## Team : mathfi

## Section: Scientific Foundations

**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

where f is the driver and is the terminal condition [91].

M.C. Quenez, N.El Karoui and S.Peng have established various
properties of BSDEs, in particular the links with stochastic control
(cf. [86], [76]). 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 [63] 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) [59]. Such a utility function
associated with a consumption rate (c_{t}, 0tT) corresponds
to the solution of a BSDE with terminal condition
which can be a function of the terminal wealth, and a
driver f(t, c_{t}, y) depending on the consumption c_{t}. The standard utility
problem corresponds to a linear driver f of the type f(t, c, y) = u(c)-_{t}y, where u is a deterministic, non decreasing, concave
function and is the discount factor.

In the case of reflected BSDEs, introduced in [80], the solution Y is forced to remain above some obstacle process. It satisfies

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. ( [81] et [84]).