Stochastic Processes in Continuous Time

We consider random processes defined on a probability space

which take values in a fixed state space. In general, the state space is at least a measurable space $(S,\mathcal{S})$; in this script, however, we will only consider the case

Most of the results in the first chapter can be transferred without substantial changes to the case where $S$ is a so-called Polish space.

This definition emphasizes the dependence on the time parameter $t$. Conversely, for fixed time points

we may consider the random vector

The distribution of $(X_{t_1},\ldots,X_{t_n})$ is called the finite-dimensional marginal distribution of $X$ at the time points $(t_1,\ldots,t_n)$.

Many interesting aspects of stochastic processes arise from the interaction between “path properties” (i.e. properties of the mappings $t\mapsto X_t(\omega)$) and properties of the marginal distributions. In general, we will impose certain regularity requirements on the paths of the stochastic processes under consideration.

Continuity is a path property of $X$ which cannot be inferred from the finite-dimensional marginal distributions of $X$ alone. Analogously, one can define right-continuous stochastic processes. Right-continuity of the path $t\mapsto X_t(\omega)$ means that at every $t\geq0$ the right-hand limit

exists and agrees with $X_t(\omega)$.

The most important example of a continuous process is the (already familiar) Brownian motion.

For the existence of Brownian motion, see Steele [2003] or Bolthausen [2009]. The requirement of continuous paths excludes some important stochastic processes which possess only right-continuous paths. An important example is the Poisson process, which can be defined as follows.

Let $(E_k)_{k\in\mathbb{N}}$ be a family of independent exponentially distributed random variables with parameter $\lambda>0$. Define

Then $(N_t)_{t\geq0}$ is called a Poisson process with rate $\lambda$. It is easy to verify that $N$ is a right-continuous but not a continuous stochastic process.