Pacific Institute for the Mathematical Sciences - PIMS

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.