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We consider a class of Backward Stochastic Differential Equations with superlinear driver process $f$ adapted to a filtration supporting at least a $d$ dimensional Brownian motion and a Poisson random measure on ${mathbb R}^m- {0}.$ We consider the following class of terminal conditions $xi_1 = infty cdot 1_{{tau_1 le T}}$ where $tau_1$ is any stopping time with a bounded density in a neighborhood of $T$ and $xi_2 = infty cdot 1_{A_T}$ where $A_t$, $t in [0,T]$ is a decreasing sequence of events adapted to the filtration ${mathcal F}_t$ that is continuous in probability at $T$. A special case for $xi_2$ is $A_T = {tau_2 > T}$ where $tau_2$ is any stopping time such that $P(tau_2 =T) =0.$ In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We further show that the first exit time from a time varying domain of a $d$-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density; therefore such exit times can be used as $tau_1$ and $tau_2$ to define the terminal conditions $xi_1$ and $xi_2.$ The proof of existence of the density is based on the classical Greens functions for the associated PDE.
We solve a class of BSDE with a power function $f(y) = y^q$, $q > 1$, driving its drift and with the terminal boundary condition $ xi = infty cdot mathbf{1}_{B(m,r)^c}$ (for which $q > 2$ is assumed) or $ xi = infty cdot mathbf{1}_{B(m,r)}$, where $B
In this paper, we provide a one-to-one correspondence between the solution Y of a BSDE with singular terminal condition and the solution H of a BSDE with singular generator. This result provides the precise asymptotic behavior of Y close to the final
In this paper, we study backward stochastic Volterra integral equations introduced in [26, 45] and extend the existence, uniqueness or comparison results for general filtration as in [31] (not only Brownian-Poisson setting). We also consider Lp-data
This paper introduces the notion of a filtration-consistent dynamic operator with a floor, by suitably formulating four axioms. It is shown that under some suitable conditions, a filtration-consistent dynamic operator with a continuous upper-bounded
In this paper, we study the solvability of anticipated backward stochastic differential equations (BSDEs, for short) with quadratic growth for one-dimensional case and multi-dimensional case. In these BSDEs, the generator, which is of quadratic growt