Numerical solutions of rough differential equations and stochastic differential equations

Abstract

In this dissertation, we investigate time-discrete numerical approximation schemes for rough differential equations and stochastic differential equations (SDE) driven by fractional Brownian motions (fBm). The dissertation is organized as follows. In Chapter 1, we introduce the basic settings and define time-discrete numerical approximation schemes. In Chapter 2, we consider the Euler scheme for SDEs driven by fBms. For a SDE driven by a fBm with Hurst parameter $H> \frac12$ it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H \rightarrow \frac12$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^o SDEs for $H=\frac12$, the convergence rate of the naive Euler scheme deteriorates for $H \rightarrow \frac12$. The new (modified Euler) approximation scheme we are introducing in this chapter is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac12$ and it has the rate of convergence $\gamma_n^{-1}$, where $ \gamma_n=n^{ 2H-\frac12}$ when $H \frac12$ it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H \rightarrow \frac12$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^o SDEs for $H=\frac12$, the convergence rate of the naive Euler scheme deteriorates for $H \rightarrow \frac12$. The new (modified Euler) approximation scheme we are introducing in this chapter is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac12$ and it has the rate of convergence $\gamma_n^{-1}$, where $ \gamma_n=n^{ 2H-\frac12}$ when $H \frac34$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t, 0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n, 0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $ \gamma_n (X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in ( \frac12, \frac34]$. In the case $H \frac 34$, we show the $L^p$ convergence of $n(X^n_t-X_t)$ and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme. In Chapter 3, we consider the Crank-Nicolson method for a SDE driven by a $m$-dimensional fBm. We consider the Crank-Nicolson method in three cases: (i) $m1$; (ii) $m=1$ and and the drift term is equal to non-zero; and (iii) $m=1$ and the drift term is equal to zero. We will show that the convergence rate of the Crank-Nicolson method is $n^{ 1/2-2H}$, $n^{-1/2-H}$ and $n^{-2H}$, respectively, in these three cases, and these convergence rates are exact in the sense that the error process for the Crank-Nicolson method converges to the solution of a linear SDE. Our main tools are the fractional calculus and the fourth moment theorem. In Chapter 4, we study two variations of the time-discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes some terms of an Taylor scheme in its recursive computation so as to reduce the computation time. The other one is to add some deterministic terms to an incomplete Taylor scheme to improve the mean rate of convergence. Almost sure rate of convergence and $L_p$-rate of convergence are obtained for the incomplete Taylor schemes. Almost sure rate is expressed in terms of the H\"older exponents of the driving signals and the $L_p$-rate is expressed by the Hurst parameters. Our explicit expressions of the convergence rates allow us to compare different incomplete Taylor schemes, and then help us construct the best incomplete schemes, depending on that one needs the almost sure convergence or one needs $L_p$-convergence. As in the smooth case, general Taylor schemes are always complicated to deal with. The incomplete Taylor scheme is even more sophisticated to analyze. A new feature of our approach is the explicit expression of the error functions which will be easier to study. Estimates for multiple integrals and formulas for the iterated vector fields are obtained to analyze the error functions and then to obtain the rates of convergence.