Wasserstein Distance Estimates for Stochastic Integrals by Forward-Backward Stochastic Calculus
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Wasserstein Distance Estimates for Stochastic Integrals by Forward-Backward Stochastic Calculus Jean-Christophe Breton1 · Nicolas Privault2 Received: 19 November 2019 / Accepted: 11 August 2020 / © Springer Nature B.V. 2020
Abstract We prove Wasserstein distance bounds between the probability distributions of stochastic integrals with jumps, based on the integrands appearing in their stochastic integral representations. Our approach does not rely on the Stein equation or on the propagation of convexity property for Markovian semigroups, and makes use instead of forward-backward stochastic calculus arguments. This allows us to consider a large class of target distributions constructed using Brownian stochastic integrals and pure jump martingales, which can be specialized to infinitely divisible target distributions with finite L´evy measure and Gaussian components. Keywords Wasserstein distance · Stochastic integrals · Forward-backward stochastic calculus · Point processes Mathematics Subject Classification (2010) 60H05 · 60H10 · 60G57 · 60G44 · 60J60 · 60J75
1 Introduction Comparison inequalities for option prices with convex payoff functions have been obtained in the literature, based on the classical Kolmogorov equation under the propagation of convexity hypothesis for Markovian semigroups. See for instance [12] in the case of continuous diffusion processes, and [4, 5, 11], in the case of jump-diffusion processes. In [7], lower and upper bounds on option prices have been obtained in one-dimensional jump-diffusion markets with point process components under different conditions. Nicolas Privault
[email protected] Jean-Christophe Breton [email protected] 1
Univ Rennes, CNRS, IRMAR - UMR 6625, University of Rennes, 263 Avenue du G´en´eral Leclerc, F-35000 Rennes, France
2
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
J.-C. Breton, N. Privault
Note however that the propagation of convexity property is not always satisfied, even in the (Markovian) jump-diffusion case, see e.g. Theorem 4.4 in [11]. Using different arguments based on forward-backward stochastic calculus, related convex ordering results have been obtained for exponential jump-diffusion processes in [7]. The case of random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component has also been treated in [1] using forward-backward stochastic calculus, extending the one-dimensional framework of [14], see also [6] for the case of Itˆo integrals and [15] for additive functionals. In [9], bounds on differences in expectation have been obtained in order to estimate the distance between the distribution L (XT ) of the terminal value XT of a stochastic integral process (Xt )t∈[0,T ] on a finite time horizon [0, T ] and a target distribution L (YT ) given by the terminal value YT of a jump-diffusion process (Yt )t∈[0,T ] solution of a stochastic diffe
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