Analysis of Financial Time Series, Third Edition

6.2 Some Continuous-Time Stochastic Processes

In mathematical statistics, a continuous-time continuous stochastic process is defined on a probability space (Ω, F, P), where Ω is a nonempty space, F is a σ field consisting of subsets of Ω, and P is a probability measure; see Chapter 1 of Billingsley (1986). The process can be written as {x(η, t)}, where t denotes time and is continuous in [0, ∞). For a given t, x(η, t) is a real-valued continuous random variable (i.e., a mapping from Ω to the real line), and η is an element of Ω. For the price of an asset at time t, the range of x(η, t) is the set of nonnegative real numbers. For a given η, {x(η, t)} is a time series with values depending on the time t. For simplicity, we write a continuous-time stochastic process as {xt} with the understanding that, for a given t, xt is a random variable. In the literature, some authors use x(t) instead of xt to emphasize that t is continuous. However, we use the same notation xt, but call it a continuous-time stochastic process.

6.2.1 Wiener Process

In a discrete-time econometric model, we assume that the shocks form a white noise process, which is not predictable. What is the counterpart of shocks in a continuous-time model? The answer is the increments of a Wiener process, which is also known as a standard Brownian motion. There are many ways to define a Wiener process {wt}. We use a simple approach that focuses on the small change Δwt = wt+Δt − wt associated with a small increment Δt in time. ...