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

ESAIM: Probability and Statistics (2014)

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

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topÉtoré, 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)\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.

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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