Backward stochastic differential equations and their applications to stochastic control problems


Thesis Type: Post Graduate

Institution Of The Thesis: Middle East Technical University, Turkey

Approval Date: 2013

Thesis Language: English

Student: Hanife Sevda Nalbant

Co-Consultant: AZİZE HAYFAVİ, YELİZ YOLCU OKUR

Abstract:

Backward stochastic di fferential equations (BSDE) were firstly introduced by Bismut in 1973. Following decades, it has been great interest all over the world and appeared in numerious areas such as pricing and hedging claims, utility theory and optimal control theory. In 1997, El Karoui, Peng and Quenez brought together their brilliant studies in the article Backward Stochastic Di fferential Equations in Finance. They considered an adapted solution pair (Y,Z) of the following BSDE: -dY_t = f(t, Y_t,Z_t)dt - Z _t^* dW_t with the terminal value Y_T =\xi . Here Z^* corresponds to the transpose of the n xn matrix Z, f is called the generator and \xi is the terminal condition. In this thesis, we study some chapter of this paper in detail. We prove the fundamental theorems of backward stochastic diff erential equations and associate them with stochastic control problems. After we prove the existence of unique solution using a Priori estimates under some restrictions, we show how to choose the optimal stochastic control that achieves the best utility or the least cost. At the end of the thesis, we o ffer an optimal choice for the solution of the BSDE in the cases of the standard generator f is concave or convex. An application for the model with consumption and an application for hedging claims with higher interest rate for borrowing are provided.