# Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift

• Volume: 18, page 686-702
• ISSN: 1292-8100

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## Abstract

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In this note we propose an exact simulation algorithm for the solution of (1)$\mathrm{d}{X}_{t}=\mathrm{d}{W}_{t}+\overline{b}\left({X}_{t}\right)\mathrm{d}t,\phantom{\rule{1.0em}{0ex}}{X}_{0}=x,$d X t = d W t + b̅ ( X t ) d t,   X 0 = x, where $\overline{b}$b̅is a smooth real function except at point 0 where $\overline{b}\left(0+\right)\ne \overline{b}\left(0-\right)$b̅(0 + ) ≠ b̅(0 −) . The main idea is to sample an exact skeleton of Xusing an algorithm deduced from the convergence of the solutions of the skew perturbed equation (2)$\mathrm{d}{X}_{t}^{\beta }=\mathrm{d}{W}_{t}+\overline{b}\left({X}_{t}^{\beta }\right)\mathrm{d}t+\beta \mathrm{d}{L}_{t}^{0}\left({X}^{\beta }\right),\phantom{\rule{1.0em}{0ex}}{X}_{0}=x$d X t β = d W t + b̅ ( X t β ) d t + β d L t 0 ( X β ) ,   X 0 = x towardsX solution of (1) as β ≠ 0 tends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in [Pierre Étoré and Miguel Martinez. Monte Carlo Methods Appl. 19 (2013) 41–71] for the solutions of (2) towards a limit algorithm. Thanks to stability properties of the rejection procedures involved as β tends to 0, we prove that this limit algorithm is an exact simulation algorithm for the solution of the limit equation (1). Numerical examples are shown to illustrate the performance of this exact simulation algorithm.

## How to cite

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Étoré, Pierre, and Martinez, Miguel. "Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift." ESAIM: Probability and Statistics 18 (2014): 686-702. <http://eudml.org/doc/273650>.

@article{Étoré2014,
abstract = {In this note we propose an exact simulation algorithm for the solution of (1)\begin\{equation\} \{\rm d\}X\_t=\{\rm d\}W\_t+\bar\{b\}(X\_t)\{\rm d\}t,\quad X\_0=x, \end\{equation\}d X t = d W t + b̅ ( X t ) d t,   X 0 = x, where $\bar\{b\}$b̅is a smooth real function except at point 0 where $\bar\{b\}(0+)\ne \bar\{b\}(0-)$b̅(0 + ) ≠ b̅(0 −) . The main idea is to sample an exact skeleton of Xusing an algorithm deduced from the convergence of the solutions of the skew perturbed equation (2)\begin\{equation\} \{\rm d\}X^\beta \_t=\{\rm d\}W\_t+\bar\{b\}(X^\beta \_t)\{\rm d\}t + \beta \{\rm d\}L^0\_t(X^\beta ),\quad X\_0=x \end\{equation\}d X t β = d W t + b̅ ( X t β ) d t + β d L t 0 ( X β ) ,   X 0 = x towardsX solution of (1) as β ≠ 0 tends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in [Pierre Étoré and Miguel Martinez. Monte Carlo Methods Appl. 19 (2013) 41–71] for the solutions of (2) towards a limit algorithm. Thanks to stability properties of the rejection procedures involved as β tends to 0, we prove that this limit algorithm is an exact simulation algorithm for the solution of the limit equation (1). Numerical examples are shown to illustrate the performance of this exact simulation algorithm.},
author = {Étoré, Pierre, Martinez, Miguel},
journal = {ESAIM: Probability and Statistics},
keywords = {exact simulation methods; brownian motion with two-valued drift; one-dimensional diffusion; skew brownian motion; local time; Brownian motion with two-valued drift; skew Brownian motion; numerical examples; algorithm; system of stochastic differential equations; convergence},
language = {eng},
pages = {686-702},
publisher = {EDP-Sciences},
title = {Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift},
url = {http://eudml.org/doc/273650},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Étoré, Pierre
AU - Martinez, Miguel
TI - Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 686
EP - 702
AB - In this note we propose an exact simulation algorithm for the solution of (1)$${\rm d}X_t={\rm d}W_t+\bar{b}(X_t){\rm d}t,\quad X_0=x,$$d X t = d W t + b̅ ( X t ) d t,   X 0 = x, where $\bar{b}$b̅is a smooth real function except at point 0 where $\bar{b}(0+)\ne \bar{b}(0-)$b̅(0 + ) ≠ b̅(0 −) . The main idea is to sample an exact skeleton of Xusing an algorithm deduced from the convergence of the solutions of the skew perturbed equation (2)$${\rm d}X^\beta _t={\rm d}W_t+\bar{b}(X^\beta _t){\rm d}t + \beta {\rm d}L^0_t(X^\beta ),\quad X_0=x$$d X t β = d W t + b̅ ( X t β ) d t + β d L t 0 ( X β ) ,   X 0 = x towardsX solution of (1) as β ≠ 0 tends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in [Pierre Étoré and Miguel Martinez. Monte Carlo Methods Appl. 19 (2013) 41–71] for the solutions of (2) towards a limit algorithm. Thanks to stability properties of the rejection procedures involved as β tends to 0, we prove that this limit algorithm is an exact simulation algorithm for the solution of the limit equation (1). Numerical examples are shown to illustrate the performance of this exact simulation algorithm.
LA - eng
KW - exact simulation methods; brownian motion with two-valued drift; one-dimensional diffusion; skew brownian motion; local time; Brownian motion with two-valued drift; skew Brownian motion; numerical examples; algorithm; system of stochastic differential equations; convergence
UR - http://eudml.org/doc/273650
ER -

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