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Translation invariant diffusions in the space of tempered distributions

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  • B. Rajeev

    (Indian Statistical Institute)

Abstract

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σ ij , b i and initial condition y in the space of tempered distributions) that may be viewed as a generalisation of Ito’s original equations with smooth coefficients. The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients σ ij ★ $$\tilde y$$ , b i ★ $$\tilde y$$ are assumed to be locally Lipshitz.Here ★ denotes convolution and $$\tilde y$$ is the distribution which on functions, is realised by the formula $$\tilde y\left( r \right): = y\left( { - r} \right)$$ . The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.

Suggested Citation

  • B. Rajeev, 2013. "Translation invariant diffusions in the space of tempered distributions," Indian Journal of Pure and Applied Mathematics, Springer, vol. 44(2), pages 231-258, April.
    • Handle: RePEc:spr:indpam:v:44:y:2013:i:2:d:10.1007_s13226-013-0012-0
    DOI: 10.1007/s13226-013-0012-0
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    Cited by:

    1. B. Rajeev & K. Suresh Kumar, 2016. "A class of stochastic differential equations with pathwise unique solutions," Indian Journal of Pure and Applied Mathematics, Springer, vol. 47(2), pages 343-355, June.
    2. Suprio Bhar, 2017. "An Itō Formula in the Space of Tempered Distributions," Journal of Theoretical Probability, Springer, vol. 30(2), pages 510-528, June.

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