Problems of Mathematical Finance by Stochastic Control Methods

Abstract

Abstract. The purpose of this paper is to present main ideas of mathematics of finance using the stochastic control methods. There is an interplay between stochastic control and mathematics of finance. On the one hand stochastic control is a powerful tool to study financial problems. On the other hand financial applications have stimulated development in several research subareas of stochastic control in the last two decades. We start with pricing of financial derivatives and modeling of asset prices, studying the conditions for the absence of arbitrage. Then we consider pricing of defaultable contingent claims. Investments in bonds lead us to the term structure modeling problems. Special attention is devoted to historical static portfolio analysis called Markowitz theory. We also briefly sketch dynamic portfolio problems using viscosity solutions to HamiltonJacobi-Bellman equation, martingale-convex analysis method or stochastic maximum principle together with backward stochastic differential equation. Finally, long time portfolio analysis for both risk neutral and risk sensitive functionals is introduced. Pricing of Financial DerivativesOne of the fundamental problems of mathematics of finance is pricing of the derivative securities (shortly derivatives) i.e. securities the value of which depends on the basic securities such as stocks or bonds. In this section we restrict ourselves to stocks, although similar problems (unfortunately much harder) concern also derivatives of bonds. Modeling of Asset PricesWe start with modeling of asset prices (stocks). We assume that we are given d assets on the market and denote the price of the i-th asset at time t by S i (t). We shall consider in parallel way two approaches: in discrete and in continuous time. We assume a given probability space (Ω , F , (F t ), P). In the case of discrete time the asset prices satisfy the relation S i (t + 1) S i (t) = ζ i (t, z(t + 1), ξ (t + 1)),Research supported by MNiSzW grant no. 1 P03A 013 28.