Geodesic
Forward and backward equations for the probability characteristics of Levy type processes in spaces of distributions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 68-78 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the correctness of equations for the probability characteristics of Levy type processes defined by stochastic differential equations. Using the Ito formula and techniques of the theory of generalized functions, we prove the following results. The forward equation for the transition probability of the process is correct on the space of compactly supported twice continuously differentiable functions under the assumptions of the theorem of existence and uniqueness of solutions to the stochastic differential equation, and the backward equation for a probability characteristic of special form is correct on the same space under additional conditions on the smoothness of the coefficients of the stochastic differential equation.
Keywords: Levy type process, Markov process, transition probability
Mots-clés : Ito formula, distribution. 
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V. A. Bovkun. Forward and backward equations for the probability characteristics of Levy type processes in spaces of distributions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 68-78. http://geodesic.mathdoc.fr/item/TIMM_2020_26_2_a4/

[1] Protter P.E., Stochastic integration and differential equations, Springer-Verlag, Berlin; Heidelberg, 2005, 415 pp. | DOI | MR

[2] Kunita H., Stochastic flows and jump-diffusions, Springer, Singapore, 2019, 352 pp. | DOI | MR | Zbl

[3] Kolokoltsov V.N., Markov processes, semigroups and generators, De Gruyter Studies in Mathematics, 38, Birkhäuser, Berlin, 2011, 430 pp. | DOI | MR

[4] Cont R., Tankov P., Financial modelling with jump processes, 1st ed., Chapman and Hall/CRC, N Y, 2003, 552 pp. | DOI | MR

[5] Liu Y., Zhang Y., Wang Q., “A stochastic SIR epidemic model with Levy jump and media coverage”, Advances in Difference Equations, 2020 (2020), 1–15 | DOI | MR

[6] Kunita H., “Ito's stochastic calculus: Its surprising power for applications”, Stochastic Processes and their Applications, 120:5 (2010), 622–652 | DOI | MR | Zbl

[7] Applebaum D., Levy processes and stochastic calculus, Cambridge University Press, Cambridge, 2009, 492 pp. | DOI | MR | Zbl

[8] Gelfand I.M., Shilov G.E., Obobschennye funktsii, v. 1, Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1959, 470 pp.