# Introduction to Finite Differences Methods

Chapter
Part of the Fields Institute Monographs book series (FIM, volume 29)

## Abstract

In this lecture, I discuss the practical aspects of designing Finite Difference methods for Hamilton–Jacobi–Bellman equations of parabolic type arising in quantitative finance. The approach is based on the very powerful and simple framework developed by Barles– Souganidis [4], see the review of the previous chapter. The key property here is the monotonicity which guarantees that the scheme satisfies the same ellipticity condition as the HJB operator. I will provide a number of examples of monotone schemes in these notes. In practice, pure finite difference schemes are only useful in 1, 2, or at most 3 spatial dimensions. One of their merits is to be quite simple and easy to implement. Also, as shown in the previous chapter, they can also be combined with Monte Carlo methods to solve nonlinear parabolic PDEs.

## Keywords

Truncation Error Directional Derivative Implicit Scheme Previous Chapter Singular Control
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
P. Bank, D. Baum, Hedging and portfolio optimization in financial markets with a large trader. Math. Finance. 14, 1–18 (2004)
2. 2.
G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi. Mathématiques & Applications (Springer, New York, 1994)Google Scholar
3. 3.
G. Barles, E.R. Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Math. Comput. 76, 1861–1893 (2007)
4. 4.
G. Barles, P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4, 271–283 (1991)
5. 5.
G. Barles, C. Daher, M. Romano, Convergence of numerical schemes for parabolic equations arising in finance theory. Math. Model Meth. Appl. Sci. 5, 125–143 (1995)
6. 6.
J.-M. Bismut, Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404 (1973)
7. 7.
F. Bonnans, H. Zidani, Consistency of generalized finite difference schemes for the stochastic HJB equation. Siam J. Numer. Anal. 41, 1008–1021 (2003)
8. 8.
U. Cetin, R. Jarrow, P. Protter, Liquidity risk and arbitrage pricing theory. Finance Stochast. 8, 311–341 (2004).
9. 9.
U. Cetin, M. Soner, N. Touzi, Option hedging under liquidity costs. Finance Stochast. 14(3), 317–341 (2010).
10. 10.
U. Cetin, R. Jarrow, P. Protter, M. Warachka, Pricing options in an extended black-scholes economy with illiquidity: theory and empirical evidence. Rev. Financ. Stud. 19, 493–529 (2006).
11. 11.
P. Cheridito, M. Soner, N. Touzi, Small time path behavior of double stochastic integrals, and application to stochastic control. Ann. Appl. Probab. 15, 2472–2495 (2005)
12. 12.
P. Cheridito, M. Soner, N. Touzi, The multi-dimensional super-replication problem under gamma constraints. Annales de l’Institut Henri Poincaré, Série C: Analyse Non-Linéaire. 22, 633–666 (2005)Google Scholar
13. 13.
P. Cheridito, M. Soner, N. Touzi, N. Victoir, Second order backward stochastic differential equations and fully non-linear parabolic pdes. Comm. Pure Appl. Math. 60, (2007)Google Scholar
14. 14.
M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)
15. 15.
J. Dugundji, Topology (Allyn and Bacon series in Advanced Mathematics, Allyn and Bacon edt.) (1966)Google Scholar
16. 16.
N. ElKaroui, M. Jeanblanc-Picqué, Contrôle de processus de markov, Séminaire de Probabilités Springer Lecture Notes in Math. vol. 1321 (1988), pp. 508–541Google Scholar
17. 17.
N. ElKaroui, R. Rouge, Pricing via utility maximization and entropy. Math. Finance. 10, 259–276 (2000)
18. 18.
N. ElKaroui, S. Peng, M.-C. Quenez, Backward stochastic differential equations in fiannce. Math. Finance. 7, 1–71 (1997)
19. 19.
A. Fahim, N. Touzi, X. Warin, A probabilistic numerical method for fully nonlinear parabolic PDEs. 21(4), 1322–1364 (2011)Google Scholar
20. 20.
W.H. Fleming, H.M. Soner, Controlled Markov Processes and Viscosity Solutions (Springer, New York, 1993)
21. 21.
H. Foellmer, P. Leukert, Quantile hedging. Finance Stochas. 3, 251–273 (1999)
22. 22.
Y. Hu, P. Imkeller, M. Müller, Utility maximization in incomplete markets. Ann. Appl. Probab. 15, 1691–1712 (2005)
23. 23.
I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)
24. 24.
I. Karatzas, S. Shreve, Methods of Mathematical Finance, 2nd edn. (Springer, New York, 1999)Google Scholar
25. 25.
N. Kazamaki, Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics (Springer, Berlin 1579)Google Scholar
26. 26.
M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Appl. Probab. 28, 558–602 (2000)
27. 27.
N.V. Krylov, Controlled Diffusion Processes (Springer, New York, 1977)
28. 28.
H. J. Kushner, P.G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time (Springer, New York, 1992)
29. 29.
R.C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time model. Rev. Econ. Stat. 51, 247–257 (1969)
30. 30.
R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theor. 3, 373–413 (1971)
31. 31.
M.-A. Morlais, Equations Différentielles Stochastiques Rétrogrades à Croissance Quadratique et Applications. Ph.D. thesis, Université de Rennes, 2007Google Scholar
32. 32.
E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Contr. Lett. 14, 55–61 (1990)
33. 33.
D. Pooley, P. Forsyth, K. Vetzal, Numerical convergence properties of option pricing pdes with uncertain volatility. IMA J. Numer. Anal. 23, 241–267 (2003)
34. 34.
P.J. Reny, On the existence of pure and mixed strategy nash equilibria in discontinuous games. Econometrica. 67, 1029–1056 (1999)
35. 35.
R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)
36. 36.
H.M. Soner, N. Touzi, Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201–236 (2002)
37. 37.
M. Soner, N. Touzi, The dynamic programming equation for second order stochastic target problems. SIAM J. Contr. Optim. 48(4), 2344–2365 (2009)
38. 38.
H.M. Soner, N. Touzi, J. Zhang, Wellposedness of second order backward sdes. Probability Theory and Related Fields 153, 149–190 (2012)
39. 39.
R. Tevzadze, Solvability of backward stochastic differential equation with quadratic growth. Stochast. Process. Appl. 118, 503–515 (2008)
40. 40.
J. Thomas, Numerical Partial Differential Equations: Finite Difference methods (Texts in Applied Mathematics) (Springer, New York, 1998)Google Scholar
41. 41.
J. Wang, P. Forsyth, Maximal use of central differencing for hamilton-jacobi-bellman pdes in finance. Siam J. Numer. Anal. 46, 1580–1601 (2008)
42. 42.
T. Zariphopoulou, A solution approach to valuation with unhedgeable risks. Finance Stochast. 5, 61–82 (2001)
43. 43.
R. Zvan, K.R. Vetzal, P.A. Forsyth PDE methods for pricing barrier options. J. Econom. Dynam. Control 24, 1563–1590 (2000)Google Scholar