Denis Bell

The Malliavin calculus and stochastic delay equations

We consider stochastic delay systems dx(t) = g(x(t − r)) dW(t) driven by multi-dimensional Brownian motion W. The diffusion coefficient g is smooth with a possible degeneracy at 0. For a large class of deterministic initial paths we show that the solution x(t) admits a smooth density with respect to Lebesgue measure. The proof is based on Malliavin calculus together with new probabilistic lower bounds on the solution x.

An extension of Hörmander’s theorem for infinitely degenerate second-order operators

We establish the hypoellipticity of a large class of highly degenerate second order differential operators of Hormander type. The hypotheses of our theorem allow Hormander’s general Lie algebra condition to fail on a collection of hypersurfaces. The proof of the theorem is probabilistic in nature. It is based on the Malliavin calculus and requires new sharp estimates for diffusion processes in Euclidean space.

On the solution of stochastic ordinary differential equations via small delays

dx(t)=g(x{t))dW(t) is proved using an approximating sequence of stochastic delay equationsGeneralizations of the approximation scheme are indicated for the Stratonovich case and when the Brownian motion W is replaced by a continuous semi-martingale.

A quasi-invariance theorem for measures on Banach spaces

We show that for a measure ? on a Banach space directional differentiability implies quasi-translation invariance. This result is shown to imply the Cameron-Martin theorem. A second application is given in which ty is the image of a Gaussian measure under a suitably regular map.

A calcium model with random absorption: A stochastic approach

Absorption of calcium, or any mineral, by the body is subject to the random fluctuations typical of diffusion through membranes. In this paper we consider the absorption of calcium from the gut as a white noise process added to the deterministic model of Sen & Mohr (1990, J. theor. Biol. 142, 179-188). The first two moments for the amount of calcium in the extracellular fluid (ECF) have been derived using the Ito Calculus. A confidence interval for the total amount of calcium in the ECF is constructed. The equations for the first two moments of the fraction of dose calcium in the ECF are also given. Suggestions are made for the collection of experimental data in a form which should be helpful in investigating the magnitude of the stochastic effect.

Transformations of Measure on an Infinite Dimensional Vector Space

Let E denote a Banach space equipped with a finite Borel measure v. For any measurable transformation T: E → E, let v T denote the measure defined by v T(B) = v(T−1(B)) for Borel sets B. A transformation theorem for v is a result which gives conditions on T under which v T is absolutely continuous with respect to v, and which gives a formula for the corresponding Radon-Nikodym derivative (RND) when these conditions hold.

The Malliavin calculus and hypoelliptic differential operators

This article is intended as an introduction to Malliavins stochastic calculus of variations and his probabilistic approach to hypoellipticity. Topics covered include an elementary derivation of the basic integration by parts formulae, a proof of the probabilistic version of Hormanders theorem as envisioned by Malliavin and completed by Kusuoka and Stroock, and an extension of Hormanders theorem valid for operators with degeneracy of exponential type due to the author and S. Mohammed.

ARBITRAGE-FREE OPTION PRICING MODELS

We describe a scheme for constructing explicitly solvable arbitrage-free models for stock price. This is used to study a model similar to one introduced by Cox and Ross, where the volatility of the stock is proportional to the square root of the stock price. We derive a formula for the value of a European call option based on this model and give a procedure for estimating parameters and for testing the validity of the model.

Stochastic Differential Equations and Hypoelliptic Operators

The first half of the twentieth century saw some remarkable developments in analytic probability theory. Wiener constructed a rigorous mathematical model of Brownian motion. Kolmogorov discovered that the transition probabilities of a diffusion process define a fundamental solution to an associated heat equation. Ito developed a stochastic calculus which made it possible to represent a diffusion with a given (infinitesimal) generator as the solution of a Stochastic Differential Equation. These developments created a link between the fields of Partial Differential Equations and stochastic analysis whereby results in the former area could be used to prove results in the latter.

Noncentral limit theorem for the generalized Hermite process

We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Rosenblatt processes