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A regression-based Monte Carlo method to solve backwardstochastic differential equations.pdf

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A regression-based Monte Carlo method to solve backwardstochastic differential equations

The Annals of Applied Probability 2005, Vol. 15, No. 3, 2172–2202 DOI 10.1214? Institute of Mathematical Statistics, 2005 A REGRESSION-BASED MONTE CARLO METHOD TO SOLVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS 1 BY EMMANUEL GOBET, JEAN-PHILIPPE LEMOR AND XAVIER WARIN Centre de Mathématiques Appliquées, Electricité de France and électricité de France We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates. 1. Introduction. In this paper we are interested in numerically approximating the solution of a decoupled forward–backward stochastic differential equation (FBSDE) St = S0 + ∫ t 0 b(s, Ss) ds + ∫ t 0 σ(s, Ss) dWs,(1) Yt =(S)+ ∫ T t f (s, Ss, Ys,Zs) ds ? ∫ T t Zs dWs.(2) In this representation, S = (St : 0 ≤ t ≤ T ) is the d-dimensional forward compo- nent and Y = (Yt : 0 ≤ t ≤ T ) the one-dimensional backward one (the extension of our results to multidimensional backward equations is straightforward). Here, W is a q-dimensional Brownian motion defined on a filtered probability space (,F ,P, (Ft )0≤t≤T ), where (Ft )t is the augmented natural filtration of W . The driver f (·, ·, ·, ·) and the terminal condition (·) are, respectively, a determin- istic function and a deterministic functional of the process S. The assumptions (H1)–(H3) below ensure the existence and the uniqueness of a solution (S,Y,Z) to such equation (1)–(2). Applications of BSDEs. Such equations, first studied by Pardoux and Peng [26] in a general form, are important tools in mathematical finance. We mention some applications and refer the reader to [10, 12] for numerous references. In a complete Received June 2004; revised Janua