60 research outputs found
Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions
In this paper we obtain Gaussian-type lower bounds for the density of
solutions to stochastic differential equations (SDEs) driven by a fractional
Brownian motion with Hurst parameter . In the one-dimensional case with
additive noise, our study encompasses all parameters , while the
multidimensional case is restricted to the case . We rely on a mix of
pathwise methods for stochastic differential equations and stochastic analysis
tools.Comment: Published at http://dx.doi.org/10.1214/14-AOP977 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme
In the present paper, we prove that the Wasserstein distance on the space of
continuous sample-paths equipped with the supremum norm between the laws of a
uniformly elliptic one-dimensional diffusion process and its Euler
discretization with steps is smaller than where
is an arbitrary positive constant. This rate is intermediate
between the strong error estimation in obtained when coupling the
stochastic differential equation and the Euler scheme with the same Brownian
motion and the weak error estimation obtained when comparing the
expectations of the same function of the diffusion and of the Euler scheme at
the terminal time . We also check that the supremum over of the
Wasserstein distance on the space of probability measures on the real line
between the laws of the diffusion at time and the Euler scheme at time
behaves like .Comment: Published in at http://dx.doi.org/10.1214/13-AAP941 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Using moment approximations to study the density of jump driven SDEs
In order to study the regularity of the density of a solution of a infinite activity jump driven stochastic differential equation we consider the following two-step approximation method. First, we use the solution of the moment problem in order to approximate the small jumps by another whose LĂ©vy measure has finite support. In a second step we replace the approximation of the first two moments by a small noise Brownian motion based on the Assmussen-Rosinski approach. This approximation needs to satisfy certain properties in order to apply the "balance" method which allows the study of densities for the solution process based on Malliavin Calculus for the Brownian motion. Our results apply to situations where the LĂ©vy measure is absolutely continuous with respect to the Lebesgue measure or purely atomic measures or combinations of them
The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion
The paper focuses on discrete-type approximations of solutions to
non-homogeneous stochastic differential equations (SDEs) involving fractional
Brownian motion (fBm). We prove that the rate of convergence for Euler
approximations of solutions of pathwise SDEs driven by fBm with Hurst index
can be estimated by ( is the diameter of
partition). For discrete-time approximations of Skorohod-type quasilinear
equation driven by fBm we prove that the rate of convergence is .Comment: 21 pages, (incorrect) weak convergence result removed, to appear in
Stochastic
Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations
In this article we consider the problem of pricing and hedging
high-dimensional Asian basket options by Quasi-Monte Carlo simulation. We
assume a Black-Scholes market with time-dependent volatilities and show how to
compute the deltas by the aid of the Malliavin Calculus, extending the
procedure employed by Montero and Kohatsu-Higa (2003). Efficient
path-generation algorithms, such as Linear Transformation and Principal
Component Analysis, exhibit a high computational cost in a market with
time-dependent volatilities. We present a new and fast Cholesky algorithm for
block matrices that makes the Linear Transformation even more convenient.
Moreover, we propose a new-path generation technique based on a Kronecker
Product Approximation. This construction returns the same accuracy of the
Linear Transformation used for the computation of the deltas and the prices in
the case of correlated asset returns while requiring a lower computational
time. All these techniques can be easily employed for stochastic volatility
models based on the mixture of multi-dimensional dynamics introduced by Brigo
et al. (2004).Comment: 16 page
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
Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise
We consider the family of stochastic partial differential equations indexed
by a parameter \eps\in(0,1], \begin{equation*} Lu^{\eps}(t,x) =
\eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*}
(t,x)\in(0,T]\times\Rd with suitable initial conditions. In this equation,
is a second-order partial differential operator with constant coefficients,
and are smooth functions and is a Gaussian noise, white
in time and with a stationary correlation in space. Let p^\eps_{t,x} denote
the density of the law of u^\eps(t,x) at a fixed point
(t,x)\in(0,T]\times\Rd. We study the existence of \lim_{\eps\downarrow 0}
\eps^2\log p^\eps_{t,x}(y) for a fixed . The results apply to a class
of stochastic wave equations with and to a class of stochastic
heat equations with .Comment: 39 pages. Will be published in the book " Stochastic Analysis and
Applications 2014. A volume in honour of Terry Lyons". Springer Verla
On the wellposedness of some McKean models with moderated or singular diffusion coefficient
We investigate the well-posedness problem related to two models of nonlinear
McKean Stochastic Differential Equations with some local interaction in the
diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with
moderate interaction, previously studied by Meleard and Jourdain in [16], under
slightly weaker assumptions, by showing the existence and uniqueness of a weak
solution using a Sobolev regularity framework instead of a Holder one. Second,
we study the construction of a Lagrangian Stochastic model endowed with a
conditional McKean diffusion term in the velocity dynamics and a nondegenerate
diffusion term in the position dynamics
- âŠ