Probability Seminar 2006

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.