Summary
We establish the existence and uniqueness of the solution to a multidimensional linear Skorohod stochastic differential equation with deterministic diffusion matrix, using the notions of Wick product andStransform. If the diffusion matrix is constant and has real eigenvalues, the solution is a stochastic process with moments of all orders, provided that the initial condition is differentiable up to a suitable order. The case of a diffusion matrix in the first Wiener chaos is discussed in the last section.
References
Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. de Gruyter Studies in Math.14, 1991
Buckdahn, R.: Quasilnnear partial stochastic differential equations without nonanticipation requirement. (Preprint)
Buckdahn, R.: Linear Skorohod stochastic differential equations. Probab. Theory Relat. Fields90, 223–240 (1991)
Buckdahn, R.: Anticipative Girsanov transformations. Probab. Theory Relat. Fields89, 221–238 (1991)
Buckdahn, R., Nualart, D.: Skorohod stochastic differential equations with boundary conditions. Stochastics Stochastics Rep. (to appear)
Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields78, 80–129 (1988)
Nualart, D., Zakai, M.: Positive and strongly positive Wiener functionals. In: Barcelona seminar on stochastic analysis. Progress in Probability, vol. 32, pp. 132–146. Basel Boston Berlin: Birkhäuser 1993
Lindstrøm, T., Øksendal, B., Ubøe, J.: Wick multiplication and Ito-Skorohod stochastic differential equations. In: S. Albeverio, J.E. Fernstad, H. Holden, T. Lindstrøm, (eds.) Ideas and methods in mathematical anlysis, stochastics and applications, pp. 183–206. Cambridge University Press 1992
Skorohod, A.V.: On a generalization of a stochastic integral. Theory Probab. Appl.20, 219–233 (1975)
Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research. Berlin Heidelberg New York: Springer 1984
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Supported by the Deutsche Forschungsgemeninschaft/Heisenberg Programm
Supported by the DGICYT grant PB 90-0452
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Buckdahn, R., Nualart, D. Linear stochastic differential equations and Wick products. Probab. Th. Rel. Fields 99, 501–526 (1994). https://doi.org/10.1007/BF01206230
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DOI: https://doi.org/10.1007/BF01206230
Mathematics Subject Classification
- 60H10
- 60H07