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C. Doleans-Dade: Stochastic processes and stochastic differential equations.- A. Friedman: Stochastic differential equations and applications.- D.W. Stroock, S.R.S. Varadhan: Theory of diffusion processes.- G.C. Papanicolaou: Wave propagation and heat conduction in a random medium.- C. Dewitt Morette: A stochastic problem in Physics.- G.S. Goodman: The embedding problem for stochastic matrices.



Stochastic Processes and Stochastic Differential Equations

Introduction. Since Ito has defined the stochastic integral with respect to the Brownian motion, mathematicians have tried to generalize it. The first step consisted of replacing the Brownian motion by a square integrable martingale. Later H. Kunita and S. Watanabe in [10] introduced the concept of local continuous martingale and stochastic integral with respect to local continuous martingales which P. A. Meyer generalized to the non continuous case.
But in many cases one observes a certain process X and there are at least two laws P and Q on \(\left({\Omega,\underline{\underline{\text{F}}}}\right)\). For the law Q, X is not a local martingale but the sum of a local martingale and a process with finite variation. We would like to talk about the stochastic integrals \(\int\limits_{\text{P}} {\Phi _{\text{S}} {\text{dX}}_{\text{S}} }\) and \(\int\limits_{\text{Q}} {\Phi _{\text{S}} {\text{dX}}_{\text{S}} }\) in the two probability spaces \(\left( {\Omega,\underline{\underline {\text{F}}},{\text{P}}} \right)\) and \(\left( {\Omega,\underline{\underline {\text{F}}},{\text{Q}}} \right)\). And of course we would like those two stochastic integrals to be the same.
C. Doleans–Dade

Stochastic Differential Equations and Applications

A stochastic process x(t), tϵI is a family of random variables x(t) defined in a measure space (Ω,ℱ) or in a probability space (Ω,ℱ P); here x(t) is either real valued or n-vector valued and I is an interval, usually [0,∞). Notice that x(t) is a function x(t,ω)), ωϵΩ.
The function t → x(t,ϵ) is called a sample path ϵ. if a.e. sample path is continuous (right continuous), we say that the process x(t) is continuous (right continuous).
Avner Friedman

Theory of Diffusion Processes

Let x(t) be a Markov process and assume that
$$\text{E}\left[\phi\left(\text{t}\,\text{ + }\,\text{h}\right) - \phi\left(\text{x}\left(\text{t}\right)\right)|\text{x}\left(\text{s}\right)\,\,\,\,\text{for}\,\,\,\text{s} \leq \text{t} \right] = \text{hL}_\text{t} \phi \left(\text{x}\left(\text{t}\right)\right) + \text{o}\left(\text{h}\right)$$
for \({{\phi }} \in {\text{C}}_0^\infty \left( {{\text{R}}^{\text{d}} } \right)\). It is not difficult to check that Lt must be a linear operator which is quasi-local (i.e., for each x ∈ Rd and ε > 0 there is a constant Cε < ∞ such that |Ltφ(x)| ≤ Cε ∥φ∥ for all \( {{\phi }} \in {\text{C}}_0^\infty \left( {{\text{R}}^{\text{d}} \backslash \overline {{\text{B}}\left( {{\text{x,}} \in } \right)} } \right) \). Here and throughout ∥·∥ denotes the uniform norm.) Moreover, Lt must satisfy the weak maximum principle in that if \( {{\phi }}\left( {\text{x}} \right) = \mathop {\max }\limits_{{\text{y}} \in {\text{R}}^{\text{d}} } \,\,{{\phi }}\left( {\text{y}} \right) \) then certainly Ltφ(x) ≤ 0. From these observations one can conclude that Lt ought to be of the form
D. Stroock, S. R. S. Varadhan

Wave Propagation and Heat Conduction in a Random Medium

We shall give a fairly selfcontained account of some results on waves in random media and related problems that we have considered in the past few years [l]–[6]. These results rcly upon properties of solutions of differential equations with random coefficients, i.e., stochastic equations. We restrict attention to one-dimensional problems so that we are dealing with stochastic ordinary differential equations. There are a few results, at present, dealing with multidimensional problems at [cf. 12} but we shall not discuss these here.
G. C. Papanicolaou

A Stochastic Problem in Physics

The world is global and stochastic and physical laws are local and deterministic, Thus the problems discussed at this Summer School are the very fabric of physics. But physics asks some questions which go beyond the territory which has been explored here. I shall present one of them, show how far physicists have gone toward its solution and mention an important problem of current interest.
Probability theory begins with a probability space (Ω,ℱ,P). The careful definition of the g-field ℱ of subsets of Ω and of the probability measure p has given us many powerful theorems. It is also possible, and often preferable in physics, to define P as a promeasure,1 namely as a projective family of bounded measures defined on the system of finite dimensional spaces Q known as the projective system of Ω. We thus start from (Ω,Q,P) rather than (Ω,ℱ,P). This is excellent for statistical mechanics. Unfortunately, in quantum mechanics, we have to deal with families of unbounded measures on the projective system Q of Ω. And this is the'key issue in the study of Feynman path integrals.
Cecile Dewitt‐Morette

The Embedding Problem for Stochastic Matrices

An n×n matrix P = [pij], with non-negative entries, is said to be stochastic if the entries along any row sum to one. In 1938, G. Elfving [2] formulated the embedding problem for stochastic matrices, essentially as follows.
For what matrices P can there be found a value t0 >0 and a continuous family of stochastic matrices P(s,t) on O⩽s⩽t⩽t0 that satisfies the functional equation
$${\text{P}}\left( {{\text{s,t}}} \right) = {\text{P}}\left( {{\text{s,}}\,{\text{u}}} \right){\text{P}}\left( {{\text{u,t}}} \right)\,\,{\text{whenever}}\,\,\left( {{\text{s}} \leqslant {\text{u}} \leqslant {\text{t}}} \right)$$
$${\text{P}}\left( {{\text{s,}}\,{\text{s}}} \right) = {\text{I}}\,\,\,\,\,\,\,\,\,\,\,\underline {{\text{for all}}} \,\,\,\,0 \leqslant {\text{s}} \leqslant {\text{t}}_{\text{o}}$$
and is such that
$${\text{P}}\left( {{\text{0, t}}_{\text{o}} } \right) = {\text{P?}}\,$$
G. S. Goodman
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