Finite variance unbiased estimation of stochastic differential equations

Abstract

We develop a new unbiased estimation method for Lipschitz continuous functions of multi-dimensional stochastic differential equations with Lipschitz continuous coefficients. This method provides a finite variance estimator based on a probabilistic representation which is similar to the recent representations obtained through the parametrix method and recursive application of the automatic differentiation formula. Our approach relies on appropriate change of variables to carefully handle the singular integrands appearing in the iterated integrals of the probabilistic representation. It results in a scheme with randomized intermediate times where the number of intermediate times has a Pareto distribution.

Unbiased Simulation Estimators for Jump-Diffusions

“…A number of unbiased diffusion estimators have been developed in the literature (Wagner, 1989;Bally and Kohatsu-Higa, 2015;Henry-Labordère et al, 2017;Agarwal and Gobet, 2017;Doumbia et al, 2017). More specifically, for a diffusion process Y and bounded function f : R n → R, those estimators feature an Euler approximation Y π and a correction functional…”

“…We remark that Ξ is not required to be an exact sample of f (X T ), and for example, Ξ may be negative all the while f is positive valued. Glynn and Rhee (2015), Bally and Kohatsu-Higa (2015), Agarwal and Gobet (2017), Doumbia et al (2017), Andersson and Kohatsu-Higa (2017), Henry-Labordère et al (2017) and Chen et al (2020) have developed and analyzed unbiased estimators for diffusion processes. These estimators are based on ideas that draw from the literature on multi-level Monte Carlo and formulations that date back to the study of partial differential equations in Levi (1907).…”

Integration by parts formula for killed processes: a point of view from approximation theory

“…We will discuss this extension in future works as well as their implementation for simulation purposes. Let X be the solution to (1). Let P denote the semigroup operator associated with the killed process.…”

“…The first step towards obtaining an IBP formula is to prove a probabilistic representation for the marginal law of the killed diffusion process, in the spirit of Bally and Kohatsu-Higa [7] which developed such a formula for multi-dimensional diffusion processes (without stopping) and some Lévy driven SDEs by means of a probabilistic perturbation argument for Markov semigroups. We also refer the reader to Labordère et al [20] and Agarwal and Gobet [1] for some recent contributions in that direction for multi-dimensional diffusion processes.…”

Probabilistic representation of integration by parts formulae for some stochastic volatility models with unbounded drift

“…Still in the case of bounded coefficients, it was then further investigated in Labordère and al. [9], Agarwal and Gobet [1] for multi-dimensional diffusion processes and in Frikha and al. [7] for one-dimensional killed processes.…”