Electronic Journal of Probability

Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators

Franziska Kühn

Full-text: Open access

Abstract

We study the existence of (Markovian) solutions to the $(A,C_{c}^{\infty }(\mathbb{R} ^{d}))$-martingale problem associated with the Lévy-type operator $A$ with symbol $q(x,\xi )$. Firstly, we establish conditions which ensure the existence of a solution. The main contribution is that our existence result allows for discontinuity in $x \mapsto q(x,\xi )$. Applying the result, we obtain new insights on the existence of weak solutions to a class of Lévy-driven SDEs with Borel measurable coefficients and on the the existence of stable-like processes with discontinuous coefficients. Secondly, we prove a Markovian selection theorem which shows that – under mild assumptions – the $(A,C_{c}^{\infty }(\mathbb{R} ^{d}))$-martingale problem gives rise to a strong Markov process. The result applies, in particular, to Lévy-driven SDEs. We illustrate the Markovian selection theorem with applications in the theory of non-local operators and equations; in particular, we establish under weak regularity assumptions a Harnack inequality for non-local operators of variable order.

Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 16, 26 pp.

Dates
Received: 4 March 2019
Accepted: 26 January 2020
First available in Project Euclid: 5 February 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1580871682

Digital Object Identifier
doi:10.1214/20-EJP424

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 35S05: Pseudodifferential operators 60G51: Processes with independent increments; Lévy processes

Keywords
martingale problem pseudo-differential operator Markovian selection existence result discontinuous coefficents Krylov estimate jump process Lévy-driven stochastic differential equation Harnack inequality viscosity solution

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kühn, Franziska. Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators. Electron. J. Probab. 25 (2020), paper no. 16, 26 pp. doi:10.1214/20-EJP424. https://projecteuclid.org/euclid.ejp/1580871682


Export citation

References

  • [1] Anulova, S., Pragarauskas, H.: Weak Markov solutions of stochastic equations. Lit. Math. J. 17 (1978), 141–155.
  • [2] Barles, G., Chasseigne, E., Imbert, C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. 13 (2011), 1–26.
  • [3] Bass, R. F., Kaßmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Amer. Math. Soc. 357 (2005), 837–850.
  • [4] Bass, R. F., Levin, D. A.: Harnack inequalities for jump processes. Potential Anal. 17 (2002), 375–388.
  • [5] Böttcher, B., Schilling, R. L., Wang, J.: Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Springer Lecture Notes in Mathematics vol. 2099, (vol. III of the “Lévy Matters” subseries). Springer, 2014.
  • [6] Costantini, C., Kurtz, T. G.: Viscosity methods giving uniqueness for martingale problems. Electron. J. Probab. 20 (2015), 1–27.
  • [7] Denk, R., Kupper, M., Nendel, M.: A semigroup approach to nonlinear Lévy processes. To appear: Stoch. Proc. Appl. Preprint arXiv:1710.08130.
  • [8] Ethier, S. N., Kurtz, T. G.: Markov Processes: Characterization and Convergence. Wiley, New York, 1986.
  • [9] Hoh, W.: The martingale problem for a class of pseudo differential operators. Math. Ann. 300 (1994), 121–148.
  • [10] Hoh, W.: Pseudo differential operators with negative definite symbols and the martingale problem. Stoch. Stoch. Rep. 55 (1995), 225–252.
  • [11] Hoh, W.: Pseudo-Differential Operators Generating Markov Processes. Habilitationsschrift. Universität Bielefeld, Bielefeld 1998.
  • [12] Hollender, J.: Lévy-Type Processes under Uncertainty and Related Nonlocal Equations. CreateSpace Independent Publishing Platform, 2016.
  • [13] Imkeller, P., Willrich, N.: Solutions of martingale problems for Lévy-type operators with discontinuous coefficients and related SDEs. Stoch. Proc. Appl. 126 (2016), 703–734.
  • [14] Jacob, N.: Pseudo Differential Operators and Markov Processes II. Imperial College Press/World Scientific, London 2002.
  • [15] Jacob, N.: Pseudo Differential Operators and Markov Processes III. Imperial College Press/World Scientific, London 2005.
  • [16] Kaßmann, M., Mimica, A.: Analysis of jump processes with nondegenerate jumping kernels. Stoch. Proc. Appl. 123 (2013), 629–650.
  • [17] Kolokoltsov, V.: On Markov processes with decomposable pseudo-differential generators. Stoch. Stoch. Rep. 76 (2004), 1045–1129.
  • [18] Krylov, N. V.: On the selection of a Markov process from a system of processes and the construction of quasi-diffusion processes. Math. USSR Izv. 7 (1973), 691–709.
  • [19] Kühn, F.: Probability and Heat Kernel Estimates for Lévy(-Type) Processes. PhD Thesis, Technische Universität Dresden 2016. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-214839
  • [20] Kühn, F.: Lévy-Type Processes: Moments, Construction and Heat Kernel Estimates. Springer Lecture Notes in Mathematics vol. 2187 (vol. VI of the “Lévy Matters” subseries). Springer, 2017.
  • [21] Kühn, F.: On Martingale Problems and Feller Processes. Eletron. J. Probab. 23 (2018), 1–18.
  • [22] Kühn, F.: Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes. Proc. Amer. Math. Soc. 146 (2018), 3591–3604.
  • [23] Kühn, F.: Perpetual integrals via random time changes. Bernoulli 25 (2019), 1755–1769.
  • [24] Kühn, F.: Random time changes of Feller processes. Preprint arXiv:1705.02830.
  • [25] Kühn, F., Schilling, R. L.: On the domain of fractional Laplacians and related generators of Feller processes. J. Funct. Anal. 276 (2019), 2397–2439.
  • [26] Kurenok, V.: Stochastic equations with multidimensional drift driven by Lévy processes. Random Oper. Stoch. Equ. 14 (2006), 311–324.
  • [27] Kurenok, V.: A note on $L^{2}$ estimates of stable integrals with drift. Trans. Amer. Math. Soc. 360 (2008), 925–938.
  • [28] Kurenok, V.: On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes. Int. J. Stoch. Anal. vol. 2012, Article ID 258415.
  • [29] Kurtz, T. G.: Equivalence of stochastic equations and martingale problems. In: Stochastic Analysis 2010. Springer, 2011, pp. 113–130.
  • [30] Schilling, R. L.: Growth and Hölder condtions for the sample paths of Feller processes. Probab. Theory Relat. Fields 112 (1998), 565–611.
  • [31] Schilling, R. L., Schnurr, A.: The Symbol Associated with the Solution of a Stochastic Differential Equation. Electron. J. Probab. 15 (2010), 1369–1393.
  • [32] Schilling, R. L., Wang, J.: Some theorems on Feller processes: Transience, local times and ultracontractivity. Trans. Amer. Math. Soc. 365 (2013), 3255–3286.
  • [33] Wada, M.: Continuity of Harmonic Functions for Non-local Markov Generators. Potential Anal. 39 (2013), 1–11.
  • [34] Zanzotto, P. A.: On stochastic differential equations driven by a Cauchy process and other stable Lévy motions. Ann. Probab. 30 (2002), 802–825.
  • [35] Zhao, G.: Weak uniqueness for SDEs driven by supercritical stable processes with Hölder drifts. Proc. Amer. Math. Soc. 147 (2019), 849–860.