Stochastic System Optimization Based On Sensitivity Analysis And Its Application In Financial Engineering

The thesis considers the issues of stochastic learning and optimization,and its application in financial engineering.This work studies the constrained linear quadratic optimal control and game problems,and optimization of risk contagion among financial institutions,by utilizing the sensitivity analysis based approach.The sensitivity analysis based approach is an significant approach to learning and optimization.The central piece of this approach is the performance difference formula.Based on this,the optimality condition and corresponding optimization algorithms such as policy iteration,can be derived.Compared with traditional approaches to optimal control,this approach has some advantages of concise and intuitive derivation,and has obtained many new results for complex problems.In this thesis,the sensitivity analysis based approach provides a unified formulation framework and powerful analysis tool for optimization problems in financial engineering,such as financial portfolio and risk management.In the aspect of financial portfolio,there always exist some market constraints in financial markets.We can transform this kind of portfolio problems into constrained linear-quadratic(LQ)optimal control and non-cooperative game problem formulations for the scalar-state stochastic system.The constrained LQ optimal control problem has various applications,especially in the financial risk management.The linear constraint on the control variable considered in our model destroys the elegant structure of the classic LQ formulation.Therefore,this work investigates optimal analytical control policy for the constrained linear quadratic optimal control and game problems.The problem is first formulated as a Markov decision problem(MDP).Based on the sensitivity analysis based approach and the state separation property induced from its structure,this work successfully derives the optimality conditions,i.e.,the extended Riccati equations,and the analytical control policy,on the basis of the performance difference formula.We reveal that the optimal control policy is a piecewise affine function of the state and can be computed off-line efficiently by solving extended Riccati equations.The result can be extended from finite horizon into infinite horizon.In the infinite horizon problem,this work studies the conditional probability parameters related to control policies.By analyzing these parameters,the relationship between the constrained stochastic LQ problem and the constrained deterministic LQ problem is studied.The sensitivity analysis based approach develops a policy iteration based algorithm,and demonstrates the convergence and efficiency in the simulation example.In the aspect of risk management,financial institutions are interconnected by holding debt claims against each other.A default bank may cause its creditors to default,and the risk may be further propagated to up-stream institutes(risk contagion).Such interconnection is a key contributing factor to the past worldwide financial crisis.We show that a good mechanism of default liquidation may improve the total wealth of the financial system and therefore may curb the risk contagion.We formulate this problem as a nonlinear optimization problem with constraints and propose an optimal liquidation policy to minimize the system's loss.We show that the problem resembles an MDP and therefore we can apply the sensitivity analysis based approach to solve this problem.Higher order directional derivatives and some optimality properties are obtained.Furthermore,we derive an iterative algorithm which combines both the policy iteration and the gradient based approach.Our work provides a new direction in curbing the risk contagion in financial networks;and it illustrates the advantages of the sensitivity analysis based approach,originated in the field of discrete event dynamic system,in nonlinear optimization problems.