1 and 3.2 we obtain

[TABLE]

Now, by the mean value theorem, we get

[TABLE]

Hence, proceeding as in $\mathbb{K}_{1}$, we obtain

[TABLE]

Taking into account the inequalities for $\mathbb{K}_{1}$ and $\mathbb{K}_{2}$, we have

[TABLE]

[TABLE]

and

[TABLE]

Finally, the desired result is achieved by direct application of the Gronwall lemma. ∎

Proof of Lemma 3.6

Proof.

Recall that $h_{1}^{n}(z,u)={\partial g^{n}(z,u)\over\partial z}$. Then, by equations (16) to (18) we obtain that

[TABLE]

where

[TABLE]

and

[TABLE]

The specific computation of the bound of the term $h_{1}^{n}(z,u)$ is left in the Appendix (Section 6). ∎

Proof of Lemma 3.7

Proof.

Let $n\in\mathbb{N}$ be fixed. We will prove Lemma 3.7 by induction on $k$ again. That is, for every $k\in\{1,\ldots,n\}$ and $t\in(t_{k-1}^{n},t_{k}^{n}]$, we have

[TABLE]

here $0<\rho<H$. As a consequence, for all $k\in\{1,\ldots,n\}$ we obtained the global bound

[TABLE]

where $C_{6}$ and $C_{7}$ are given in (24) and (25), respectively.

First for $k=1$ and $t\in(t_{0}^{n},t_{1}^{n}]$, equations (16) to (18) imply

[TABLE]

where

[TABLE]

Equality (17) and the triangle inequality allow us to write

[TABLE]

Therefore, the Lipschitz property on $b$ and the mean value theorem yield

[TABLE]

Consequently, Lemma 3.3 lead us to

[TABLE]

Proceeding similarly as in $\mathbb{F}_{1}$,

[TABLE]

Hence, using Lemma 3.6, we can establish

[TABLE]

Now, we deal with $\mathbb{F}_{3}$. From Lemma 6.1 (Section 6),

[TABLE]

where

[TABLE]

and

[TABLE]

Note that (19) implies

[TABLE]

Hence, (40) implies

[TABLE]

Since $H<1/2$, then the previous estimations for $\mathbb{F}_{1}$, $\mathbb{F}_{2}$ and $\mathbb{F}_{3}$ give

[TABLE]

Then by the Gronwall lemma and $t\in(t_{0}^{n},t_{1}^{n}]$, we conclude

[TABLE]

Now we show that (39) is true for $k+1$ if it holds for $k$. So we choose $t\in(t_{k}^{n},t_{k+1}^{n}]$. Towards this end, we proceed as in the case $k=1$:

[TABLE]

Therefore, using the Gronwall lemma again and $t\in(t_{k}^{n},t_{k+1}^{n}]$

[TABLE]

Therefore (39) is true for any $k\leq n$.

Finally, for all $t\in[0,T]$, there exits $k\in\{1,\ldots,n\}$ such that $t^{n}_{k-1}<t\leq t^{n}_{k}$. Thus (39) implies

[TABLE]

and the proof is complete. ∎

6 Appendix

Here, we consider the following useful result for the analysis of the convergence of the scheme.

Lemma 6.1**.**

Let $\left\{u_{i}^{l}\right\}$ be a partion of the interval $[-R,R]$ given by $-R=u^{l}_{-l}<\ldots<u^{l}_{-1}<u^{l}_{0}=0<u^{l}_{1}<\ldots<u^{l}_{l}=R$ and $f:[-R,R]\rightarrow\mathbb{R}$ a $C^{2}([u_{j}^{l},u_{j+1}^{l}])$-function for each $j\in\{-l,\ldots,l-1\}$. Also let $f$ be continuous on $[-R,R]$, $C$ a constant such that

[TABLE]

and $x\in(u_{j}^{l},u^{l}_{j+1}]$ and $y\in(u_{j+k}^{l},u^{l}_{j+k+1}]$. Then,

[TABLE]

where

[TABLE]

Proof.

We will prove that (41) holds via induction on $k$.

We start our induction with $k=1$. That is, we consider two consecutive intervals. If $x\in(u_{j},u_{j+1}]$ and $y\in(u_{j+1},u_{j+2}]$. Then,

[TABLE]

It means, (41) holds for $k=1$.

It remains to prove that the inequality (41) is true for its successor, $k+1$ assuming that until $k$ is satisfied. To do so, choose $x\in[u_{j},u_{j+1}]$ and $y\in[u_{j+k+1},u_{j+k+2}]$. Then,

[TABLE]

Hence, our induction hypothesis implies

[TABLE]

Therefore, (41) is satisfied for $k+1$ and the proof is complete. ∎

Acknowledgments We would like to thank anonymous referee for useful comments and suggestions that improved the presentation of the paper. Part of this work was done while Jorge A. León was visiting CIMFAV, Chile, and Héctor Araya and Soledad Torres were visiting Cinvestav-IPN, Mexico. The authors thank both Institutions for their hospitality and economical support. Jorge A. León was partially supported by the CONACYT grant 220303. Soledad Torres was partially supported by Proyecto ECOS C15E05; Fondecyt 1171335, REDES 150038 and Mathamsud 16MATH03. Héctor Araya was partially supported by Beca CONICYT-PCHA/Doctorado Nacional/2016-21160138; Proyecto ECOS C15E05, REDES 150038 and Mathamsud 16MATH03.