Abstract
We prove Itô’s formula for the L p -norm of a stochastic \({W^{1}_{p}}\) -valued processes appearing in the theory of SPDEs in divergence form.
References
Brzeźniak Z., van Neerven J.M.A.M., Veraar M.C., Weis L.: Ito’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differ. Equ. 245, 30–58 (2008)
Kallianpur G., Striebel C.: Stochastic differential equations occurring in the estimation of continuous parameter stochastic processes. Teor. Verojatnost. i Primenen. 14(4), 597–622 (1969)
Krylov, N.V.: An analytic approach to SPDEs. In: Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, vol. 64, pp. 185–242. AMS, Providence, RI (1999)
Krylov, N.V.: On parabolic PDEs and SPDEs in Sobolev spaces \({W^{2}_{p}}\) without and with weights. In: Chow, P.-L., Mordukhovich, B., Yin, G. (eds.) Topics in Stochastic Analysis and Nonparametric Estimation. IMA Volumes in Mathematics and its Applications, vol. 145, pp. 151–198. Springer, New York (2008)
Krylov N.V.: On divergence form SPDEs with VMO coefficients. SIAM J. Math. Anal. 40(6), 2262–2285 (2009)
Krylov, N.V.: Filtering equations for partially observable diffusion processes with Lipschitz continuous coefficients. In: The Oxford Handbook of Nonlinear Filtering. Oxford University Press, Oxford (To appear)
Krylov, N.V., Rozovsky, B.L.: Stochastic evolution equations, “Itogy nauki i tekhniki” 14, VINITI, Moscow, 71–146 (1979). In Russian; English translation in J. Soviet Math. 16(4), 1233–1277 (1981)
Rozovskii B.L.: Stochastic Evolution Systems. Kluwer, Dordrecht (1990)
van Neerven, J.M.A.M., Veraar, M.: On the stochastic Fubini theorem in infinite dimensions. In: Stochastic Partial Differential Equations and Applications—VII. Volume 245 of Lect. Notes Pure Appl. Math., pp. 323–336. Chapman & Hall/CRC, Boca Raton, FL (2006)
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The work was partially supported by NSF Grant DMS-0653121.
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Krylov, N.V. Itô’s formula for the L p -norm of stochastic \({W^{1}_{p}}\) -valued processes. Probab. Theory Relat. Fields 147, 583–605 (2010). https://doi.org/10.1007/s00440-009-0217-7
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DOI: https://doi.org/10.1007/s00440-009-0217-7
Keywords
- Stochastic partial differential equations
- Divergence equations
- Itô’s formula
Mathematics Subject Classification (2000)
- 60H15
- 35R60