Open Access
2020 Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators
Franziska Kühn
Electron. J. Probab. 25: 1-26 (2020). DOI: 10.1214/20-EJP424


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.


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Franziska Kühn. "Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators." Electron. J. Probab. 25 1 - 26, 2020.


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

zbMATH: 1448.60162
Digital Object Identifier: 10.1214/20-EJP424

Primary: 60J35
Secondary: 35S05 , 45K05 , 60G51 , 60H10 , 60J25 , 60J75

Keywords: discontinuous coefficents , Existence result , Harnack inequality , jump process , Krylov estimate , Lévy-driven stochastic differential equation , Markovian selection , Martingale problem , ‎pseudo-differential operator , viscosity solution

Vol.25 • 2020
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