4,695 research outputs found
Anticipated backward stochastic differential equations
In this paper we discuss new types of differential equations which we call
anticipated backward stochastic differential equations (anticipated BSDEs). In
these equations the generator includes not only the values of solutions of the
present but also the future. We show that these anticipated BSDEs have unique
solutions, a comparison theorem for their solutions, and a duality between them
and stochastic differential delay equations.Comment: Published in at http://dx.doi.org/10.1214/08-AOP423 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Anticipated BSDEs driven by fractional Brownian motion with time-delayed generator
This paper discusses a new type of anticipated backward stochastic
differential equation with a time-delayed generator (DABSDEs, for short) driven
by fractional Brownian motion, also known as fractional BSDEs, with Hurst
parameter , which extends the results of the anticipated backward
stochastic differential equation to the case of the drive is fractional
Brownian motion instead of a standard Brownian motion and in which the
generator considers not only the present and future times but also the past
time. By using the fixed point theorem, we will demonstrate the existence and
uniqueness of the solutions to these equations. Moreover, we shall establish a
comparison theorem for the solutions
Optimal distributed control of a stochastic Cahn-Hilliard equation
We study an optimal distributed control problem associated to a stochastic
Cahn-Hilliard equation with a classical double-well potential and Wiener
multiplicative noise, where the control is represented by a source-term in the
definition of the chemical potential. By means of probabilistic and analytical
compactness arguments, existence of an optimal control is proved. Then the
linearized system and the corresponding backward adjoint system are analysed
through monotonicity and compactness arguments, and first-order necessary
conditions for optimality are proved.Comment: Key words and phrases: stochastic Cahn-Hilliard equation, phase
separation, optimal control, linearized state system, adjoint state system,
first-order optimality condition
Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media
We explore the propagation and transformation of electromagnetic waves
through spatially homogeneous yet smoothly time-dependent media within the
framework of classical electrodynamics. By modelling the smooth transition,
occurring during a finite period {\tau}, as a phenomenologically realistic and
sigmoidal change of the dielectric permittivity, an analytically exact solution
to Maxwell's equations is derived for the electric displacement in terms of
hypergeometric functions. Using this solution, we show the possibility of
amplification and attenuation of waves and associate this with the decrease and
increase of the time-dependent permittivity. We demonstrate, moreover, that
such an energy exchange between waves and non-stationary media leads to the
transformation (or conversion) of frequencies. Our results may pave the way
towards controllable light-matter interaction in time-varying structures.Comment: 5 figure
Asymptotic Implied Volatility at the Second Order with Application to the SABR Model
We provide a general method to compute a Taylor expansion in time of implied
volatility for stochastic volatility models, using a heat kernel expansion.
Beyond the order 0 implied volatility which is already known, we compute the
first order correction exactly at all strikes from the scalar coefficient of
the heat kernel expansion. Furthermore, the first correction in the heat kernel
expansion gives the second order correction for implied volatility, which we
also give exactly at all strikes. As an application, we compute this asymptotic
expansion at order 2 for the SABR model.Comment: 27 pages; v2: typos fixed and a few notation changes; v3: published
version, typos fixed and comments added. in Large Deviations and Asymptotic
Methods in Finance, Springer (2015) 37-6
Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes
The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is
a long-standing problem with no known closed-form solution except in specific
cases. Taking this as a starting-point, and extending to a general
mean-reverting process, we investigate the long- and short-time asymptotics
using a combination of Hopf-Cole and Laplace transform techniques. As a result
we are able to give a single formula that is correct in both limits, as well as
being exact in certain special cases. We demonstrate the results using a
variety of other models
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