Generalized backward doubly stochastic differential equations driven by Lévy processes with continuous coefficients

Abstract

A new class of generalized backward doubly stochastic differential equations (GBDS-DEs in short) driven by Teugels martingales associated with Lévy process are investigated. We establish a comparison theorem which allows us to derive an existence result of solutions under continuous and linear growth conditions. MSC:Primary: 60F05, 60H15; Secondary: 60J30

Lp - estimates of solutions of backward doubly stochastic differential equations

“…In present paper, we extend the previous results of stability to nonhomogeneous BDSDE (2). The main idea of this paper is to study the L p -stability, p ≥ 2, for Eq.…”

“…The following propositions give the existence and uniqueness results for the homogenous and nonhomogeneous Eqs. (1) and (2).…”

“…Since then, many authors tried to weaken the conditions for the functions f and and to give more general results. For example, Lin [14], Aman [1], Aman and Owo [2], N'zi and Owo [18,19], Boufoussi, Casteren and Mrhardy [4], Ren, Lin and Hu [26], Hu and Ren [12]. All previous papers deal with BDSDEs which have only the process z in the forward integral, i.e.…”

“…In order to estimate J 2 , it should be noted that from hypothesis (H 3 ) it follows that there exists a constant ε 0 > 0 such that || (t, y, z)|| 2 ≤ (1 + ε 0 )ρ 2 p (|y| 2 ) + α(1 + ε 0 )||z|| 2…”

Generalized Backward doubly SDEs driven by Lévy processes with discontinuous and linear growth coefficients

“…The proof is strongly linked to the comparison theorem which does not hold in general for BDSDEs with jumps (see the counter-example in Buckdahn et al [5]). To overcome this difficulty, they assumed an additional relation between the generator f and the jumps size of the Lévy process L. Note that, in the previous works on GBDSDEs driven by Lévy process, the generators are at least continuous (see Hu and Ren [10] for Lipschitz continuous, and Aman and Owo [1] for continuous and linear growth). But, unfortunately, the continuous conditions can not be satisfied in certain models that makes the results in [10] and [1] not applied for several applications (finance, stochastic control, stochastic games, SPDEs, etc,...).…”