Unbiased Monte Carlo estimate of stochastic differential equations expectations
Doumbia, Mahamadou1 ; Oudjane, Nadia1 ; Warin, Xavier1
1 EDF R&D & FiME, Laboratoire de Finance des Marchés de l’Energie (www.fime-lab.org), 7 boulevard Gaspard Monge, 91120 Palaiseau, France.
ESAIM: Probability and Statistics, Tome 21 (2017), pp. 56-87.

We propose an unbiased Monte Carlo method to compute E(g(X T )) where g is a Lipschitz function and X an Ito process. This approach extends the method proposed in [16] to the case where X is solution of a multidimensional stochastic differential equation with varying drift and diffusion coefficients. A variance reduction method relying on interacting particle systems is also developed.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2017001
Classification : 65C05, 60J60, 60J85, 35K10
Mots clés : Unbiased estimate, linear parabolic PDEs, interacting particle systems
Doumbia, Mahamadou 1 ; Oudjane, Nadia 1 ; Warin, Xavier 1

1 EDF R&D & FiME, Laboratoire de Finance des Marchés de l’Energie (www.fime-lab.org), 7 boulevard Gaspard Monge, 91120 Palaiseau, France. 
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     title = {Unbiased {Monte} {Carlo} estimate of stochastic differential equations expectations},
     journal = {ESAIM: Probability and Statistics},
     pages = {56--87},
     publisher = {EDP-Sciences},
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Doumbia, Mahamadou; Oudjane, Nadia; Warin, Xavier. Unbiased Monte Carlo estimate of stochastic differential equations expectations. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 56-87. doi : 10.1051/ps/2017001. http://www.numdam.org/articles/10.1051/ps/2017001/

P. Andersson and A. Kohatsu-Higa, Unbiased simulation of stochastic differential equations using parametrix expansions. To appear in Bernoulli [Online] Available on: http://www.bernoulli-society.org/index.php/publications/bernoulli-journal-papers (2016). | MR

V. Bally and A. Kohatsu-Higa, A probabilistic interpretation of the parametrix mehod. Ann. Appl. Probab. 25 (2015) 3095–3138. | DOI | MR | Zbl

A. Beskos and G. Roberts, Exact simulation of diffusions. Ann. Appl. Probab. 15 (2005) 2422–2444. | DOI | MR | Zbl

M. Broadie and P. Glasserman, Estimating security prices using simulation. Manage. Sci. 42 (1996) 269–285. | DOI | Zbl

K. Burrage, P. M. Burrage and P. Tian, Numerical methods for strong solutions of stochastic differential equations: an overview. Proc. R. Soc. London, Series A 460 (2004) 373–402. | DOI | MR | Zbl

N. Chen and Z. Huang, Localization and exact simulation of brownian motion-driven stochastic differential equations. Math. Oper. Res. 38 (2013) 591–616. | DOI | MR | Zbl

P. Del Moral, Feynman-Kac formulae. Genealogical and interacting particle systems with applications. Probability and its Applications (New York). Springer Verlag, New York (2004). | MR | Zbl

P. Del Moral, Mean field simulation for Monte Carlo integration. Monographs on Statistics and applied Probability. Chapman and Hall (2013). | MR | Zbl

D. Duffie and P.W. Glynn, Efficient Monte Carlo simulation of security prices. Ann. Appl. Probab. 5 (1995) 897–905. | DOI | MR | Zbl

E. Fournié, J.M. Lasry, J. Lebuchoux, P.L. Lions and N. Touzi, Applications of Malliavin calculus to Monte Carlo methods in finance. Fin. Stoch. 3 (1999) 391–412. | DOI | MR | Zbl

E. Fournié, J.M. Lasry, J. Lebuchoux and N. Touzi, Some applications of malliavin calculus to Monte Carlo methods in finance. Fin. Stoch. 3 (1999) 391–412. | MR

M.B. Giles, Multilevel Monte Carlo path simulation. Oper. Res. 56 (2008) 607–617. | DOI | MR | Zbl

P. Henry-Labordère, Countreparty risk valuation: A marked branching diffusion approach. Available at SSRN: http://ssrn.com/abstract=1995503 or (2012). | DOI

P. Henry-Labordère, N. Oudjane, X. Tan, N. Touzi and X. Warin, Branching diffusion representation of semilinear pdes and Monte Carlo approximations. Preprint (2016). | arXiv | MR

P. Henry-Labordère, X. Tan and N. Touzi, A numerical algorithm for a class of bsde via branching process. Stoch. Process. Appl. 124 (2014) 1112–1140. | DOI | MR | Zbl

P. Henry-Labordère, X. Tan and N. Touzi, Unbiased simulation of stochastic differential equations. To appear in Ann. Appl. Probab. (2016). | MR

B. Jourdain and M. Sbai, Exact retrospective Monte Carlo computation of arithmetic average asian options. Monte Carlo Meth. Appl. 13 (2007) 135–171. | DOI | MR | Zbl

P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Vol. 23 of Applications of Mathematics (New York). Springer Verlag, Berlin (1992). | MR | Zbl

G.N. Milstein, Approximate integration of stochastic differential equations. Theory Probab. Appl. 19 (1975) 557–600. | DOI | MR | Zbl

C.H. Rhee and P.W. Glynn, Unbiased estimation with square root convergence for sde models. Oper. Res. 63 (2015) 1026–1043. | DOI | MR | Zbl

D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8 (1990) 483–509. | DOI | MR | Zbl

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