Backward Euler method for stochastic differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

Abstract

We study the traditional backward Euler method for stochastic differential equation driven by fractional Brownian motion whose drift coefficient satisfies the one-sided Lipschitz condition. The backward Euler scheme is proved to be of order one and this rate is optimal by showing the asymptotic error distribution result. Numerical experiments are performed to validate our claims about the optimality of the rate of convergence.

7 Appendix

In this section, we consider the differential equation

$$\begin{aligned} x_{t} = x_{t_{0}} +\int _{t_{0}}^{t} b(s, x_{s}) {\text{ d }}s\,, \qquad t \ge t_0\,, \end{aligned}$$

(7.1)

where we assume that

• (i) The function $b: [t_0,\infty )\times {\mathbb {R}}^{m}\rightarrow {\mathbb {R}}^{m}$ is one-sided Lipschitz: there is a constant $\kappa >0$ such that

  $$\begin{aligned} \langle x-y,~ b(s, x) - b(s, y) \rangle \le \kappa |x-y|^{2}\,, \qquad \forall x, y \in {\mathbb {R}}^{m} \text{ and } s\ge t_0\,. \end{aligned}$$

• (ii) Both b and $\partial b$ are continuous functions in $[t_0,\infty )\times {\mathbb {R}}^{m}$. Here $\partial b$ denotes by the Jacobian $\frac{\partial b(s, x)}{\partial x}$ of b as we defined in Eq. (3.1).

Example 7.1

It is easy to show that the following two equations are examples of (7.1) and the conditions (i)–(ii) are satisfied.

• (1) The equation

  $$\begin{aligned} y_{t}' = b(g_t+y_{t} ) \,, \end{aligned}$$

  where b is one-sided Lipschitz and g is a continuous function. In fact, in this case $b(s, x)=b(g_s+x)$ and we have

  $$\begin{aligned} \langle x-y, b(s, x) - b(s, y) \rangle =&{} \langle (g_s+ x)-(g_s+ y), b(g_s+ x) - b(g_s+ y) \rangle \\\le&{} \kappa |x-y|^{2}\,, \qquad \forall x, y \in {\mathbb {R}}^{m} \text{ and } s\ge t_0\,. \end{aligned}$$

• (2) The linear equation

  $$\begin{aligned} x_{t}' = U_{t} x_{t} \,, \end{aligned}$$

  where the eigenvalues of $U_{t}$ are uniformly bounded for all $t\ge 0$.

The following existence and uniqueness results about Eq. (7.1) are mostly well-known. We did not find the exact explicit result for (7.1) in the literature and so we include a proof for the sake of completeness.

Proposition 7.1

There exists a unique solution x to Eq. (7.1), and the solution satisfies the relation

$$\begin{aligned} |x_{t} |^{2} \le \left( |x_{t_{0}}|^{2}+ \int _{t_{0}}^{t} |b(s, 0)|^{2}\textrm{d} s \right) \cdot e^{(2\kappa +1)(t-t_{0})}\,, \qquad \ t\in [t_0,\infty )\,. \end{aligned}$$

Proof

We start by defining

$$\begin{aligned} \tau = \sup \big \{ t: \text {there exists a solution to equation (7.1) on } [t_{0}, t] \big \}. \end{aligned}$$

We first note that by Peano’s theorem it is easy to see that $\tau >t_{0}$. On the other hand, the one-sided Lipschitz condition implies that for any $t<\tau$ there exists a unique solution on the interval $[t_{0}, t]$ (see e.g. [19, Lemma 12.1]).

In the following we show that $\tau =\infty$ by contradiction, which then concludes the existence of the solution on $[t_{0}, \infty )$.

We calculate

$$\begin{aligned} |x_{t}|^{2} =&\, |x_{t_{0}}|^{2} +2 \int _{t_{0}}^{t} \langle x_{s}, b(s, x_{s}) \rangle {\text {d}}s \\ =&\, |x_{t_{0}}|^{2} +2 \int _{t_{0}}^{t} \langle x_{s}, b(s, x_{s}) - b(s, 0) \rangle {\text {d}}s + 2 \int _{t_{0}}^{t} \langle x_{s}, b(s, 0) \rangle {\text {d}}s \\ \le&\, |x_{t_{0}}|^{2} + 2 \kappa \int _{t_{0}}^{t} |x_{s}|^{2}{\text {d}}s + \int _{t_{0}}^{t} |x_{s}|^{2}{\text {d}}s + \int _{t_{0} }^{t} |b(s, 0)|^{2}{\text {d}}s \,. \end{aligned}$$

Applying Gronwall’s inequality yields

$$\begin{aligned} |x_{t}|^{2} \le&\left( |x_{t_{0}}|^{2}+ \int _{t_{0}}^{t} |b(s, 0)|^{2}{\text {d}}s \right) e^{(2\kappa +1)(t-t_{0})} \\ \le&\left( |x_{t_{0}}|^{2}+ \int _{t_{0}}^{\tau } |b(s, 0)|^{2}{\text {d}}s \right) e^{(2\kappa +1)(\tau -t_{0})}\,. \end{aligned}$$

This implies that $x_{t}$ is bounded on $[t_{0}, \tau )$. Since $x_{t}' = b(t, x_{t})$ we obtain that x is uniformly continuous on the interval $[t_{0}, \tau )$. In particular, the limit $\lim _{t\rightarrow \tau -}x_{t}$ exists. This shows that x is a solution to Eq. (7.1) on $[t_{0}, \tau ]$.

Now consider the equation $x_{t}' = b(t, x_{t})$ with initial value $x_{\tau }$. Peano’s theorem implies that the equation has a solution on the interval $[\tau , \tau +\varepsilon ]$ for some $\varepsilon >0$. Combining the two functions $(x_{t}, t\in [t_{0}, \tau ])$ and $(x_{t}, t\in [\tau ,\tau +\varepsilon ])$, we obtain a solution to Eq. (7.1) on the interval $[t_{0}, \tau +\varepsilon ]$. This contradicts the definition of $\tau$. We conclude that $\tau = \infty$. The proof is complete. $\square$

Our next result addresses the Malliavin differentiability of Eq. (1.1). The proof is an application of [20, Lemma 1.2.3]. Recall that the Malliavin derivative is defined in Sect. 2.1.

Lemma 7.1

Let $X_t$ be the solution of Eq. (1.1). For any $t\ge 0$, the Malliavin derivative $DX_t$ of $X_{t}$ exists in ${\mathbb {D}}^{1,2}$.

Proof

Let $Y_t^\pi$ be defined by (4.5). Then for any fixed $t\in [0,T]$, we can show that $Y_t^\pi$ converges to $X_t$ in $L^2$. This can be done by following the method of Sect. 4 without the use of Malliavin calculus. Indeed, bounding the right-hand side of (4.46) we get that ${\mathbb {E}}|R^t_j|^2\le C|\pi |^{2\alpha +2}$ for $0<\alpha <H$. Applying this bound and also the bound $|A_j^t|\le C$ in (4.43) to the expression of $Z_t=X_t-Y^{\pi }_t$ in (4.42), we obtain that

$$\begin{aligned} Y_t^{\pi } \rightarrow X_t \qquad \hbox {in} L^2 \hbox {as} |\pi |\rightarrow 0\,. \end{aligned}$$

(7.2)

From the first inequality of (4.15) it follows that

$$\begin{aligned} \sup _{|\pi |} {\mathbb {E}}\left[ \Vert DY_t^\pi \Vert _{\mathscr {H}}^2\right] <\infty \,. \end{aligned}$$

(7.3)

Applying [20, Lemma 1.2.3] with $F=X_t$ and $F_n=Y^{\pi }_t$ and taking into account (7.2) and (7.3), we conclude that $DX_t$ exists in ${\mathbb {D}}^{1,2}$. $\square$