Diffusions, Markov processes and Martingales, vol 2: Ito calculus

Diffusions, martingales, and Markov processes are each particular types of stochastic processes. A stochastic process, in a state space E, with parameter set T, is a family (Xt)t∈T of E-valued random variables, or equivalently, a random variable X that takes its values in a space of functions from T to E. Usually, the parameter set T is a subset of R, often [0,∞) or {0, 1, 2, 3, . . .}, the parameter is thought of as time, and the functions from T to E are thought of as paths in E. A stochastic process thus describes the evolution in time of a system for which we do not know the path in state space that the system will follow, but only the probability (usually “infinitesimal”) of each possible path that the system might follow. A Markov process is a stochastic process whose future evolution at any given time t depends only on the state of the system at the present time t and not on the states of the system at past times s < t. A diffusion is a Markov process whose paths are continuous functions of time. Brownian motion is the quintessential example of a diffusion, and the Poisson process is the quintessential example of a Markov process that is not a diffusion. A martingale is a stochastic process that models the fortune of a gambler as a function of time if the gambler is playing a fair game. Martingale theory turns out to be a powerful tool for the study of Markov processes, because a Markov process has many martingales that are naturally associated to it. The evolution in time of a Markov process in E with time set [0,∞) may be described by a transition function Ps,t(x, dy), 0 ≤ s < t <∞, x, y ∈ E, which gives the conditional probability that Xt ∈ dy given that Xs = x. By considering the space-time process t 7→ (Xt, t), one may reduce to the case where Ps,s+t does not depend on s. Hence it is usual to assume that one is in this time-homogeneous case and to write Pt for Ps,s+t. When E = R , the special case where Pt is also spatially homogeneous is fundamental. In this case, Pt(x, dy) = Pt(z + x, z + dy), and the family μt(dx) = Pt(0, dx), 0 < t < ∞, is a convolution semigroup of probability measures on R. To avoid uninteresting pathology, it is customary to assume that μt converges weakly to the unit point mass at the origin as t tends to 0+. The process X is then called a Lévy process. In the theory of stochastic differential equations, the Lévy processes play a role analogous to the role played by straight lines in ordinary calculus. In the case of Brownian motion in R, μt(dx) has a normal density (2πt)−d/2 exp{−|x|/(2t)} with respect to Lebesgue measure and X can be taken to have continuous paths. Conversely, if X is a Lévy process in R with continuous paths, then for each t, since the random variable Xt −X0 is the sum of the independent increments Xkt/n −X(k−1)t/n, k = 1, . . . , n, and since the