Abstract
A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper.
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References
Applebaum, D: Lévy processes and stochastic calculus. Cambridge University Press, Cambridge (2009)
Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.): Levy Processes. Theory and Applicationś. Birkhäuser, Cambridge (2001)
Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications, lecture notes of the unione matematica italiana. Springer International Publishing, Switzerland (2016)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7-9), 1245–1260 (2007)
Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Inventiones Math. 171(2), 425–461 (2008)
Chang, T., Lee, K.: On a stochastic partial differential equation with a fractional Laplacian operator. Stoch. Process. Appl. 122(9), 3288–3311 (2012)
Chen, Z., Kim, K.: An Lp-theory for non-divergence form SPDEs driven by Lévy processes. Forum Mathematicum 26(5), 1381–1411 (2014)
Cont, R., Tankov, P.: Financial modelling with jump processes. Chapman & Hall CRC Press Company, London (2004)
Dareiotis, K.: A note on degenerate stochastic integro–differential equations arXiv:1406.5649v1 (2014)
Freidlin, M.I.: On the factorization of nonnegative definite matrices. Teor. Veroyatn. Primen. 33(2), 375–378 (1968)
Gordon, W.B.: On the diffeomorphisms of euclidean space. Am. Math. Monthly 79(7), 755–759 (1972)
Garroni, M.G., Menaldi, J.L.: Second order elliptic integro-differential problems. Chapman & Hall CRC Press Company, London (2002)
Garroni, M.G., Menaldi, J.L.: Green functions for second order parabolic integro-differential problems. Longman, UK (1992)
Gerencsér, M., Gyöngy, I.: Finite difference schemes for stochastic partial differential equations in Sobolev spaces. Appl. Math. Optim. 72(1), 77–100 (2015)
Gerencsér, M., Gyöngy, I., Krylov, N.V.: On the solvability of degenerate stochastic partial differential equations in Sobolev spaces. Stoch. Partial Differ. Equ. Anal. Comput. 3(1), 52–83 (2015)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes North-Holland (2011)
Jacob, N.: Pseudo-Differential Operators and markov processes: generators and potential theory imperial college press (2002)
Krylov, N.V.: On Lp-theory of stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 27(2), 313–340 (1996)
Kim, K., Kim, P.: An Lp-theory of a class of stochastic equations with the random fractional Laplacian driven by lévy processes. Stoch. Process. Appl. 122(12), 3921–3952 (2012)
Kim, K., Lee, K.: A note on \(W^{\gamma }_p\)-theory of linear stochastic parabolic partial differential systems. Stoch. Process. Appl. 123, 76–90 (2013)
Krylov, N.V.: Lectures on elliptic and parabolic equations in sobolev spaces graduate studies in mathematics, vol. 96. AMS, Providence, Rhode Island (2008)
Krylov, N.V.: Itô’s formula for the Lp-norm of stochastic \(W_{p}^{1}\)-valued processes. Probab. Theory Relat. Fields 147, 583–605 (2010)
Krylov, N.V., Rozovskii, B.L.: On the Cauchy problem for linear stochastic partial differential equations. Math. USSR Izvestija 11(6), 1267–1284 (1977)
Krylov, N.V., Rozovskii, B.L.: Characteristics of degenerating second-order parabolic Itô equations. J. Soviet Maths. 32, 336–348 (1986). (Translated from Trudy Seminara imeni I.G. Petrovskogo, No. 8. pp.. 153-168, 1982.)
Leahy, J.-M., Mikulevicius, R.: On classical solutions of linear stochastic integro-differential equations. Stoch. Partial Differ. Equ. Anal. Comput. 4(3), 535–591 (2016)
Leahy, J.-M., Mikulevicius, R.: On degenerate linear stochastic evolution equations driven by jump processes. Stoch. Process. Appl. 125(10), 3748–3784 (2015)
Mikulevičius, R., Phonsom, C.: On Lp-theory for parabolic and elliptic integro-differential equations with scalable operators in the whole space. Stoch. Partial Differ. Equ. Anal. Comput. 5(4), 472–519 (2017)
Mikulevičius, R., Phonsom, C.: On the Chauchy problem for stochastic integro-differential parabolic equations in the scale of Lp-spaces of generalised smoothness arXiv:1805.03232v1 (2018)
Mikulevičius, R., Pragarauskas, H.: On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces. Liet. Mat. Rink. 32(2), 299–331 (1992). translation in Lithuanian Math. J. 32 (1992), no.2 238–264 (1993)
Mikulevičius, R., Pragarauskas, H.: On Classical Solutions of Certain Nonlinear Integro-Differential Equations. Stochastic Processes and Optimal Control (Friedrichroda, 1992), 151-163 Stochastics Monogr., vol. 7. Gordon and Breach, Montreux (1993)
Mikulevicius, R., Pragarauskas, H.: On Lp-theory for stochastic parabolic integro-differential equations. Stoch. Partial Differ. Equ. Anal. Comput. 1(2), 282–324 (2013)
Mikulevičius, R., Xu, F.: On the Cauchy proble for parabolic integro-differential equations in generalised Hölder spaces, arXiv:1806.07019v1 (2018)
Murray, J.D.: Mathematical biology: I. An Introduction, 3rd edn. Springer, New York (2007)
Oleı̆nik, O.A.: Alcuni risultati sulle equazioni lineari e quasi lineari ellittico-paraboliche a derivate parziali del secondo ordine, (Italian). Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat Natur. 40(8), 775–784 (1966)
Oleı̆nik, O.A.: On the smoothness of solutions of degenerating elliptic and parabolic equations. Dokl Akad. Nauk SSSR 163, 577–580 (1965). in Russian; English translation in Soviet Mat. Dokl. Vol, 6 (1965), No. 3, 972–976
Oleı̆nik, O.A., Radkevič, E.V.: Second order equations with nonnegative characteristic form, Mathematical Analysis, 1969, pp. 7-252. (errata insert) Akad. Nauk SSSR, Vsesojuzn. Inst. Naučn. i Tehn. Informacii, Moscow, 1971 in Russian, English translation: Plenum Press New York-London (1973)
Olejnik, O.A, Radkevich, E.V.: Second order equations with nonnegative characteristic form, AMS, Providence (1973)
Pascucci, A.: PDE And martingale methods in option pricing. Springer, Italia (2011)
Pragarauskas, G.: The first boundary value problem for a certain class of integro-differential equations. Litovsk. Mat. Sb. 14(4), 195–200 (1974)
Pragarauskas, G.: On the Bellman equation for weakly degenerate general random processes. Litovsk. Mat. Sb. 20(2), 129–136 (1980)
Rozovskii, B.L., Systems, Stochastic Evolution: Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht (1990)
Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Differ. Equ. 35, 2092–2122 (2010)
Triebel, H.: Interpolation Theory - Function Spaces - Differential Operators, North holland publishing company Amsterdam-New York-Oxford (1978)
Acknowledgments
A generalisation of Theorem 2.1 to stochastic integro-differential equations was presented at the conference on “Harmonic Analysis for Stochastic PDEs” in Delft, 10-13 July, 2018 and at the “9th International Conference on Stochastic Analysis and Its Applications” in Bielefeld, 3-7 September, 2018. The authors are grateful to the organisers of these conferences for the invitation and for discussions. They thank Stefan Geiss and Mark Veraar for useful discussions and Alexander Davie for correcting some mistakes. The authors are also thankful to the referees for their comments which helped to improve the presentation of the paper.
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De León-Contreras, M., Gyöngy, I. & Wu, S. On Solvability of Integro-Differential Equations. Potential Anal 55, 443–475 (2021). https://doi.org/10.1007/s11118-020-09864-2
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DOI: https://doi.org/10.1007/s11118-020-09864-2