Rate of Convergence for Wong–Zakai-Type Approximations of Itô Stochastic Differential Equations
- 11 Downloads
We consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and we investigate the rate of convergence for Wong–Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the pointwise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Itô equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise toward the original Brownian motion. We also prove, in analogy with a well-known property for exact solutions, that the solutions of approximated Itô equations solve approximated Stratonovich equations with a certain correction term in the drift.
KeywordsStochastic differential equations Wong–Zakai theorem Wick product
Mathematics Subject Classification (2010)60H10 60H30 60H05
The author acknowledges the support of the Italian INDAM-GNAMPA.
- 8.Gyöngy, I., Stinga, P.R.: Rate of convergence of Wong–Zakai approximations for stochastic partial differential equations. In: Seminar on Stochastic Analysis, Random Fields and Applications VII, vol. 67. Birkhäuser, Basel (2013)Google Scholar
- 14.Hu, Y., Øksendal, B.: Wick approximation of quasilinear stochastic differential equations. In: Körezlioğlu, H., Øksendal, B., Üstünel, A.S. (eds.) Stochastic Analysis and Related Topics, vol. V, pp. 203–231, Birkhäuser, Basel (1996)Google Scholar
- 19.Londono, J.A., Villegas, A.M.: Numerical performance of some Wong–Zakai type approximations for stochastic differential equations. Int. J. Pure Appl. Math. 107, 301–315 (2016)Google Scholar
- 22.Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings 6th Berkeley Symposium Mathematical Statistics and Probability, vol. 3, pp. 333–359. University of California Press, Berkeley (1972)Google Scholar