Approximation of heat equation and backward SDEs using random walk : convergence rates
This thesis addresses questions related to approximation arising from the ﬁelds of stochastic analysis and partial diﬀerential equations. Theoretical results regarding convergence rates are obtained by using discretization schemes where the limiting process, the Brownian motion, is approximated by a simple discrete-time random walk. The rate of convergence is derived for a ﬁnite-diﬀerence approximation of the solution of a terminal value problem for the backward heat equation. This weak approximation result is proved for a terminal function which has bounded variation on compact sets. The sharpness of the according rate is achieved by applying some new results related to the ﬁrst exit time behavior of Brownian bridges. In addition, convergence rates in the L2-norm are proved for Markovian forward-backward stochastic diﬀerential equations, where the underlying forward process is either Brownian motion or a more general Itô diﬀusion.
PublisherUniversity of Jyväskylä
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Geiss, Christel; Labart, Céline; Luoto, Antti (Cambridge University Press (CUP), 2020)Let (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk from the underlying Brownian motion B by Skorokhod embedding, one can show -convergence of ...
Toivola, Anni (University of Jyväskylä, 2009)
Briand, Philippe; Geiss, Christel; Geiss, Stefan; Labart, Céline (International Statistical Institute, 2021)In this paper, we study in the Markovian case the rate of convergence in Wasserstein distance when the solution to a BSDE is approximated by a solution to a BSDE driven by a scaled random walk as introduced in Briand, ...
Geiss, Christel; Labart, Céline; Luoto, Antti (International Statistical Institute, 2020)In this paper, we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally Hölder continuous function of the Brownian motion. We state the rate of the L2-convergence ...
Laukkarinen, Eija (University of Jyväskylä, 2012)