Linear inverse problems for Markov processes and their regularisation

Cetin, Umut ORCID: 0000-0001-8905-853X (2019) Linear inverse problems for Markov processes and their regularisation. Stochastic Processes and Their Applications. ISSN 0304-4149

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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.

Item Type:	Article
Official URL:	https://www.journals.elsevier.com/stochastic-proce...
Divisions:	Statistics
Subjects:	H Social Sciences > HA Statistics
Date Deposited:	26 Nov 2019 11:48
Last Modified:	06 Apr 2024 22:24
URI:	http://eprints.lse.ac.uk/id/eprint/102633

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