Optimal Control And Large Deviations In Matrix Valued Factor Models

This paper mainly introduces risk sensitive control problem of portfolio under matrix valued factor models.On the basis,the large deviation control problem of stochastic optimal control is discussed.We calculate down-side risk probability minimization problem in detail,and we explain it is difficult to solve up-side chance probability maximization problem under the matrix valued factor models.By the dual method and the large deviation control theory of the linear Gaussian model which has been studied by previous researchers,we can reasonably conclude that the large deviation control of the matrix value factor model is related to the risk sensitive control problem of that.Besides,based on the risk sensitive control problem of the matrix value factor model,I calculate the large deviation control problem under the Wishart factor model.The paper first introduces one of matrix valued factor model,the Wishart autoregressive factor model,and the optimal control problem in a finite time horizon in the model.Under the linear Gaussian model,the effective solution to the finite horizon risk sensitive control problem is establishing the HJB equation.While under a Wishart autoregressive factor model,we can solve this problem by the corelation between the HJB equation and the Riccati ordinary differential equation.The factor model is extended to the generalized Wishart process,and the risk sensitive control problem in the finite horizon is introduced in this model.Besides we compare it with the above results.Similarly,under the linear Gaussian model,the long-term behavior can correspond to the HJB equation in the ergodic case,and the optimal control is solved by solving the quadratic equation.The method we introduce here is to solve the ergodic risk sensitive control problem by establishing an algebraic Riccati equation and traversing the HJB equation.The large deviation control problem is an important part of stochastic optimal control.Thanks to duality method,the dual problem of the large deviation control problem is an ergodic risk sensitive control problem.While the traditional method of seeking optimal control is known as to maximize the utility function.In the case,we need to assume the investor's risk preference which is corresponding to utility function but subjective and difficult to determine in the market.Therefore,it is meaningful to study the large deviation control problem in the matrix valued stochastic control model.