$${{\varvec{L}}}^{{\varvec{p}}}$$ L p -
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Lp -Solutions and Comparison Results for Lévy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting Stefan Kremsner1
· Alexander Steinicke2
Received: 29 January 2020 / Revised: 3 September 2020 / Accepted: 30 October 2020 © The Author(s) 2020
Abstract We present a unified approach to L p -solutions ( p > 1) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke. Keywords Backward stochastic differential equation · Lévy process · L p -solutions · Predictable version Mathematics Subject Classification (2020) 60H10
1 Introduction The existence and uniqueness of solutions to a backward stochastic differential equation (BSDE) have been extensively investigated in many, but also various specifically chosen settings, partly due to certain applications in practice and partly also for theoretically interesting reasons. In this paper, we both unify and simplify the approach for a general BSDE framework driven by a Lévy process with a straightforward extension to more general filtrations. We show new comparison results and relax the assumptions known so far for guaranteeing unique L p -solutions, p > 1, to a BSDE with terminal
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Stefan Kremsner [email protected] Alexander Steinicke [email protected]
1
Department of Mathematics, University of Graz, Graz, Austria
2
Department of Mathematics, Montanuniversitaet Leoben, Leoben, Austria
123
Journal of Theoretical Probability
condition ξ and generator f that satisfies a monotonicity condition. An L p -solution is a triplet of processes (Y , Z , U ) from suitable L p -spaces (defined in Sect. 2) which satisfies a.s. T T f (s, Ys , Z s , Us )ds − Z s dWs − Us (x) N˜ (ds, d x), Yt = ξ + t
t
]t,T ]×Rd \{0}
(1) for each t ∈ [0, T ], where W is a Brownian motion and N˜ is a compensated Poisson random measure independent of W . The BSDE (1) itself will be denoted by (ξ, f ). 1.1 Related Works For nonlinear BSDEs (ξ , f ) driven by Brownian motion, existence and uniqueness results were first systematically studied by Pardoux and Peng [21] with (ω, y, z) → f (ω, y, z) Lipschitz in (z, y) and ξ square integrable. The importance of BSDEs in mathematical finance and stochastic optimal control was further elaborated by various works, e.g., by El Karoui et al. [7] which consider Lipschitz generators, L p solutions and Malliavin derivatives of BSDEs in the Brownian setting. The ambition to weaken the assumptions on f and ξ to still guarantee a unique solution gave birth to a large number of contributions, where—in the case of a generator with Lipschitz dependence on the z-variable—at least a few should be mentioned herein: Pardou
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