12,740 research outputs found
A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications
We formulate a new class of stochastic partial differential equations
(SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which
allow the high-order integral-partial differential operators into both drift
and diffusion coefficients. Under certain type of Lipschitz and linear growth
conditions, we develop a method to prove the existence and uniqueness of
adapted solution to these B-SPDEs with jumps. Comparing with the existing
discussions on conventional backward stochastic (ordinary) differential
equations (BSDEs), we need to handle the differentiability of adapted triplet
solution to the B-SPDEs with jumps, which is a subtle part in justifying our
main results due to the inconsistency of differential orders on two sides of
the B-SPDEs and the partial differential operator appeared in the diffusion
coefficient. In addition, we also address the issue about the B-SPDEs under
certain Markovian random environment and employ a B-SPDE with strongly
nonlinear partial differential operator in the drift coefficient to illustrate
the usage of our main results in finance.Comment: 22 pagea, 1 figur
Maximum principle for a stochastic delayed system involving terminal state constraints
We investigate a stochastic optimal control problem where the controlled
system is depicted as a stochastic differential delayed equation; however, at
the terminal time, the state is constrained in a convex set. We firstly
introduce an equivalent backward delayed system depicted as a time-delayed
backward stochastic differential equation. Then a stochastic maximum principle
is obtained by virtue of Ekeland's variational principle. Finally, applications
to a state constrained stochastic delayed linear-quadratic control model and a
production-consumption choice problem are studied to illustrate the main
obtained result.Comment: 16 page
Feynman-Kac representation of fully nonlinear PDEs and applications
The classical Feynman-Kac formula states the connection between linear
parabolic partial differential equations (PDEs), like the heat equation, and
expectation of stochastic processes driven by Brownian motion. It gives then a
method for solving linear PDEs by Monte Carlo simulations of random processes.
The extension to (fully)nonlinear PDEs led in the recent years to important
developments in stochastic analysis and the emergence of the theory of backward
stochastic differential equations (BSDEs), which can be viewed as nonlinear
Feynman-Kac formulas. We review in this paper the main ideas and results in
this area, and present implications of these probabilistic representations for
the numerical resolution of nonlinear PDEs, together with some applications to
stochastic control problems and model uncertainty in finance
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