980 research outputs found

    Valuation of boundary-linked assets

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    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

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    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

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    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 E(Fn4)−3\sqrt{\mathbb{E}(F_n^4)-3} in the central limit theorem for elements {Fn}n≄1\{F_n\}_{n\ge 1} of Wiener chaos of any fixed order such that E(Fn2)=1\mathbb{E}(F_n^2) = 1. 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

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    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

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    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

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    We prove strong convergence of order 1/4−ϔ1/4-\epsilon for arbitrarily small Ï”>0\epsilon>0 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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