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We consider a financial market with memory effects in which wealth processes are driven by mean-field stochastic Volterra equations. In this financial market, the classical dynamic programming method can not be used to study the optimal investment problem, because the solution of mean-field stochastic Volterra equation is not a Markov process. In this paper, a new method through Malliavin calculus introduced in [1], can be used to obtain the optimal investment in a Volterra type financial market....
We combine Stein’s method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of gaussian fields. Among several examples, we provide an application to a functional version of the Breuer–Major CLT for fields subordinated to a fractional brownian motion.
We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process (i.e. , where is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process ) and of an arbitrary Lévy process independent of , that the drift coefficient is continuous (but not...
Using a representation as an infinite linear combination of chi-square independent random variables, it is shown that some Wiener functionals, appearing in empirical characteristic process asymptotic theory, have densities which are tempered in the properly infinite case and exponentially decaying in the finite case.
Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of Fournier (2000) on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by Guérin and Méléard (2003), some simulations of the solution of the Landau equation will be given. This result...
Using probabilistic tools, this work states a pointwise convergence of
function solutions of the 2-dimensional Boltzmann equation to the function
solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of
Fournier (2000) on the Malliavin calculus for the Boltzmann
equation. Moreover, using the particle system introduced by Guérin and
Méléard (2003), some simulations of the solution of the Landau equation will be given. This result...
We consider the random vector , where are distinct points of and denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for . We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize the set of...
We consider the random vector , where t > 0, x1,...,xd are
distinct points of
and u denotes the stochastic process solution to a stochastic wave
equation driven by
a noise white in time and correlated in space. In a recent paper by
Millet and Sanz–Solé
[10], sufficient conditions are given ensuring existence and
smoothness of
density for . We study here the positivity of such
density. Using
techniques developped in [1] (see also [9]) based
on Analysis on an
abstract Wiener space, we characterize...
We give a generalization in the non-compact case to various positivity theorems obtained by Malliavin Calculus in the compact case.
With the pioneering work of [Pardoux and Peng,
Syst. Contr. Lett.14 (1990) 55–61; Pardoux and Peng,
Lecture Notes in Control and Information Sciences176
(1992) 200–217]. We have at our disposal
stochastic processes which solve the so-called backward stochastic
differential equations. These processes provide us with a Feynman-Kac
representation for the solutions of a class of nonlinear partial differential equations (PDEs) which appear
in many applications in the field of Mathematical Finance....
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