65,348 research outputs found
Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise
Unique existence of solutions to porous media equations driven by continuous
linear multiplicative space-time rough signals is proven for initial data in
on bounded domains . The generation of a
continuous, order-preserving random dynamical system on and
the existence of a random attractor for stochastic porous media equations
perturbed by linear multiplicative noise in space and time is obtained. The
random attractor is shown to be compact and attracting in norm. Uniform bounds and uniform space-time continuity of
the solutions is shown. General noise including fractional Brownian motion for
all Hurst parameters is treated and a pathwise Wong-Zakai result for driving
noise given by a continuous semimartingale is obtained. For fast diffusion
equations driven by continuous linear multiplicative space-time rough signals,
existence of solutions is proven for initial data in .Comment: Published in at http://dx.doi.org/10.1214/13-AOP869 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Functional Approach to FBSDEs and Its Application in Optimal Portfolios
In Liang et al (2009), the current authors demonstrated that BSDEs can be
reformulated as functional differential equations, and as an application, they
solved BSDEs on general filtered probability spaces. In this paper the authors
continue the study of functional differential equations and demonstrate how
such approach can be used to solve FBSDEs. By this approach the equations can
be solved in one direction altogether rather than in a forward and backward
way. The solutions of FBSDEs are then employed to construct the weak solutions
to a class of BSDE systems (not necessarily scalar) with quadratic growth, by a
nonlinear version of Girsanov's transformation. As the solving procedure is
constructive, the authors not only obtain the existence and uniqueness theorem,
but also really work out the solutions to such class of BSDE systems with
quadratic growth. Finally an optimal portfolio problem in incomplete markets is
solved based on the functional differential equation approach and the nonlinear
Girsanov's transformation.Comment: 26 page
Random attractors for degenerate stochastic partial differential equations
We prove the existence of random attractors for a large class of degenerate
stochastic partial differential equations (SPDE) perturbed by joint additive
Wiener noise and real, linear multiplicative Brownian noise, assuming only the
standard assumptions of the variational approach to SPDE with compact
embeddings in the associated Gelfand triple. This allows spatially much rougher
noise than in known results. The approach is based on a construction of
strictly stationary solutions to related strongly monotone SPDE. Applications
include stochastic generalized porous media equations, stochastic generalized
degenerate p-Laplace equations and stochastic reaction diffusion equations. For
perturbed, degenerate p-Laplace equations we prove that the deterministic,
infinite dimensional attractor collapses to a single random point if enough
noise is added.Comment: 34 pages; The final publication is available at
http://link.springer.com/article/10.1007%2Fs10884-013-9294-
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