Mean-reverting θ Process With Time Delay And The Convergence Of Its Numerical Solutions

The mean-revertingθprocess is used as a model for interest rates and volatility as well as other financial quantities. Some statistical data of stock price and phenomenon of financial market indicate that the financial quantities depend on their past behavior. Stochastic differential delay equation (SDDE) can be used to describe the dynamics. In this thesis, we study a class of mean-revertingθprocess with time delay which can be expressed as a solution of the following SDDE: dS(t)=λ(μ-S(t))dt+σSθ(t)Sγ(t-τ)dw(t). where the changing of S(t) is dependent on its past behavior as well as itself.It is a natural requirement of the existence and non-negativity of the solution because the mean-revertingθprocess with time delay is frequently used as a model for financial quantities which are past level dependent. Since the diffusion coefficient of mean-reverting withθprocess with time delay is nonlinear and non-Lipschitz, some standard results of SDDE cannot be appealed. For 1/2≤θ< 1, by using of the technique of Ito formula, some important inequalities, this paper proves the existence and uniqueness as well as nonnegativity of the solution.Since the corresponding stochastic delay differential equation has no explicit so-lution, it is very important to study numerical methods for approximations. Only if an approximation converges to the solution of equation, we can study the property of solution through studying that of the approximation. Therefore, the convergence is the most important issue on the numerical solution of SDDE. For 1/2≤θ< 1, by using of the technique of stopping time and Burkholder-Davis-Gundy inequality as well as the theory of stochastic differential delay equation, this paper proves the the strong con-vergence of Euler-Maruyama approximation solution and Caratheodory approximation in sense of p-th moment(p≥2).