# 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

## Access Full Article

top## Abstract

top## How to cite

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 -

## References

top- [1] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 (1996) 43–60. Zbl0838.60051MR1367666
- [2] V.E. Beneš, L.A. Shepp and H.S. Witsenhausen, Some solvable stochastic control problems. In Analysis and optimisation of stochastic systems (Proc. Internat. Conf., Univ. Oxford, Oxford, 1978), Academic Press, London (1980) 3–10. Zbl0451.93068MR592971
- [3] A. Beskos, O. Papaspiliopoulos and G.O. Roberts, Retrospective exact simulation of diffusion sample paths with applications. Bernoulli12 (2006) 1077–1098. Zbl1129.60073MR2274855
- [4] A. Beskos, O. Papaspiliopoulos and G.O. Roberts, A factorisation of diffusion measure and finite sample path constructions. Methodol. Comput. Appl. Probab.10 (2008) 85–104. Zbl1152.65013MR2394037
- [5] A. Beskos, G. Roberts, A. Stuart and J. Voss, MCMC methods for diffusion bridges. Stoch. Dyn.8 (2008) 319–350. Zbl1159.65007MR2444507
- [6] A. Beskos and G.O. Roberts, Exact simulation of diffusions. Ann. Appl. Probab.15 (2005) 2422–2444. Zbl1101.60060MR2187299
- [7] P. Étoré and M. Martinez, Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process. Monte Carlo Methods Appl.19 (2013) 41–71. Zbl1269.65007MR3039402
- [8] S.E. Graversen and A.N. Shiryaev, An extension of P. Lévy’s distributional properties to the case of a Brownian motion with drift. Bernoulli 6 (2000) 615–620. Zbl0965.60077MR1777686
- [9] I. Karatzas and S.E. Shreve, Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control. Ann. Probab.12 (1984) 819–828. Zbl0544.60069MR744236
- [10] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, 2nd edn. vol. 113. In Grad. Texts Math. Springer-Verlag, New York (1991) 440–441. Zbl0734.60060MR1121940
- [11] J.-F. Le Gall, One-dimensional stochastic differential equations involving the local times of the unknown process. In Stochastic analysis and applications (Swansea, 1983), Lecture Notes in Math., vol. 1095. Springer, Berlin (1984) 51–82. Zbl0551.60059MR777514
- [12] V. Reutenauer and E. Tanré, Exact simulation of prices and greeks: application to cir. Preprint (2008).
- [13] D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd edn. Springer-Verlag (1999). Zbl0804.60001MR1725357
- [14] M. Sbai, Modélisation de la dépendance et simulation de processus en finance. Ph.D. thesis, CERMICS – Centre d’Enseignement et de Recherche en Mathématiques et Calcul Scientifique (2009).