Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

Abstract

We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. In both cases, we develop new theoretical tools to provide explicit sufficient conditions under which our probabilistic representations hold. As an application, we consider several examples including multi-dimensional semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo method.

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

“…In the following we also add some comments on shortcomings and possible generalizations of Theorem 1.1. In particular, we observe that the driver f : Ê → Ê in the BSDEs in (3) does only depend on the solution processes…”

“…In Theorem 1.1 we do not assume that the filtererd probability space (Ω, F , È, (F t ) t∈[0,T ] ) satisfies the usual conditions in the sense that for all t ∈ [0, T ) it holds that {A ∈ F : È(A) = 0} ⊆ F t = ∩ s∈(t,T ] F s . The (F t ) t∈[0,T ] -predictable stochastic processes Y d : [0, T ] × Ω → Ê, d ∈ AE, in the last but fifth line of Theorem 1.1 are the solution processes of the BSDEs in (3).…”

“…In (1)-(2) in Theorem 1.1 we specify the Monte Carlo-type approximation algorithm which we propose to approximate the solution processes of the BSDEs in (3). To formulate the proposed Monte Carlo-type approximation algorithm in (1)-(2) we need, roughly speaking, sufficiently many independent random quantities which are indexed over a sufficiently large index set.…”

Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

“…The validity of this method in the path-dependent case, and for further analytic nonlinearities which are of the power type in the triple (u, ∂ x u, ∂ 2 xx T u) is analyzed in [13,14,15,16]. We also refer to Agarwal and Claisse [1] for the extension to elliptic semilinear PDEs, and Bouchard, Tan, Warin & Zou [5] for Lipschitz nonlinearity in the pair (v, ∂ x v). A critical ingredient for the extension is the use of Galton-Watson trees weighted by some Malliavin automatic differentiation weights.…”

Branching diffusions with jumps, and valuation with systemic counterparties

“…We refer in particular to [12] for the zeroth-order case and [11] for the rst-order case. We also refer to [4] for an extension of the branching di usion approach to the case of locally analytic nonlinearities, [3] for the case of Lipschitz nonlinearities, [13] for higher-order partial di erential equations, and [1] for an extension to elliptic equations. The main contribution of this article is to extend the branching di usion approach to the case of nonlocal terms inside the nonlinearity.…”