The Annals of Applied Probability

A strong order $1/2$ method for multidimensional SDEs with discontinuous drift

Gunther Leobacher and Michaela Szölgyenyi

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Abstract

In this paper, we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order $1/2$. Our result is the first one that shows existence and uniqueness as well as strong convergence for such a general class of SDEs.

The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler–Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE.

As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle and we present an application from stochastic optimal control.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 4 (2017), 2383-2418.

Dates
Received: July 2016
First available in Project Euclid: 30 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1504080036

Digital Object Identifier
doi:10.1214/16-AAP1262

Mathematical Reviews number (MathSciNet)
MR3693529

Zentralblatt MATH identifier
1373.60102

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 65C30: Stochastic differential and integral equations 65C20: Models, numerical methods [See also 68U20]
Secondary: 65L20: Stability and convergence of numerical methods

Keywords
Stochastic differential equations discontinuous drift degenerate diffusion existence and uniqueness of solutions numerical methods for stochastic differential equations strong convergence rate

Citation

Leobacher, Gunther; Szölgyenyi, Michaela. A strong order $1/2$ method for multidimensional SDEs with discontinuous drift. Ann. Appl. Probab. 27 (2017), no. 4, 2383--2418. doi:10.1214/16-AAP1262. https://projecteuclid.org/euclid.aoap/1504080036


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References

  • [1] Berkaoui, A. (2004). Euler scheme for solutions of stochastic differential equations with non-Lipschitz coefficients. Port. Math. (N.S.) 61 461–478.
  • [2] Étoré, P. and Martinez, M. (2013). Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process. Monte Carlo Methods Appl. 19 41–71.
  • [3] Étoré, P. and Martinez, M. (2014). Exact simulation for solutions of one-dimensional stochastic differential equations with discontinuous drift. ESAIM Probab. Stat. 18 686–702.
  • [4] Foote, R. L. (1984). Regularity of the distance function. Proc. Amer. Math. Soc. 92 153–155.
  • [5] Gyöngy, I. (1998). A note on Euler’s approximations. Potential Anal. 8 205–216.
  • [6] Halidias, N. and Kloeden, P. E. (2008). A note on the Euler–Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient. BIT 48 51–59.
  • [7] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2012). Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 1611–1641.
  • [8] Itô, K. (1951). On stochastic differential equations. Mem. Amer. Math. Soc. 4 1–57.
  • [9] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [10] Kloeden, P. E. and Platen, E. (1992). Numerical Solutions of Stochastic Differential Equations. Springer, Berlin–Heidelberg.
  • [11] Kohatsu-Higa, A., Lejay, A. and Yasuda, K. (2013). Weak approximation errors for stochastic differential equations with non-regular drift. Preprint, Inria, hal-00840211.
  • [12] Krantz, S. G. and Parks, H. R. (1981). Distance to $C^{k}$ hypersurfaces. J. Differential Equations 40 116–120.
  • [13] Leobacher, G. and Szölgyenyi, M. (2016). A numerical method for SDEs with discontinuous drift. BIT 56 151–162.
  • [14] Leobacher, G., Szölgyenyi, M. and Thonhauser, S. (2014). Bayesian dividend optimization and finite time ruin probabilities. Stoch. Models 30 216–249.
  • [15] Leobacher, G., Szölgyenyi, M. and Thonhauser, S. (2015). On the existence of solutions of a class of SDEs with discontinuous drift and singular diffusion. Electron. Commun. Probab. 20 no. 6, 1–14.
  • [16] Ruzhansky, M. and Sugimoto, M. (2015). On global inversion of homogeneous maps. Bull. Math. Sci. 5 13–18.
  • [17] Sabanis, S. (2013). A note on tamed Euler approximations. Electron. Commun. Probab. 18 no. 47, 1–10.
  • [18] Shardin, A. A. and Szölgyenyi, M. (2016). Optimal control of an energy storage facility under a changing economic environment and partial information. Int. J. Theor. Appl. Finance 19 1650026, 1–27.
  • [19] Shardin, A. A. and Wunderlich, R. (2017). Partially observable stochastic optimal control problems for an energy storage. Stochastics 89 280–310.
  • [20] Szölgyenyi, M. (2016). Dividend maximization in a hidden Markov switching model. Stat. Risk Model. 32 143–158.
  • [21] Veretennikov, A. Yu. (1983). Criteria for the existence of a strong solution of a stochastic equation. Theory Probab. Appl. 27 441–449.
  • [22] Veretennikov, A. Y. U. (1981). On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR, Sb. 39 387–403.
  • [23] Veretennikov, A. Y. U. (1984). On stochastic equations with degenerate diffusion with respect to some of the variables. Math. USSR, Izv. 22 173–180.
  • [24] Zvonkin, A. K. (1974). A transformation of the phase space of a diffusion process that removes the drift. Math. USSR, Sb. 22 129–149.