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Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory

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  • Arnold, Ludwig
  • Imkeller, Peter

Abstract

Let u(t, x), t [epsilon] R, be an adapted process parametrized by a variable x in some metric space X, [mu]([omega], dx) a probability kernel on the product of the probability space [Omega] and the Borel sets of X. We deal with the question whether the Stratonovich integral of u(., x) with respect to a Wiener process on [Omega] and the integral of u(t,.) with respect to the random measure [mu](., dx) can be interchanged. This question arises, for example, in the context of stochastic differential equations. Here [mu](., dx) may be a random Dirac measure [delta][eta](dx), where [eta] appears as an anticipative initial condition. We give this random Fubini-type theorem a treatment which is mainly based on ample applications of the real variable continuity lemma of Garsia, Rodemich and Rumsey. As an application of the resulting "uniform Stratonovich calculus" we give a rigorous verification of the diagonalization algorithm of a linear system of stochastic differential equations.

Suggested Citation

  • Arnold, Ludwig & Imkeller, Peter, 1996. "Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 19-54, March.
    • Handle: RePEc:eee:spapps:v:62:y:1996:i:1:p:19-54
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    References listed on IDEAS

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    1. Imkeller, Peter, 1994. "On the perturbation problem for occupation densities," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 41-55, January.
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    Cited by:

    1. Nakahiro Yoshida, 2011. "Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(3), pages 431-479, June.
    2. Mirzaee, Farshid & Hadadiyan, Elham, 2017. "Solving system of linear Stratonovich Volterra integral equations via modification of hat functions," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 254-264.
    3. Mohammed, Salah & Zhang, Tusheng, 2009. "Anticipating stochastic differential systems with memory," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2773-2802, September.

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