Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Stochastic equations in infinite dimensions

Stochastic Functional Differential Equations

In this article we develop an explicit formula for pricing European options when the underlying stock price follows nonlinear stochastic functional differential equations with fixed and variable delays. We believe that the proposed models are sufficiently flexible to fit real market data, and yet simple enough to allow for a closed-form representation of the option price. Furthermore, the models maintain the no-arbitrage property and the completeness of the market. The derivation of the option-pricing formula is based on an equivalent local martingale measure.

/pdf/a-delayed-black-and-scholes-formula-ijwm92qwwt.pdf

A Delayed Black and Scholes Formula

The purpose of this article is to introduce the reader to certain aspects of stochastic differential systems whose evolution depends on the past history of the state

/pdf/stochastic-differential-systems-with-memory-theory-examples-mvnejw53ku.pdf

Stochastic Differential Systems With Memory: Theory, Examples and Applications

This article is a sequel to [A.H.M.P]. In [A.H.M.P], we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic delay equation with fixed delays in the drift and diffusion terms. In this article, we look at models of the stock price described by stochastic functional differential equations with variable delays. We present a class of examples of stock dynamics with variable delays that permit an explicit form for the option pricing formula. As in [A.H.M.P], the market is complete with no arbitrage. This is achieved through the existence of an equivalent martingale measure. In subsequent work, the authors intend to test the models in [A.H.M.P] and the present article against real market data.

A Delayed Black and Scholes Formula II

The main objective of this paper is to characterize the pathwise local structure of
solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations
(spde’s) near stationary solutions. Such characterization is realized through the long-term
behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part
1, we prove general existence and compactness theorems for Ck-cocycles of semilinear see’s and
spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with
Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic
Burgers equation with additive infinite-dimensional noise. In Part 2, stationary solutions are
viewed as cocycle-invariant random points in the infinite-dimensional state space. The pathwise
local structure of solutions of semilinear see’s and spde’s near stationary solutions is described in
terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary
solution. More specifically, we establish local stable manifold theorems for semilinear see’s
and spde’s (Theorems 2.4.1-2.4.4). These results give smooth stable and unstable manifolds in the
neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable
and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary
solution and are asymptotically invariant under the stochastic semiflow of the see/spde. Furthermore,
the local stable and unstable manifolds intersect transversally at the stationary point, and the
unstable manifolds has fixed finite dimension. The proof uses infinite-dimensional multiplicative
ergodic theory techniques, interpolation and perfection arguments (Theorem 2.2.1).

/pdf/the-stable-manifold-theorem-for-semilinear-stochastic-13rq9to3nw.pdf

Salah-Eldin A. Mohammed

Papers

Stochastic Functional Differential Equations

A Delayed Black and Scholes Formula

Stochastic Differential Systems With Memory: Theory, Examples and Applications

A Delayed Black and Scholes Formula II

The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations