980 research outputs found
Valuation of boundary-linked assets
This article studies the valuation of boundary-linked assets and their derivatives in continuous-time markets. Valuing boundary-linked assets requires the solution of a stochastic differential equation with boundary conditions, which, often, is not Markovian. We propose a wavelet-collocation algorithm for solving a Milstein approximation to the stochastic boundary problem. Its convergence properties are studied. Furthermore, we value boundary-linked derivatives using Malliavin calculus and Monte Carlo methods. We apply these ideas to value European call options of boundary-linked asset
Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient
We provide new regularity results for the solutions of the Kolmogorov
equation associated to a SPDE with nonlinear diffusion coefficients and a
Burgers type nonlinearity. This generalizes previous results in the simpler
cases of additive or affine noise. The basic tool is a discrete version of a
two sided stochastic integral which allows a new formulation for the
derivatives of these solutions. We show that this can be used to generalize the
weak order analysis performed in [16]. The tools we develop are very general
and can be used to study many other examples of applications
Stein meets Malliavin in normal approximation
Stein's method is a method of probability approximation which hinges on the
solution of a functional equation. For normal approximation the functional
equation is a first order differential equation. Malliavin calculus is an
infinite-dimensional differential calculus whose operators act on functionals
of general Gaussian processes. Nourdin and Peccati (2009) established a
fundamental connection between Stein's method for normal approximation and
Malliavin calculus through integration by parts. This connection is exploited
to obtain error bounds in total variation in central limit theorems for
functionals of general Gaussian processes. Of particular interest is the fourth
moment theorem which provides error bounds of the order
in the central limit theorem for elements
of Wiener chaos of any fixed order such that
. This paper is an exposition of the work of Nourdin and
Peccati with a brief introduction to Stein's method and Malliavin calculus. It
is based on a lecture delivered at the Annual Meeting of the Vietnam Institute
for Advanced Study in Mathematics in July 2014.Comment: arXiv admin note: text overlap with arXiv:1404.478
VALUATION OF BOUNDARY-LINKED ASSETS
This article studies the valuation of boundary-linked assets and their derivatives in continuous-time markets. Valuing boundary-linked assets requires the solution of a stochastic differential equation with boundary conditions, which, often, is not Markovian. We propose a wavelet-collocation algorithm for solving a Milstein approximation to the stochastic boundary problem. Its convergence properties are studied. Furthermore, we value boundary-linked derivatives using Malliavin calculus and Monte Carlo methods. We apply these ideas to value European call options of boundary-linked assets.
Skorohod and Stratonovich integration in the plane
This article gives an account on various aspects of stochastic calculus in
the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of
variable formula for a path indexed by a square, satisfying some H\"older
regularity conditions with a H\"older exponent greater than 1/3. (ii) Get some
Skorohod change of variable formulas for a large class of Gaussian processes
defined on the suqare. (iii) Compare the bidimensional integrals obtained with
those two methods, computing explicit correction terms whenever possible. As a
byproduct, we also give explicit forms of corrections in the respective change
of variable formulas
Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient
We prove strong convergence of order for arbitrarily small
of the Euler-Maruyama method for multidimensional stochastic
differential equations (SDEs) with discontinuous drift and degenerate diffusion
coefficient. The proof is based on estimating the difference between the
Euler-Maruyama scheme and another numerical method, which is constructed by
applying the Euler-Maruyama scheme to a transformation of the SDE we aim to
solve
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