Fredholm Determinant of an Integral Operator Driven by a Diffusion Process

Adrian P. C. Lim

International Journal of Stochastic Analysis, 2008, vol. 2008, 1-17

Abstract:

This article aims to give a formula for differentiating, with respect to 𠑇 , an expression of the form 𠜆 ( 𠑇 , ð ‘¥ ) ∶ = ð ”¼ ð ‘¥ [ ð ‘“ ( ð ‘‹ 𠑇 ) ð ‘’ − ∫ 𠑇 0 𠑉 ( ð ‘‹ ð ‘ ) ð ‘‘ ð ‘ ( d e t ( ð ¼ + ð ¾ ð ‘‹ , 𠑇 ) ) 𠑃 ] , where ð ‘ â‰¥ 0 and ð ‘‹ is a diffusion process starting from ð ‘¥ , taking values in a manifold, and the expectation is taken with respect to the law of this process. ð ¾ ð ‘‹ , 𠑇 ∶ ð ¿ 2 ( [ 0 , 𠑇 ) → â„ ð ‘ ) → ð ¿ 2 ( [ 0 , 𠑇 ) → â„ ð ‘ ) is a trace class operator defined by ð ¾ ð ‘‹ , 𠑇 ∫ ð ‘“ ( ð ‘ ) = 𠑇 0 â‹€ ð » ( ð ‘ ð ‘¡ ) Γ ( ð ‘‹ ( ð ‘¡ ) ) ð ‘“ ( ð ‘¡ ) ð ‘‘ ð ‘¡ , where ð » , Γ are locally Lipschitz, positive ð ‘ Ã— ð ‘ matrices.

Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:130940

DOI: 10.1155/2008/130940

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