2,629 research outputs found
Pathwise large deviations for white noise chaos expansions
We consider a family of continuous processes
which are measurable with respect to a
white noise measure, take values in the space of continuous functions
, and have the Wiener chaos expansion
We provide sufficient conditions for the large deviations principle of
to hold in , thereby
refreshing a problem left open by P\'erez-Abreu (1993) in the Brownian motion
case. The proof is based on the weak convergence approach to large deviations:
it involves demonstrating the convergence in distribution of certain
perturbations of the original process, and thus the main difficulties lie in
analysing and controlling the perturbed multiple stochastic integrals.
Moreover, adopting this representation offers a new perspective on pathwise
large deviations and induces a variety of applications thereof.Comment: 21 pages. New applications are provide
A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion
In this thesis, we mainly study some properties for certain stochastic diāµer-ential equations.The types of stochastic diāµerential equations we are interested in are (i) stochastic diāµerential equations driven by Brownian motion, (ii) stochastic functional diāµerential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic diāµerential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic diāµerential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic diāµerential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic diāµerential equations driven by fractional Brownian motion, and then the Bismut formula of Lionās derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic diāµerential equation model, we ļ¬rst construct the moderate deviation principle for the law of the approxima-tion stochastic diāµerential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic diāµerential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic diāµer-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we ļ¬rst establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model
Short-time near-the-money skew in rough fractional volatility models
We consider rough stochastic volatility models where the driving noise of volatility has
fractional scaling, in the "rough" regime of Hurst parameter H < Ā½. This regime recently attracted
a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we
sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around
the money while maintaining full analytical tractability. More precisely, this amounts to proving higher
order moderate deviation estimates, only recently introduced in the option pricing context. This in turn
allows us to push the applicability range of known at-the-money skew approximation formulae from CLT
type log-moneyness deviations of order t1/2 (recent works of Al{\`o}s, Le{\'o}n & Vives and Fukasawa) to the
wider moderate deviations regime
The instanton method and its numerical implementation in fluid mechanics
A precise characterization of structures occurring in turbulent fluid flows
at high Reynolds numbers is one of the last open problems of classical physics.
In this review we discuss recent developments related to the application of
instanton methods to turbulence. Instantons are saddle point configurations of
the underlying path integrals. They are equivalent to minimizers of the related
Freidlin-Wentzell action and known to be able to characterize rare events in
such systems. While there is an impressive body of work concerning their
analytical description, this review focuses on the question on how to compute
these minimizers numerically. In a short introduction we present the relevant
mathematical and physical background before we discuss the stochastic Burgers
equation in detail. We present algorithms to compute instantons numerically by
an efficient solution of the corresponding Euler-Lagrange equations. A second
focus is the discussion of a recently developed numerical filtering technique
that allows to extract instantons from direct numerical simulations. In the
following we present modifications of the algorithms to make them efficient
when applied to two- or three-dimensional fluid dynamical problems. We
illustrate these ideas using the two-dimensional Burgers equation and the
three-dimensional Navier-Stokes equations
- ā¦