Stochastic control of hereditary systems and applications / Mou-Hsiung Chang

This book deals with optimal control of hereditary stochastic differential equations. These are stochastic differential equations whose coefficients depend on a bounded patch of the past trajectory (bounded memory), or an infinite past, but with appropriately decreasing dependence on distant past (infinite fading memory). The theory of such equations is developed starting from basics of stochastic calculus. A classical discounted cost control problem is analyzed in terms of its infinite dimensional Hamilton-Jacobi-Bellman equation, for which a viscosity solution and a verification theorem are established. For the bounded memory case, an optimal stopping problem is analyzed in terms of the associated infinite dimensional variational inequalities, for which a viscosity solution and a verification theorem are established. Various approximation schemes based on discretization and Markov chain approximations are presented. The results are applied to option pricing and portfolio optimization. (Zentralblatt)