Backward stochastic differential equations with quadratic growth and their applications.
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文摘
This dissertation discusses backward stochastic differential equations (BSDE's in short) with quadratic growth, their applications to stochastic optimal control problems, and the numerical schemes for the high dimensional cases mainly focusing on linear quadratic regulator (LQR in short) problems.;First, we consider the stochastic LQR problem where the state dynamics doesn't depend on the control and the coefficients are deterministic. We characterize a probabilistic representation of the Hamilton-Jacobi-Bellman equation via a decoupled forward-backward SDE (FBSDE) system and discuss the well-posedness of the solutions. By means of an exponential transformation, we obtain the existence-uniqueness of the solutions to this FBSDE system, extend the findings to higher dimensions, give results for the PDE counterparts and provide some comparison theorems. We also consider more general LQR problems which are related to the Riccati BSDE's.;Next, we discuss the numerical discretization of the decoupled FBSDE's arising from LQR problem relying on the Markov structure of the model and show that this approximation yields a deterministic sequence that converges strongly to the value function of the LQR problem. Then we apply a regression approximation which is based on Malliavin calculus to standard LQR problems in order estimate the conditional expectations involved. We also provide some regularity results and a three dimensional example.;Finally, we include two applications: An application to economics which is a terminal value modification of an infinite time impulse control of a central bank; and a mean-variance portfolio selection problem which is solved by both LQR approach that is adapted from literature with slight changes and martingale duality method, in a continuous time complete market setting. The dissertation ends with comparing this methods and mentioning other major applications.