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Probabilistic methods for semilinearpartial differential equations.Applications to finance

Published online by Cambridge University Press:  26 August 2010

Dan Crisan  and
Konstantinos Manolarakis
Dan Crisan
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, UK. d.crisan@imperial.ac.uk; km3@imperial.ac.uk
Konstantinos Manolarakis
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, UK. d.crisan@imperial.ac.uk; km3@imperial.ac.uk
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Abstract

With the pioneering work of [Pardoux and Peng, Syst. Contr. Lett.14 (1990) 55–61; Pardoux and Peng,Lecture Notes in Control and Information Sciences176 (1992) 200–217]. We have at our disposalstochastic processes which solve the so-called backward stochasticdifferential equations. These processes provide us with a Feynman-Kacrepresentation for the solutions of a class of nonlinear partial differential equations (PDEs) which appearin many applications in the field of Mathematical Finance. Therefore thereis a great interest among both practitioners and theoreticians to developreliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods forapproximating solutions of semilinear PDEs all based on the correspondingFeynman-Kac representation. We also include a general introduction tobackward stochastic differential equations and their connection with PDEsand provide a generic framework that accommodates existing probabilisticalgorithms and facilitates the construction of new ones.

Keywords

Probabilistic methodssemilinear PDEsBSDEsMonte Carlo methodsMalliavin calculuscubature methods
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 5: Special Issue on Probabilistic methods and their applications , September 2010 , pp. 1107 - 1133
Copyright
© EDP Sciences, SMAI, 2010

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References

Bally, V. and Pagès, G., Error analysis of the quantization algorithm for obstacle problems. Stochastic Processes their Appl. 106 (2003) 1–40. CrossRef
Bally, V. and Pagès, G., A quantization algorithm for solving multi dimensional discrete-time optional stopping problems. Bernoulli 6 (2003) 1003–1049. CrossRef
Becherer, D., Bounded solutions to backward SDE's with jumps for utility optimization and indifference pricing. Ann. Appl. Prob. 16 (2006) 2027–2054. CrossRef
J.M. Bismut, Théorie probabiliste du contrôle des diffusions, Mem. Amer. Math. Soc. 176 . Providence, Rhode Island (1973).
Bouchard, B. and Touzi, N., Discrete time approximation and Monte Carlo simulation for Backward Stochastic Differential Equations. Stochastic Processes their Appl. 111 (2004) 175–206. CrossRef
Bouchard, B., Ekeland, I. and Touzi, N., On the Malliavin approach to Monte Carlo methods of conditional expectations. Financ. Stoch. 8 (2004) 45–71. CrossRef
Briand, P. and BSDE, Y. Hu with quadratic growth and unbounded terminal value. Probab. Theor. Relat. Fields 136 (2006) 604–618. CrossRef
Chen, K.-T., Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math. 65 (1957) 163–178. CrossRef
Cheridito, P., Soner, M., Touzi, N. and Victoir, N., Second-order backward stochastic differential equations and fully non linear parabolic pdes. Commun. Pure Appl. Math. 60 (2007) 1081–1110. CrossRef
D. Crisan and K. Manolarakis, Numerical solution for a BSDE using the Cubature method. Preprint available at http://www2.imperial.ac.uk/ dcrisan/ (2007).
Crisan, D., Manolarakis, K. and Touzi, N., On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights. Stochastic Processes their Appl. 120 (2010) 1133–1158. CrossRef
Cvitanic, J. and Karatzas, I., Hedging contingent claims with constrained portfolios. Ann. Appl. Prob. 3 (1993) 652–681. CrossRef
Duffy, D. and Epstein, L., Asset pricing with stochastic differential utility. Rev. Financ. Stud. 5 (1992) 411–436. CrossRef
Duffy, D. and Epstein, L., Stochastic differential utility. Econometrica 60 (1992) 353–394. CrossRef
N. El Karoui and S.J. Huang, A general result of existence and uniqueness of backward stochastic differential equations, in Backward Stochastic Differential Equations, N. El Karoui and L. Mazliak Eds., Longman (1996).
El Karoui, N. and Quenez, M., Dynamic programming and pricing of contigent claims in incomplete markets. SIAM J. Contr. Opt. 33 (1995) 29–66. CrossRef
N. El Karoui and M. Quenez, Non linear pricing theory and Backward Stochastic Differential Equations, in Financial Mathematics 1656, Springer (1995) 191–246.
El Karoui, N., Kapoudjan, C., Pardoux, E., Peng, S. and Quenez, M.C., Reflected solutions of backward SDEs and related obstacle problems. Annals Probab. 25 (1997) 702–737.
N. El Karoui, E. Pardoux and M. Quenez, Reflected backward SDEs and American Options, in Numerical Methods in Finance, Chris Rogers and Denis Talay Eds., Cambridge University Press, Cambridge (1997).
El Karoui, N., Peng, S. and Quenez, M., Backward Stochastic Differential Equations in finance. Mathematical Finance 7 (1997) 1–71. CrossRef
Feynman, R., Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20 (1948) 367–387. CrossRef
Föllmer, H. and Schied, A., Convex measures of risk and trading constraints. Financ. Stoch. 6 (2002) 429–447. CrossRef
P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and applications. Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge (2010).
Gobet, E. and Labart, C., Error expansion for the discretization of Backward Stochastic Differential Equations. Stochastic Processes their Appl. 117 (2007) 803–829. CrossRef
Gobet, E., Lemor, J.P. and Warin, X., A regression based Monte Carlo method to solve Backward Stochastic Differential Equations. Ann. Appl. Prob. 15 (2005) 2172–2202. CrossRef
Gobet, E., Lemor, J.P. and Warin, X., Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12 (2006) 889–916.
Jouini, E. and Kallal, H., Arbitrage in securities markets with short sales constraints. Mathematical Finance 5 (1995) 178–197. CrossRef
Kac, M., On distributions of certain Wiener functionals. Trans. Amer. Math. Soc. 65 (1949) 1–13. CrossRef
I. Karatzas and S. Schreve, Brownian Motion and Stochastic Calculus. Springer Verlag, New York (1991).
Kobylanski, M., Backward Stochastic Differential Equations and Partial Differential Equations. Ann. Appl. Prob. 28 (2000) 558–602. CrossRef
Lepeltier, J.-P. and San Martin, J., Backward Stochastic Differential Equations with continuous coefficients. Stat. Probab. Lett. 32 (1997) 425–430. CrossRef
Longstaff, F. and Schwartz, E.S., Valuing American options by simulation: a simple least squares approach. Rev. Financ. Stud. 14 (2001) 113–147. CrossRef
T. Lyons and Z. Qian, System Control and Rough Paths. Oxford Science publication, Oxford University Press, Oxford (2002).
Lyons, T. and Victoir, N., Cubature on Wiener space. Proc. Royal Soc. London 468 (2004) 169–198. CrossRef
T. Lyons, M. Caruana and T. Levy, Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics 1908. Springer (2004).
Ma, J. and Zhang, J., Representation theorems for Backward Stochastic Differential Equations. Ann. Appl. Prob. 12 (2002) 1390–1418.
Ma, J. and Zhang, J., Representation and regularities for solutions to BSDEs with reflections. Stochastic Processes their Appl. 115 (2005) 539–569. CrossRef
Ma, J., Protter, P. and Yong, J., Solving Forward-Backward SDEs expicitly – A four step scheme. Probab. Theor. Relat. Fields 122 (1994) 163–190.
D. Nualart, The Malliavin calculus and related topics. Springer-Verlag (1996).
Pardoux, E. and Peng, S., Adapted solution to Backward Stochastic Differential Equations. Syst. Contr. Lett. 14 (1990) 55–61. CrossRef
E. Pardoux and S. Peng, Backward Stochastic Differential Equations and quasi linear parabolic partial differential equations, in Lecture Notes in Control and Information Sciences 176, Springer, Berlin/Heidelberg (1992) 200–217.
Pardoux, E. and Tang, S., Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theor. Relat. Fields 114 (1999) 123–150. CrossRef
S. Peng, Backward SDEs and related g-expectations, in Pitman Research Notes in Mathematics Series 364, Longman, Harlow (1997) 141–159.
S. Peng, Non linear expectations non linear evaluations and risk measures 1856. Springer-Verlag (2004).
S. Peng, Modelling derivatives pricing mechanisms with their generating functions. Preprint, arxiv:math/0605599v1 (2006).
Rosazza Giannin, E., Risk measures via g expectations. Insur. Math. Econ. 39 (2006) 19–34.
Tang, S. and Necessary, X. Li conditions for optimal control of stochastic systems with random jumps. SIAM J. Contr. Opt. 32 (1994) 1447–1475. CrossRef
J. Zhang, Some fine properties of backward stochastic differential equations. Ph.D. Thesis, Purdue University, USA (2001).
Zhang, J., A numerical scheme for BSDEs. Ann. Appl. Prob. 14 (2004) 459–488. CrossRef