Abstract
We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this requires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Convergence of our algorithm is proved without any limitation on the time horizon. We also provide numerical simulations to illustrate the performance of the algorithm.
Highlights
The aim of this paper is to extend the branching process based numerical algorithm proposed in Bouchard et al [3] to general BSDEs in form: T
In the case where the driver depends on the Z component of the BSDE, a similar truncation has to be performed on the gradient itself
We first approximate the BSDE (1) by a BSDE with a local polynomial generator. To solve the latter BSDE, we suggest a Picard iteration together with the truncation and face-lifting technique on the value function
Summary
The aim of this paper is to extend the branching process based numerical algorithm proposed in Bouchard et al [3] to general BSDEs in form: T. where W is a standard d-dimensional Brownian motion, f : Rd × R × Rd → R is the driver function, g : Rd → R is the terminal condition, and X is the solution of. In the case where the driver depends on the Z component of the BSDE, a similar truncation has to be performed on the gradient itself It can not be done by projecting Z on a suitable compact set at certain time steps, since Z only maters up to an equivalent class of ([0, T ] × Ω, dt × dP). For a map (t, x) → ψ(t, x), we denote by ∂tψ is derivative with respect to its first variable and by Dψ and D2ψ its Jacobian and Hessian matrix with respect to its second component
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