Nigel J. Cutland

Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model

In mathematical finance, semimartingales are traditionally viewed as the largest class of stochastic processes which are economically reasonable models for stock price movements. This is mainly because stochastic integrals play a crucial role in the modern theory of finance, and semimartingales represent the largest class of stochastic processes for which a general theory of stochastic integration exists. However, some empirical evidence from actual stock price data suggests stochastic models that are not covered by the semimartingale setting.

Nonstandard methods for stochastic fluid mechanics

Standard Preliminaries Nonstandard Preliminaries Weak Solutions of Navier-Stokes Equations Statistical Solutions of Navier-Stokes Equations Stochastic Navier-Stokes Equations Other Equations of Hydromechanics Euler Equation.

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n≤4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.

Infinitesimal methods in control theory: Deterministic and stochastic

In Part I, methods of nonstandard analysis are applied to deterministic control theory, extending earlier work of the author. Results established include compactness of relaxed controls, continuity of solution and cost as functions of the controls, and existence of optimal controls. In Part II, the methods are extended to obtain similar results for partially observed stochastic control. Systems considered take the form:where the feedback control u depends on information from a digital read-out of the observation process y. The noise in the state equation is controlled along with the drift. Similar methods are applied to a Markov system in the final section.

A Simple Proof of Existence of Weak and Statistical Solutions of Navier-Stokes Equations

The Galerkin approximation to the Navier–Stokes equations in dimension N, where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

From discrete to continuous stochastic calculus

We formulate the notion of D2-convergence of discrete time entities (based on simple random walks) to their continuous counterparts on Wiener space, and show that D2-convergence is preserved under stochastic integration and differentiation, and for chaos decomposition.

Global Attractors for 3-Dimensional Stochastic Navier–Stokes Equations

Sells approach 35 to the construction of attractors for the Navier-Stokes equations in 3-dimensions is extended to the 3D stochastic equations with a general multiplicative noise. The new notion of a process attractor is defined as a set A of processes, living on a single filtered probability space, that is a set of solutions and attracts all solution processes in a given class. This requires the richness of a Loeb probability space. Non-compactness results for A and a characterization in terms of two-sided solutions are given.

A Nonstandard Approach to the Malliavin Calculus

We outline an intuitive approach to the Malliavin calculus for the classical Wiener space, showing that the basic operators of this calculus have natural descriptions as classical differential operators on a nonstandard space *R N for an infinite natural number N.

Navier-Stokes equations with multiplicative noise

The authors use the methods of nonstandard analysis to give a solution to stochastic Navier-Stokes equations in dimension <or=4 with noise depending in a specific way on the first-order derivatives of the solution. Uniqueness holds for the two-dimensional case.

An extension of the ventcel- freidlin large deviation principle

The large deviation principle obtained by Ventcel-Freidlin for the measures associated with time homogeneous Markov diffusions is extended to the measures given by non-homogeneous functional stocha...