Numerical methods for backward stochastic differential equations.

*(English)*Zbl 0898.90031
Rogers, L. C. G. (ed.) et al., Numerical methods in finance. Session at the Isaac Newton Institute, Cambridge, GB, 1995. Cambridge: Cambridge Univ. Press. 232-244 (1997).

Backward stochastic differential equations (BSDEs) were introduced by Pardoux and Peng (1990) to give a probabilistic representation for the solutions of certain nonlinear partial differential equations, thus generalizing the Feynman-Kac formula.

This sort of equation has also found many applications in finance, notably in contingent claim valuation when there are constraints on the hedging portfolios (see El Karoui and Quenez 1995, El Karoui, Peng and Quenez 1994, Cvitanic and Karatzas 1992) and in the definition of stochastic differential utility (see Duffi and Epstein 1992, El Karoui, Peng and Quenez 1994). A financial application of forward-backward SDEs can be found in Duffie, Ma and Yong (1991).

However little research has yet been performed on numerical methods for BSDEs. Here we give a review of three different contributions in that field.

In Section 2, we present a random time discretization scheme introduced by V. Bally to approximate BSDEs. The advantage of Bally’s scheme is that one can get a convergence result with virtually no other regularity assumption than the ones needed for the existence of a solution to the equation. However that scheme is not fully numerical and its actual implementation would require further approximations.

In Section 3, we give an account of a four step algorithm developed by J. Ma, P. Protter and J. Yong [Probab. Theory Related Fields 98, 339-359 (1994; Zbl 0794.60056)] to solve a class of more general equations called forward-backward SDEs. It is based on solving the associated PDE by a deterministic-type method and also makes use of the Euler scheme for stochastic differential equations. The convergence rate turns out to be as good as for the solution of a simple SDE.

Finally, in Section 4, we describe a numerical method examined in the author’s Ph. D. Thesis (Univ. Provence, 1996) for solving a BSDE associated with a forward SDE. It involves the discretization of the BSDE both in time and in space. This leads to an algorithm which can be implemented in practice, but there one needs much stronger regularity assumptions than in Bally’s scheme.

For the entire collection see [Zbl 0867.00036].

This sort of equation has also found many applications in finance, notably in contingent claim valuation when there are constraints on the hedging portfolios (see El Karoui and Quenez 1995, El Karoui, Peng and Quenez 1994, Cvitanic and Karatzas 1992) and in the definition of stochastic differential utility (see Duffi and Epstein 1992, El Karoui, Peng and Quenez 1994). A financial application of forward-backward SDEs can be found in Duffie, Ma and Yong (1991).

However little research has yet been performed on numerical methods for BSDEs. Here we give a review of three different contributions in that field.

In Section 2, we present a random time discretization scheme introduced by V. Bally to approximate BSDEs. The advantage of Bally’s scheme is that one can get a convergence result with virtually no other regularity assumption than the ones needed for the existence of a solution to the equation. However that scheme is not fully numerical and its actual implementation would require further approximations.

In Section 3, we give an account of a four step algorithm developed by J. Ma, P. Protter and J. Yong [Probab. Theory Related Fields 98, 339-359 (1994; Zbl 0794.60056)] to solve a class of more general equations called forward-backward SDEs. It is based on solving the associated PDE by a deterministic-type method and also makes use of the Euler scheme for stochastic differential equations. The convergence rate turns out to be as good as for the solution of a simple SDE.

Finally, in Section 4, we describe a numerical method examined in the author’s Ph. D. Thesis (Univ. Provence, 1996) for solving a BSDE associated with a forward SDE. It involves the discretization of the BSDE both in time and in space. This leads to an algorithm which can be implemented in practice, but there one needs much stronger regularity assumptions than in Bally’s scheme.

For the entire collection see [Zbl 0867.00036].

Reviewer: Reviewer (Berlin)

##### MSC:

91B24 | Microeconomic theory (price theory and economic markets) |

65Z05 | Applications to the sciences |

91B28 | Finance etc. (MSC2000) |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |