28,332 research outputs found
Solutions of Backward Stochastic Differential Equations with Jumps
Given , we study -solutions of a multi-dimensional
backward stochastic differential equation with jumps (BSDEJ) whose generator
may not be Lipschitz continuous in variables. We show that such a BSDEJ
with a p-integrable terminal data admits a unique solution by
approximating the monotonic generator by a sequence of Lipschitz generators via
convolution with mollifiers and using a stability result.Comment: Keywords: Backward stochastic differential equations with jumps,
solutions, monotonic generators, convolution with mollifier
On Zero-Sum Stochastic Differential Games
We generalize the results of Fleming and Souganidis (1989) on zero sum
stochastic differential games to the case when the controls are unbounded. We
do this by proving a dynamic programming principle using a covering argument
instead of relying on a discrete approximation (which is used along with a
comparison principle by Fleming and Souganidis). Also, in contrast with Fleming
and Souganidis, we define our pay-off through a doubly reflected backward
stochastic differential equation. The value function (in the degenerate case of
a single controller) is closely related to the second order doubly reflected
BSDEs.Comment: Key Words: Zero-sum stochastic differential games, Elliott-Kalton
strategies, dynamic programming principle, stability under pasting, doubly
reflected backward stochastic differential equations, viscosity solutions,
obstacle problem for fully non-linear PDEs, shifted processes, shifted SDEs,
second-order doubly reflected backward stochastic differential equation
Quadratic Reflected BSDEs with Unbounded Obstacles
In this paper, we analyze a real-valued reflected backward stochastic
differential equation (RBSDE) with an unbounded obstacle and an unbounded
terminal condition when its generator has quadratic growth in the
-variable. In particular, we obtain existence, comparison, and stability
results, and consider the optimal stopping for quadratic -evaluations. As an
application of our results we analyze the obstacle problem for semi-linear
parabolic PDEs in which the non-linearity appears as the square of the
gradient. Finally, we prove a comparison theorem for these obstacle problems
when the generator is convex or concave in the -variable.Comment: Key Words: Quadratic reflected backward stochastic differential
equations, convex/concave generator, -difference method, Legenre-Fenchel
duality, optimal stopping problems for quadratic -evaluations, stability,
obstacle problems for semi-linear parabolic PDEs, viscosity solution
On Quadratic g-Evaluations/Expectations and Related Analysis
In this paper we extend the notion of g-evaluation, in particular
g-expectation, to the case where the generator g is allowed to have a quadratic
growth. We show that some important properties of the g-expectations, including
a representation theorem between the generator and the corresponding
g-expectation, and consequently the reverse comparison theorem of quadratic
BSDEs as well as the Jensen inequality, remain true in the quadratic case. Our
main results also include a Doob-Meyer type decomposition, the optional
sampling theorem, and the up-crossing inequality. The results of this paper are
important in the further development of the general quadratic nonlinear
expectations.Comment: 27 page
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