Risk-sensitive stochastic control and its singular limit

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(a) It is important to know the conditions for no breakdown in risk-sensitive stochastic control problems since the value function has not always a finite value. In this research we have obtained the condition as that of the size of risk-sensitive parameter in the case of a finite time horizon and also shown the solvability of the corresponding Bellman equation under the condition. It is applicable for the case that the risk-sensitive parameter is large and has great meaning in application. In the case of infinite time horizon we have shown existence and uniqueness of the corresponding ergodic type Bellman equation under a similar condition by noticing the relationships between the problems and the eigenvalue problems for Schrodinger operator. Besides, we have derived a first order partial differential equation relating to game theoretical approach to nonlinear H_∞ control as its singular limit.(b) We have considered portfolio optimization problem for a factor model as application to m … More athematical finance of risk-sensitive stochastic control and got the results giving the explicit representation to optimal portfolio for the problem in the case of partial information. In the case of infinite time horizon we obtained the condition under which the solution of corresponding ergodic type Bellman equation defines the optimal portfolio and constructed it by the solution. We have also found that the solution does not always give the optimal portfolio without any condition.(c) We have shown existence of the spectral gap of Schrodinger opeartor by using log Sobolev inequality and gave the estimate.(d) We have shown that the large deviation principle holds for additive functional of Brownian motion corresponding to measures in Kato class. As its application we obtained necessary and sufficient condition under which additive functionals converges exponentially fast.(e) We have shown Trotter product formula with respect to L_p and trace norm and their error estimates for the Schrodinger operator with a potential bounded below.(f) We have shown that the sequence of symmetric statistics defined by Weyl transformation on infinite dimensional torus converges to the limit represented by multiple Wiener integral under the probability measures. Less