Abstract
The connections between stochastic differential equations in which continuous and discontinuous random processes serve as sources of randomness and deterministic equations for the probabilistic characteristics of solutions of these stochastic equations are studied. In the study, we use various approaches based on the stochastic change of variables formula (Itô’s formula), on the analysis of local infinitesimal characteristics of the process, and on the theory of semigroups of operators in combination with the generalized Fourier transform. This allows us to obtain direct and inverse integro-differential equations for various probabilistic characteristics.
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Notes
\(A\in \mathcal {B}(\mathbb {R}\setminus \{0\})\) and \(0\notin \overline {A}\).
As is customary in stochastic analysis, a strong solution is a process that a.s. satisfies relation (3) and is consistent with the filtering \((\mathcal {F}_t)_{t\geq 0}\) generated by random perturbations entering the equation.
Further, to be concise, instead of the “transition probability density function” we write “density” everywhere where this does not lead to confusion.
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This work was supported by Act no. 211 of the Government of the Russian Federation, contract no. 02.A03.21.0006.
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Translated by V. Potapchouck
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Melnikova, I.V., Bovkun, V.A. & Alekseeva, U.A. Integro-Differential Equations Generated by Stochastic Problems. Diff Equat 57, 379–390 (2021). https://doi.org/10.1134/S0012266121030101
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DOI: https://doi.org/10.1134/S0012266121030101