## Probability Seminar 2006

- Date: 05/03/2006

Matthias Mueller

University of British Columbia

The first part of the talk gives a survey about BSDE. Linear BSDE have

been introduced by Bismuth (1973) in Control Theory. The existence of

solutions for Lipschitz BSDE was proven by Peng, and for quadratic BSDE

by Lepeltier/San Martin and by Kobylanski. A nonlinear Feynman-Kac

formula allows the representation of semilinear PDE by BSDE and vise

versa. Furthermore, the Malliavin derivatives of BSDE can be

represented as the solution of a linear BSDE. We apply BSDE to an

economical problem. The goal is the pricing of a bond that depends on a

non-financial risk factor, e.g. weather. An equilibrium price can be

calculated using a quadratic BSDE. Price means here a probability

measure Q equivalent to the 'real world' measure P. Random payouts are

then priced by the expectation under Q. Prices at intermediate times

are taken as conditional expectations under Q. We show that or bond

completes the market. In mathematical terms: the process gained by the

successive conditioned Q-expectations of the random variable modelling

the payout of the bond have the representation property: every random

variable in L1(Q) can be written as stochastic integral with respect to

the price process.