200 research outputs found
Intersection local times of independent fractional Brownian motions as generalized white noise functionals
In this work we present expansions of intersection local times of fractional
Brownian motions in , for any dimension , with arbitrary Hurst
coefficients in . The expansions are in terms of Wick powers of white
noises (corresponding to multiple Wiener integrals), being well-defined in the
sense of generalized white noise functionals. As an application of our
approach, a sufficient condition on for the existence of intersection local
times in is derived, extending the results of D. Nualart and S.
Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional
Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and
more general Hurst coefficients.Comment: 28 page
Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations
In this paper we prove the law of large numbers and central limit theorem for
trajectories of a particle carried by a two dimensional Eulerian velocity
field. The field is given by a solution of a stochastic Navier--Stokes system
with a non-degenerate noise. The spectral gap property, with respect to
Wasserstein metric, for such a system has been shown in [9]. In the present
paper we show that a similar property holds for the environment process
corresponding to the Lagrangian observations of the velocity. In consequence we
conclude the law of large numbers and the central limit theorem for the tracer.
The proof of the central limit theorem relies on the martingale approximation
of the trajectory process
The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps
Consider jump-type stochastic differential equations with the drift,
diffusion and jump terms. Logarithmic derivatives of densities for the solution
process are studied, and the Bismut-Elworthy-Li type formulae can be obtained
under the uniformly elliptic condition on the coefficients of the diffusion and
jump terms. Our approach is based upon the Kolmogorov backward equation by
making full use of the Markovian property of the process.Comment: 29 pages, to appear in Journal of Theoretical Probabilit
Multidimensional Quasi-Monte Carlo Malliavin Greeks
We investigate the use of Malliavin calculus in order to calculate the Greeks
of multidimensional complex path-dependent options by simulation. For this
purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the
multidimensional case. The multidimensional setting shows the convenience of
the Malliavin Calculus approach over different techniques that have been
previously proposed. Indeed, these techniques may be computationally expensive
and do not provide flexibility for variance reduction. In contrast, the
Malliavin approach exhibits a higher flexibility by providing a class of
functions that return the same expected value (the Greek) with different
accuracies. This versatility for variance reduction is not possible without the
use of the generalized integral by part formula of Malliavin Calculus. In the
multidimensional context, we find convenient formulas that permit to improve
the localization technique, introduced in Fourni\'e et al and reduce both the
computational cost and the variance. Moreover, we show that the parameters
employed for variance reduction can be obtained \textit{on the flight} in the
simulation. We illustrate the efficiency of the proposed procedures, coupled
with the enhanced version of Quasi-Monte Carlo simulations as discussed in
Sabino, for the numerical estimation of the Deltas of call, digital Asian-style
and Exotic basket options with a fixed and a floating strike price in a
multidimensional Black-Scholes market.Comment: 22 pages, 6 figure
On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples
We show a concise extension of the monotone stability approach to backward
stochastic differential equations (BSDEs) that are jointly driven by a Brownian
motion and a random measure for jumps, which could be of infinite activity with
a non-deterministic and time inhomogeneous compensator. The BSDE generator
function can be non convex and needs not to satisfy global Lipschitz conditions
in the jump integrand. We contribute concrete criteria, that are easy to
verify, for results on existence and uniqueness of bounded solutions to BSDEs
with jumps, and on comparison and a-priori -bounds. Several
examples and counter examples are discussed to shed light on the scope and
applicability of different assumptions, and we provide an overview of major
applications in finance and optimal control.Comment: 28 pages. Added DOI
https://link.springer.com/chapter/10.1007%2F978-3-030-22285-7_1 for final
publication, corrected typo (missing gamma) in example 4.1
Fractional smoothness and applications in finance
This overview article concerns the notion of fractional smoothness of random
variables of the form , where is a certain
diffusion process. We review the connection to the real interpolation theory,
give examples and applications of this concept. The applications in stochastic
finance mainly concern the analysis of discrete time hedging errors. We close
the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages
Gaussian density estimates for the solution of singular stochastic Riccati equations
summary:Stochastic Riccati equation is a backward stochastic differential equation with singular generator which arises naturally in the study of stochastic linear-quadratic optimal control problems. In this paper, we obtain Gaussian density estimates for the solutions to this equation
Stationary distributions for diffusions with inert drift
Consider reflecting Brownian motion in a bounded domain in that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential
Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization
Let be an underlying space with a non-atomic measure on it (e.g.
and is the Lebesgue measure). We introduce and study a
class of non-commutative generalized stochastic processes, indexed by points of
, with freely independent values. Such a process (field),
, , is given a rigorous meaning through smearing out
with test functions on , with being a
(bounded) linear operator in a full Fock space. We define a set
of all continuous polynomials of , and then define a con-commutative
-space by taking the closure of in the norm
, where is the vacuum in the Fock
space. Through procedure of orthogonalization of polynomials, we construct a
unitary isomorphism between and a (Fock-space-type) Hilbert space
, with
explicitly given measures . We identify the Meixner class as those
processes for which the procedure of orthogonalization leaves the set invariant. (Note that, in the general case, the projection of a
continuous monomial of oder onto the -th chaos need not remain a
continuous polynomial.) Each element of the Meixner class is characterized by
two continuous functions and on , such that, in the
space, has representation
\omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag\di^2_t,
where \di_t^\dag and \di_t are the usual creation and annihilation
operators at point
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