Inria / Raweb 2004
Team: Mathfi

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Keywords: BSDE.

## Backward Stochastic Differential equations

Participants: M.C. Quenez, M. Kobylanski (University of Marne la Vallée).

Backward Stochastic Differential equations (DSDE) are related to the stochastic maximum principle for stochastic control problems. They also provide the prices of contingent claims in complete and incomplete markets.

The solution of a BSDE is a pair of adapted processes (Y, Z) which satisfy (1)

where f is the driver and is the terminal condition .

M.C. Quenez, N.El Karoui and S.Peng have established various properties of BSDEs, in particular the links with stochastic control (cf. , ). There are numerous applications in finance. For example in the case of a complete market, the price of a contingeant claim B satisfies a BSDE with a linear driver and a terminal condition equal to B. This is a dynamic way of pricing which provides the price of B at all time and not only at 0. In incomplete markets, the price process as defined by Föllmer and Schweizer (1990) in  corresponds to the solution of a linear BSDE. The selling price process can be approximated by penalised prices which satisfy nonlinear BSDEs. Moreover nonlinear BSDEs appear in the case of big investors whose strategies affect market prices. Another application in finance concerns recursive utilities as introduced by Duffie and Epstein (1992) . Such a utility function associated with a consumption rate (ct, 0 t T) corresponds to the solution of a BSDE with terminal condition which can be a function of the terminal wealth, and a driver f(t, ct, y) depending on the consumption ct. The standard utility problem corresponds to a linear driver f of the type f(t, c, y) = u(c)- ty, where u is a deterministic, non decreasing, concave function and is the discount factor.

In the case of reflected BSDEs, introduced in , the solution Y is forced to remain above some obstacle process. It satisfies (2)

where K is a nondecreasing process.

For example the price of an American option satisfies a reflected BSDE where the obstacle is the payoff. The optimal stopping time is the first time when the prices reaches the payoff. (  et ).