Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter $H>1/2$). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motion and the corresponding underlying standard Brownian motion. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need to develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motion while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.Comment: 44 page

Anticipated BSDEs driven by fractional Brownian motion with time-delayed generator

This paper discusses a new type of anticipated backward stochastic differential equation with a time-delayed generator (DABSDEs, for short) driven by fractional Brownian motion, also known as fractional BSDEs, with Hurst parameter $H\in(1/2,1)$, which extends the results of the anticipated backward stochastic differential equation to the case of the drive is fractional Brownian motion instead of a standard Brownian motion and in which the generator considers not only the present and future times but also the past time. By using the fixed point theorem, we will demonstrate the existence and uniqueness of the solutions to these equations. Moreover, we shall establish a comparison theorem for the solutions

Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes

In this dissertation, I investigate two types of stochastic differential equations driven by fractional Brownian motion and backward stochastic differential equations. Malliavin calculus is a powerful tool in developing the main results in this dissertation. This dissertation is organized as follows. In Chapter 1, I introduce some notations and preliminaries on Malliavin Calculus for both Brownian motion and fractional Brownian motion. In Chapter 2, I study backward stochastic differential equations with general terminal value and general random generator. In particular, the terminal value has not necessary to be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation neither. Motivated from applications to numerical simulations, first the Lp-H¨older continuity of the solution is obtained. Then, several numerical approximation schemes for backward stochastic differential equations are proposed and the rate of convergence of the schemes is established based on the obtained Lp-H¨older continuity results. Chapter 3 is concerned with a singular stochastic differential equation driven by an additive one-dimensional fractional Brownian motion with Hurst parameter H > 1 2 . Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time t > 0. In Chapter 4, I am interested in some approximation solutions of a type of stochastic differential equations driven by multi-dimensional fractional Brownian motion BH with Hurst parameter H > 1 2 . In order to obtain an optimal rate of convergence, some techniques are developed in the deterministic case. Some work in progress is contained in this chapter. The results obtained in Chapter 2 are accepted by the Annals of Applied Probability, and the material contained in Chapter 3 has been published in Statistics and Probability Letters 78 (2008) 2075-2085

Maximum Principle for Control System driven by Mixed Fractional Brownian Motion

In this paper, we study the optimal control problem for system driven by mixed fractional Brownian motion (including a fractional Brownian motion with Hurst parameter $H>1/2$ and the underlying standard Brownian motion). By using Malliavin calculus, we obtain the necessary condition the optimal control should satisfy. Through martingale representation theorem and the properties of the transforms operator, we give out the adjoint backward stochastic differential equation in a natural way. As a straightforward consequence, the maximum principle for control system driven by fractional Brownian motion and an independent Brownian motion is also deduced, which is different to the underlying case. As an application, the linear quadratic case is investigated to illustrate the main results

Fractional backward stochastic differential euqations and fractional backward variational inequalities

In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as: [{[c]{l}% -dY(t)= f(t,\eta(t),Y(t),Z(t))dt-Z(t)\delta B^{H}(t), \quad t\in[0,T], Y(T)=\xi,.] where $\eta$ is a stochastic process given by $\eta(t)=\eta(0) +\int_{0}^{t}\sigma(s) \delta B^{H}(s)$, $t\in[0,T]$, and $B^{H}$ is a fractional Brownian motion with Hurst parameter greater than 1/2. The stochastic integral used in above equation is the divergence-type integral. Based on Hu and Peng's paper, \textit{BDSEs driven by fBm}, SIAM J Control Optim. (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equation [{[c]{l} -dY(t)+\partial\varphi(Y(t))dt\ni f(t,\eta(t),Y(t),Z(t))dt-Z(t)\delta B^{H}(t),\quad t\in[0,T], Y(T)=\xi,.] where $\partial\varphi$ is a multivalued operator of subdifferential type associated with the convex function $\varphi$.Comment: 41 page