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Solving equations with semimartingale noise

Jonathan Gutierrez-Pavón and Carlos G. Pacheco

Abstract

In this work we focus on a method for solving equations with a coefficient formally given in terms of the derivative of a continuous semimartingale. This generalizes the case of coefficients being the white noise. The idea for solving the equation is to find explicitly the inverse of the ill-posed differential operator, which boils down to finding the associated Green kernel. To find the kernel we give explicitly two homogeneous solutions in terms of the so-called Dolean–Dade exponential. The general idea to define rigorously differential operators lies on dealing with them through bilinear forms. We give several examples with explicit calculations.

MSC 2010: 60K37; 60H25

Communicated by Anatoly F. Turbin


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Received: 2020-04-10
Accepted: 2021-09-04
Published Online: 2022-01-23
Published in Print: 2022-03-01

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