$L_{2}$-variation of Lévy driven BSDEs with non-smooth terminal conditions

Abstract

We consider the L 2 -regularity of solutions to backward stochastic differential equations (BSDEs) with Lipschitz generators driven by a Brownian motion and a Poisson random measure associated with a Lévy process (X t ) t∈ [0,T ] . The terminal condition may be a Borel function of finitely many increments of the Lévy process which is not necessarily Lipschitz but only satisfies a fractional smoothness condition. The results are obtained by investigating how the special structure appearing in the chaos expansion of the terminal condition is inherited by the solution to the BSDE.

Lp solutions of backward stochastic differential equations with jumps

“…By (H5) and (H6), it holds for any (y 1 , z 1 , u 1 ), (y 2 , z 2 , u 2 ) ∈ R l ×R l×d ×L p ν that 36) where h n t,ω (ỹ,z) := f t, ω, Y n t (ω)− 1 nỹ , π n (Z n t (ω)− 1 nz ), ζ n (U n t (ω)) . Next, we fix (ỹ,z) ∈ R l ×R l×d with |(ỹ,z)| < 1 and set ( y m,n , z m,n ) :…”

“…As to BSDEs driven by other discontinuous random sources, Xia [72] and Bandini [6] studied BSDEs driven by a random measure; Confortola et al [25,26] considered BSDEs driven by a marked point process; [61,5,66,36] analyzed BSDEs driven by Lévy processes; [2,68,46] discussed BSDEs driven by a process with a finite number of marked jumps.…”

Simulation of BSDEs with jumps by Wiener Chaos expansion

“…In case of BSDEs driven by a Poisson random measure, Bouchard and Elie [3] have proposed a scheme based on the dynamic programming equation and studied the rate of convergence of the method when the terminal condition is given by ξ = g(X T ), where g is a Lipschitz function and X is a forward process. More recently, Geiss and Steinicke [9] have extended this result to the case of a terminal condition which may be a Borel function of finitely many increments of the Lévy forward process X which is not necessarily Lipschitz but only satisfies a fractional smoothness condition. In the case of jumps driven by a compensated Poisson process, Lejay, Mordecki and Torres [15] have developed a fully implementable scheme based on a random binomial tree, following the approach proposed by Briand, Delyon and Mémin [5].…”

Malliavin smoothness on the Lévy space with Hölder continuous or BV functionals

“…The L p -variation of the solution of certain BSDEs depends on the Malliavin fractional smoothness of the terminal condition f (Y 1 ). This was shown with more general terminal conditions for the Brownian motion by C. Geiss, S. Geiss and Gobet [9] and S. Geiss and Ylinen [16] and for p = 2 for general L 2 -Lévy processes by C. Geiss and Steinicke [12].…”