Weak regularity of BSDEs for semilinear PDEs: the key ingredient for Euler scheme's type approximations
by
Bruno Bouchard
PMA, University Paris 6

We present recent results on the path regularity of BSDEs of the form

Yt = g(XT) + ó õ T t  f(Xs, Ys, Zs) ds - ó õ T t  Zs dWs      ,   t ∈ [0, T]  ,

where

Xt = X0 + ó õ t 0  b(Xs) ds + ó õ t 0  s(Xs) dWs   .

When the coefficients are Lipschitz continuous, it has been shown that the quantities

ℜ(Y)S2: = max i < n  E é ë sup t ∈ [ti, ti+1]  |Yt-Yti|2 ù û and ℜ(Z)H2:= å i < n  E é ë ó õ ti+1 ti  ∥Zt- ^ Z   ti  ∥2dt ù û

where t_i=iT/n, n ∈ N^*, and

^ Z   ti  := n T E é ë ó õ ti+1 ti  Zs ds | Fti ù û    for   i < n  ,

satisfy

ℜ(Y)S2+ℜ(Z)H2 ≤ O(n-1)  .

This can be seen as a weak regularity result on the solution u of the associated PDE and its gradient. We then introduce an Euler scheme type appoximation of (Y, Z) of the form

- Y   ti  := E é ë - Y   ti+1   | Fti ù û +   T n   f( - X   ti  , - Y   ti  , - Z   ti  ) - Z   ti  := n T E é ë - Y   ti+1  (Wti+1-Wti) | Fti ù û     ,   i < n  ,

with the terminal condition [`Y]<sub>T</sub>=g( [`X]_T), and where [`X] is the standard Euler scheme of X. We show that the approximation error is of order of ℜ(Y)_S²+ℜ(Z)_H². Various extensions will be considered.

Date received: July 3, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavj-81.