In practice, other than for a small number of special processes, it is not possible to write down transition functions explicitly. Equivalently, it is a Markov process with the transition function Alternatively forms a semigroup.Īs an example, standard Brownian motion, B, has the defining property that is normal with mean 0 and variance t– s independently of, for times. The identity is known as the Chapman-Kolmogorov equation, and is required so that the transition probabilities are consistent with the tower rule for conditional expectations. Ī process X is Markov with transition function, and with respect to a filtered probability space if it is adapted and This is not much of a restriction because, given an inhomogeneous Markov process X, it is always possible to look at its space-time process taking values in, which will be homogeneous Markov.ĭefinition 2 A homogeneous transition function on is a collection of transition probabilities on such that for all. I only consider the homogeneous case here, meaning that depends only on the size t– s of the time increment and not explicitly on the start or end times s, t, so the notation can be replaced by. This is just the tower rule for conditional expectation,Ī Markov process is defined by a collection of transition probabilities, one for each, describing how it goes from its state at time s to a distribution at time t. Then, the combination MN describes how it transitions from its state at time s to u. Suppose that M, N are transition probabilities describing how a process X goes from its state at time s to its conditional distribution at time t and from its state at time t to a distribution at time u respectively, for. Two kernels M and N can be combined by first applying N followed by M to get the kernel MN, In the case where N is a transition probability, is well-defined and bounded for all bounded and measurable. This extends to positive linear combinations of such indicator functions (the simple functions) and, then, by monotone convergence, measurability of extends to all nonnegative measurable f. For the case of an indicator function of any, is measurable by definition. Then, is itself a measurable function from E to. Given any such kernel N and, we denote the integral of a measurable function with respect to the measure by This can be used to describe how the conditional distribution of a process at a time t depends on its value at an earlier time s If, furthermore, for all, then N is a transition probability.Ī transition probability, then, associates to each a probability measure on. To state the definition of transition functions, it is necessary to introduce the concept of transition probabilities.ĭefinition 1 A (transition) kernel N on a measurable space is a map These specify how the distribution of is determined by its value at an earlier time s. Ĭontinuous-time Markov processes are usually defined in terms of transition functions. A process X is Markov with respect to if it is adapted and ( 1) holds for times. More generally, this idea makes sense with respect to any filtered probability space. The Markov property then says that, for any times and bounded measurable function, the expected value of conditional on is a function of. To make this precise, let us suppose that X takes values in a measurable space and, to denote the past, let be the sigma-algebra generated by. Intuitively speaking, a process X is Markov if, given its whole past up until some time s, the future behaviour depends only its state at time s. In fact, all of the special processes considered ( Brownian motion, Poisson processes, Lévy processes, Bessel processes) satisfy the much stronger property of being Feller processes, which I will define in the next post. Although I do not take the second approach, all of the special processes considered in the current section are Markov, so it seems like a good idea to introduce the basic definitions and properties now. An alternative starting point would be to consider Markov processes. In these notes, the approach taken to stochastic calculus revolves around stochastic integration and the theory of semimartingales.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |