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Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise

Author

Listed:
  • Barbu, Viorel
  • Brzeźniak, Zdzisław
  • Hausenblas, Erika
  • Tubaro, Luciano

Abstract

The solution Xn to a nonlinear stochastic differential equation of the form dXn(t)+An(t)Xn(t)dt−12∑j=1N(Bjn(t))2Xn(t)dt=∑j=1NBjn(t)Xn(t)dβjn(t)+fn(t)dt, Xn(0)=x, where βjn is a regular approximation of a Brownian motion βj, Bjn(t) is a family of linear continuous operators from V to H strongly convergent to Bj(t), An(t)→A(t), {An(t)} is a family of maximal monotone nonlinear operators of subgradient type from V to V′, is convergent to the solution to the stochastic differential equation dX(t)+A(t)X(t)dt−12∑j=1NBj2(t)X(t)dt=∑j=1NBj(t)X(t)dβj(t)+f(t)dt, X(0)=x. Here V⊂H≅H′⊂V′ where V is a reflexive Banach space with dual V′ and H is a Hilbert space. These results can be reformulated in terms of Stratonovich stochastic equation dY(t)+A(t)Y(t)dt=∑j=1NBj(t)Y(t)∘dβj(t)+f(t)dt.

Suggested Citation

  • Barbu, Viorel & Brzeźniak, Zdzisław & Hausenblas, Erika & Tubaro, Luciano, 2013. "Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 934-951.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:3:p:934-951
    DOI: 10.1016/j.spa.2012.10.008
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    References listed on IDEAS

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    1. Viorel Barbu, 2012. "Optimal Control Approach to Nonlinear Diffusion Equations Driven by Wiener Noise," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 1-26, April.
    2. Duncan, T.E. & Maslowski, B. & Pasik-Duncan, B., 2005. "Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise," Stochastic Processes and their Applications, Elsevier, vol. 115(8), pages 1357-1383, August.
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    Cited by:

    1. Tölle, Jonas M., 2020. "Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3220-3248.

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