The Annals of Applied Probability

Approximations of stochastic partial differential equations

Giulia Di Nunno and Tusheng Zhang

Full-text: Open access


In this paper, we show that solutions of stochastic partial differential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applications to stochastic Burgers equations are discussed.

Article information

Ann. Appl. Probab., Volume 26, Number 3 (2016), 1443-1466.

Received: February 2014
Revised: April 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 93E20: Optimal stochastic control 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Stochastic partial differential equations approximations jump noise tightness weak convergence stochastic Burgers equations


Di Nunno, Giulia; Zhang, Tusheng. Approximations of stochastic partial differential equations. Ann. Appl. Probab. 26 (2016), no. 3, 1443--1466. doi:10.1214/15-AAP1122.

Export citation


  • [1] Albeverio, S., Wu, J.-L. and Zhang, T.-S. (1998). Parabolic SPDEs driven by Poisson white noise. Stochastic Process. Appl. 74 21–36.
  • [2] Asmussen, S. and Rosiński, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38 482–493.
  • [3] Benth, F. E., Di Nunno, G. and Khedher, A. (2011). Robustness of option prices and their deltas in markets modelled by jump-diffusions. Commun. Stoch. Anal. 5 285–307.
  • [4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [5] Chow, P.-L. (2007). Stochastic Partial Differential Equations. Chapman & Hall/CRC, Boca Raton, FL.
  • [6] da Silva, J. L. and Erraoui, M. (2011). The $\alpha$-dependence of stochastic differential equations driven by variants of $\alpha$-stable processes. Comm. Statist. Theory Methods 40 3465–3478.
  • [7] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [8] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. Cambridge Univ. Press, Cambridge.
  • [9] Di Nunno, G., Khedher, A. and Vanmaele, M. (2015). Robustness of quadratic hedging in finance via backward stochastic differential equations. Appl. Math. Optim. 72 353–389.
  • [10] Evans, L. C. (1998). Partial Differential Equations. Amer. Math. Soc., Providence, RI.
  • [11] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin.
  • [12] Gyöngy, I. (1982). On stochastic equations with respect to semimartingales. III. Stochastics 7 231–254.
  • [13] Gyöngy, I. and Krylov, N. V. (1980/81). On stochastic equations with respect to semimartingales. I. Stochastics 4 1–21.
  • [14] Gyöngy, I. and Krylov, N. V. (1981/82). On stochastics equations with respect to semimartingales. II. Itô formula in Banach spaces. Stochastics 6 153–173.
  • [15] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [16] Jakubowski, A. (1986). On the Skorokhod topology. Ann. Inst. Henri Poincaré Probab. Stat. 22 263–285.
  • [17] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [18] Krylov, N. V. and Rozovskiĭ, B. L. (1979). Stochastic evolution equations. In Current Problems in Mathematics, Vol. 14 (Russian) 71–147. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow.
  • [19] Liu, W. and Röckner, M. (2013). Local and global well-posedness of SPDE with generalized coercivity conditions. J. Differential Equations 254 725–755.
  • [20] Pardoux, E. (1979). Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 127–167.
  • [21] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin.
  • [22] Röckner, M. and Zhang, T. (2007). Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. Potential Anal. 26 255–279.
  • [23] Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam.