# Probabilistic methods for semilinear partial differential equations. Applications to finance

Dan Crisan; Konstantinos Manolarakis

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 5, page 1107-1133
- ISSN: 0764-583X

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topCrisan, Dan, and Manolarakis, Konstantinos. "Probabilistic methods for semilinear partial differential equations. Applications to finance." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 1107-1133. <http://eudml.org/doc/250847>.

@article{Crisan2010,

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 disposal
stochastic processes which solve the so-called backward stochastic
differential equations. These processes provide us with a Feynman-Kac
representation for the solutions of a class of nonlinear partial differential equations (PDEs) which appear
in many applications in the field of Mathematical Finance. Therefore there
is a great interest among both practitioners and theoreticians to develop
reliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods for
approximating solutions of semilinear PDEs all based on the corresponding
Feynman-Kac representation. We also include a general introduction to
backward stochastic differential equations and their connection with PDEs
and provide a generic framework that accommodates existing probabilistic
algorithms and facilitates the construction of new ones.
},

author = {Crisan, Dan, Manolarakis, Konstantinos},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Probabilistic methods; semilinear PDEs; BSDEs; Monte Carlo methods; Malliavin calculus; cubature methods; probabilistic methods; BSDEs Monte Carlo methods},

language = {eng},

month = {8},

number = {5},

pages = {1107-1133},

publisher = {EDP Sciences},

title = {Probabilistic methods for semilinear partial differential equations. Applications to finance},

url = {http://eudml.org/doc/250847},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Crisan, Dan

AU - Manolarakis, Konstantinos

TI - Probabilistic methods for semilinear partial differential equations. Applications to finance

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/8//

PB - EDP Sciences

VL - 44

IS - 5

SP - 1107

EP - 1133

AB -
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 disposal
stochastic processes which solve the so-called backward stochastic
differential equations. These processes provide us with a Feynman-Kac
representation for the solutions of a class of nonlinear partial differential equations (PDEs) which appear
in many applications in the field of Mathematical Finance. Therefore there
is a great interest among both practitioners and theoreticians to develop
reliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods for
approximating solutions of semilinear PDEs all based on the corresponding
Feynman-Kac representation. We also include a general introduction to
backward stochastic differential equations and their connection with PDEs
and provide a generic framework that accommodates existing probabilistic
algorithms and facilitates the construction of new ones.

LA - eng

KW - Probabilistic methods; semilinear PDEs; BSDEs; Monte Carlo methods; Malliavin calculus; cubature methods; probabilistic methods; BSDEs Monte Carlo methods

UR - http://eudml.org/doc/250847

ER -

## References

top- V. Bally and G. Pagès, Error analysis of the quantization algorithm for obstacle problems. Stochastic Processes their Appl.106 (2003) 1–40. Zbl1075.60523
- V. Bally and G. Pagès, A quantization algorithm for solving multi dimensional discrete-time optional stopping problems. Bernoulli6 (2003) 1003–1049. Zbl1042.60021
- D. Becherer, Bounded solutions to backward SDE's with jumps for utility optimization and indifference pricing. Ann. Appl. Prob.16 (2006) 2027–2054. Zbl1132.91457
- J.M. Bismut, Théorie probabiliste du contrôle des diffusions, Mem. Amer. Math. Soc. 176. Providence, Rhode Island (1973). Zbl0323.93046
- B. Bouchard and N. Touzi, Discrete time approximation and Monte Carlo simulation for Backward Stochastic Differential Equations. Stochastic Processes their Appl.111 (2004) 175–206. Zbl1071.60059
- B. Bouchard, I. Ekeland and N. Touzi, On the Malliavin approach to Monte Carlo methods of conditional expectations. Financ. Stoch.8 (2004) 45–71. Zbl1051.60061
- P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value. Probab. Theor. Relat. Fields136 (2006) 604–618. Zbl1109.60052
- K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math.65 (1957) 163–178. Zbl0077.25301
- P. Cheridito, M. Soner, N. Touzi and N. Victoir, Second-order backward stochastic differential equations and fully non linear parabolic pdes. Commun. Pure Appl. Math.60 (2007) 1081–1110. Zbl1121.60062
- D. Crisan and K. Manolarakis, Numerical solution for a BSDE using the Cubature method. Preprint available at dcrisan/ (2007). Zbl1259.65005URIhttp://www2.imperial.ac.uk/
- D. Crisan, K. Manolarakis and N. Touzi, On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights. Stochastic Processes their Appl.120 (2010) 1133–1158. Zbl1193.65005
- J. Cvitanic and I. Karatzas, Hedging contingent claims with constrained portfolios. Ann. Appl. Prob.3 (1993) 652–681. Zbl0825.93958
- D. Duffy and L. Epstein, Asset pricing with stochastic differential utility. Rev. Financ. Stud.5 (1992) 411–436.
- D. Duffy and L. Epstein, Stochastic differential utility. Econometrica60 (1992) 353–394. Zbl0763.90005
- 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). Zbl0887.60064
- N. El Karoui and M. Quenez, Dynamic programming and pricing of contigent claims in incomplete markets. SIAM J. Contr. Opt.33 (1995) 29–66. Zbl0831.90010
- N. El Karoui and M. Quenez, Non linear pricing theory and Backward Stochastic Differential Equations, in Financial Mathematics1656, Springer (1995) 191–246. Zbl0904.90010
- N. El Karoui, C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez, Reflected solutions of backward SDEs and related obstacle problems. Annals Probab.25 (1997) 702–737. Zbl0899.60047
- 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). Zbl0898.90033
- N. El Karoui, S. Peng and M. Quenez, Backward Stochastic Differential Equations in finance. Mathematical Finance7 (1997) 1–71. Zbl0884.90035
- R. Feynman, Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys.20 (1948) 367–387.
- H. Föllmer and A. Schied, Convex measures of risk and trading constraints. Financ. Stoch.6 (2002) 429–447. Zbl1041.91039
- P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and applications. Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge (2010). Zbl1193.60053
- E. Gobet and C. Labart, Error expansion for the discretization of Backward Stochastic Differential Equations. Stochastic Processes their Appl.117 (2007) 803–829. Zbl1117.60058
- E. Gobet, J.P. Lemor and X. Warin, A regression based Monte Carlo method to solve Backward Stochastic Differential Equations. Ann. Appl. Prob.15 (2005) 2172–2202. Zbl1083.60047
- E. Gobet, J.P. Lemor and X. Warin, Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli12 (2006) 889–916. Zbl1136.60351
- E. Jouini and H. Kallal, Arbitrage in securities markets with short sales constraints. Mathematical Finance5 (1995) 178–197. Zbl0866.90032
- M. Kac, On distributions of certain Wiener functionals. Trans. Amer. Math. Soc.65 (1949) 1–13. Zbl0032.03501
- I. Karatzas and S. Schreve, Brownian Motion and Stochastic Calculus. Springer Verlag, New York (1991).
- M. Kobylanski, Backward Stochastic Differential Equations and Partial Differential Equations. Ann. Appl. Prob.28 (2000) 558–602. Zbl1044.60045
- J.-P. Lepeltier and J. San Martin, Backward Stochastic Differential Equations with continuous coefficients. Stat. Probab. Lett.32 (1997) 425–430. Zbl0904.60042
- F. Longstaff and E.S. Schwartz, Valuing American options by simulation: a simple least squares approach. Rev. Financ. Stud.14 (2001) 113–147.
- T. Lyons and Z. Qian, System Control and Rough Paths. Oxford Science publication, Oxford University Press, Oxford (2002). Zbl1029.93001
- T. Lyons and N. Victoir, Cubature on Wiener space. Proc. Royal Soc. London468 (2004) 169–198. Zbl1055.60049
- T. Lyons, M. Caruana and T. Levy, Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics1908. Springer (2004). Zbl1176.60002
- J. Ma and J. Zhang, Representation theorems for Backward Stochastic Differential Equations. Ann. Appl. Prob.12 (2002) 1390–1418. Zbl1017.60067
- J. Ma and J. Zhang, Representation and regularities for solutions to BSDEs with reflections. Stochastic Processes their Appl.115 (2005) 539–569. Zbl1076.60049
- J. Ma, P. Protter and J. Yong, Solving Forward-Backward SDEs expicitly – A four step scheme. Probab. Theor. Relat. Fields122 (1994) 163–190.
- D. Nualart, The Malliavin calculus and related topics. Springer-Verlag (1996). Zbl1099.60003
- E. Pardoux and S. Peng, Adapted solution to Backward Stochastic Differential Equations. Syst. Contr. Lett.14 (1990) 55–61. Zbl0692.93064
- E. Pardoux and S. Peng, Backward Stochastic Differential Equations and quasi linear parabolic partial differential equations, in Lecture Notes in Control and Information Sciences176, Springer, Berlin/Heidelberg (1992) 200–217. Zbl0766.60079
- E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theor. Relat. Fields114 (1999) 123–150. Zbl0943.60057
- S. Peng, Backward SDEs and related g-expectations, in Pitman Research Notes in Mathematics Series364, Longman, Harlow (1997) 141–159. Zbl0892.60066
- S. Peng, Non linear expectations non linear evaluations and risk measures1856. Springer-Verlag (2004).
- S. Peng, Modelling derivatives pricing mechanisms with their generating functions. Preprint, v1 (2006). URIarxiv:math/0605599
- E. Rosazza Giannin, Risk measures via g expectations. Insur. Math. Econ.39 (2006) 19–34. Zbl1147.91346
- S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Contr. Opt.32 (1994) 1447–1475. Zbl0922.49021
- J. Zhang, Some fine properties of backward stochastic differential equations. Ph.D. Thesis, Purdue University, USA (2001).
- J. Zhang, A numerical scheme for BSDEs. Ann. Appl. Prob.14 (2004) 459–488. Zbl1056.60067

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