Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 686-702.

In this note we propose an exact simulation algorithm for the solution of (1)

dX t =dW t +b ¯(X t )dt,X 0 =x,
d X t = d W t + b̅ ( X t ) d t,   X 0 = x, where b ¯ is a smooth real function except at point 0 where b ¯(0+)b ¯(0-) (0 + ) ≠ (0 -) . The main idea is to sample an exact skeleton of X using an algorithm deduced from the convergence of the solutions of the skew perturbed equation (2)
dX t β =dW t +b ¯(X t β )dt+βdL t 0 (X β ),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 towards X 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.

DOI : 10.1051/ps/2013053
Classification : 65C05, 65U20, 65C30, 65C20
Mots-clés : exact simulation methods, brownian motion with two-valued drift, one-dimensional diffusion, skew brownian motion, local time
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     author = {\'Etor\'e, Pierre and Martinez, Miguel},
     title = {Exact simulation for solutions of one-dimensional {Stochastic} {Differential} {Equations} with discontinuous drift},
     journal = {ESAIM: Probability and Statistics},
     pages = {686--702},
     publisher = {EDP-Sciences},
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     year = {2014},
     doi = {10.1051/ps/2013053},
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     url = {http://archive.numdam.org/articles/10.1051/ps/2013053/}
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Étoré, Pierre; Martinez, Miguel. Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 686-702. doi : 10.1051/ps/2013053. http://archive.numdam.org/articles/10.1051/ps/2013053/

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