Stochastic Calculus and Financial Applications

Abstract

Library of Congress Cataloging-in-Publication Data Steele, J. Michael. Stochastic calculus and financial applications I J. Michael Steele. p. em. -(Applications of mathematics ; 45) Includes bibliographical references and index.

Martingale option pricing

“…Our main result is that the generalization of the Black-Scholes partial differential equation (pde) to the case of variable diffusion D(x,t) describes a Martingale in the risk neutral discounted stock price. This was proven by Harrison and Kreps [7] for the original Black-Scholes model [8], where D=constant (see also [9,10]) so that returns are Gaussian. Our interest is in classes of nonGaussian returns models that reflect the empirical data faithfully.…”

“…Girsanov's theorem is often stated in financial math texts [9,18] as transforming a Wiener process plus a drift term into another Wiener process. This is wrong: when the drift depends on a random variable x and is not merely t-dependent, then the resulting process is not Wiener.…”

“…Extending the original Black-Scholes result to a purely time dependent diffusion D(t) is trivial, the returns in that case are still Gaussian. Steele [9] discusses Black-Scholes pdes with Martingale solutions for the case where D(x) depends on x alone, but that restriction eliminates the market.…”

Quantum Risk-Sensitive Estimation and Robustness

“…If we let go to infinity and to 0, but keep the product constant, then it easily follows from Donsker's invariance principle (see e.g., [26]) that converges weakly to a classical Brownian motion. Note that the relation becomes in the limit (see e.g., [37]). In physics the observable is known as a field quadrature, see e.g., [12], [18].…”

A Lévy-Ciesielski Expansion for Quantum Brownian Motion and the Construction of Quantum Brownian Bridges

“…standard normals. A beautiful recent textbook account is given in [23] (see also [15] for an older version before the language of wavelets was in vogue). The idea of constructing Brownian motion in this way was originally due to Paul Lévy and later, Z.Cielsielski (see also the comments on pp.18-19 of [18]).…”