Linear inverse problems for Markov processes and their regularisation

Umut Cetin

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: We study the solutions of the inverse problem g(z)=∫f(y)P T(z,dy)for a given g, where (P t(⋅,⋅)) t≥0 is the transition function of a given symmetric Markov process, X, and T is a fixed deterministic time, which is linked to the solutions of the ill-posed Cauchy problem u t+Au=0,u(0,⋅)=g,where A is the generator of X. A necessary and sufficient condition ensuring square integrable solutions is given. Moreover, a family of regularisations for above problems is suggested. We show in particular that these inverse problems have a solution when X is replaced by ξX+(1−ξ)J, where ξ is a Bernoulli random variable and J is a suitably constructed jump process. The probability of success for ξ can be chosen arbitrarily close to 1 and thereby leading to a jump component whose jumps are rarely visible in the practical implementations of the regularisation.

JEL-codes: C1 (search for similar items in EconPapers)
Pages: 21 pages
Date: 2019-12-04
New Economics Papers: this item is included in nep-ore
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Published in Stochastic Processes and Their Applications, 4, December, 2019. ISSN: 0304-4149

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