Numerical Computation For HJB Partial Differential Equations

E. Pardoux and S. Peng, in 1990 firstly proved the uniqueness and existence of the solution to backward stochastic differential equation [9], that is, there exists a unique pair of F_t-adapted process (Y_t,Z_t)∈L²(0,T;R)×H²(0,T;R^d), satisfied the following equationwhere g satisfied (i) g : [0,T]×R×R, and g(·,0,0)∈L², (ii) the Lipschitz condition: (?)(y₁,z₁),(y₂,z₂)∈R^d, there exists a constant C＞0, satisfyingandξ∈L_T². Using the backward stochastic differential equations, Professor Shige Peng in 1997 found that the solutions of the BSDE equipped very good properties when generatorssatisfy special condition: g(t,y,0)=0, for (t,y)∈[0,T]×R, and then he defined a kind of nonlinear mathematical expectation [14]: g-expectation, which extend the definitionof the classical expectation.However, g-expectation is a quasi-linear mathematical expectation, the fully nonlinearsituation cannot be covered. Recently, Professor Shige Peng proposed more general F_t consistent nonlinear expectations and nonlinear Markovian chains in [15], later he gave the definition and properties of G-expectation. G-expectation satisfies the propertiesof monotonicity, preserving of constants, sub-additivity, positive homogeneity and constant translatability. And so G-expectation is equivalent to the notion of coherent risk measure: p(X) = E[-X], we refer the readers [1-3] for details. G-expectation with the related conditional G-expectation makes a dynamic risk measure. We know that the G-expectation is generated by a specific fully nonlinear heat equation, which is a kind of nonlinear HJB equation. But we cannot get explicit solutions of such equations generally, and in most cases, we can only rely on numerical methods for solving them.In this thesis we discretize the corresponding HJB equation of the G-expectation and propose four numerical schemes for the HJB equation. Then we solve the HJB equation numerically and analyze the error of the numerical solution.The thesis is organized as follows:Chapter 1 presents the background and applications of SDE, BSDE, g-expectation and G-expectation.In chapter 2 we mainly discuss the relationship between the HJB equation and optimalcontrol, applications of HJB equation in finance are also included.In chapter 3 we propose the discretized scheme of HJB equation of G-expectation.In chapter 4 we use some numerical experiments to demonstrate the convergence of our schemes.