Study On Optimal Investment And Optimal Stopping Problems Under Stochastic Volatility Models

This thesis studies the optimal investment and optimal stopping problems under the stochastic volatility models,including the optimal investment problem for general utilities under the local stochastic volatility models,the high-dimensional optimal pairs trading problem for general utilities under the local stochastic volatility models,and the optimal investment stopping problem for power utility under the stochastic volatility models.First,we study the optimal investment problem for general utilities under the local stochastic volatility models.The expected utility maximization problem can be transformed into a fully nonlinear partial differential equation problem,whose coefficients are related to state variable and terminal condition changes with the utility function.We propose a new transformation to rewrite the original problem into an equivalent nonlinear equation problem with more complex equation and fixed terminal condition.Using model coefficient expansion techniques for the equivalent problem,we derive explicit approximations for both the value function and the optimal investment strategy under general utility functions.We verify the reliability of our approximations theoretically and numerically,and give numerical examples with a CRRA utility,two IRRA utilities and a DRRA utility,respectively.Then,we study the high-dimensional optimal pairs trading problem for general utilities under the local stochastic volatility models,that is,the optimal investment problem of co-integrated portfolio.The expected utility maximization problem can be transformed into a fully nonlinear partial differential equation problem,whose coefficients are related to both state variable and time.Combining the transformation and coefficient expansion techniques,we derive explicit approximations for both the value function and the optimal investment strategy based on a clever guess for high-order terms.For a special case,we give a rigorous accuracy error bound theoretically for the approximation of the value function.We verify the reliability of our approximations numerically,and verify the effectiveness of our pairs trading model by numerical simulation.Finally,we study the optimal investment stopping problem for power utility under the stochastic volatility models.The expected utility maximization problem is a mixed optimal control and stopping problem,which can be transformed into a variational inequality problem that involves a free boundary problem of a two-dimensional nonlinear partial differential equation.We apply a transformation to covert the original problem into an equivalent variational inequality problem of a one-dimensional linear partial differential equation.On this basis,we apply the standard partial differential equation method to prove the existence and uniqueness of the solution for the equivalent problem,and analyze the existence and monotonicity of the free boundary.Numerical examples of free boundary are provided to support our theoretical results.