Probability Distributions and Stochastic Processes
We introduce some basic facts on probability distributions and stochastic processes. Probability distributions and their characteristic functions are discussed in Sect. 1.1. Criteria for smooth densities of distributions are given by their characteristic functions. In Sect. 1.2, we consider Gaussian, Poisson and infinitely divisible distributions and give criteria for these distributions to have smooth densities. Concerning the density problem of an infinitely divisible distribution, we study the Lévy measure in detail. Regarding that the origin 0 is the center of the Lévy measure, we will give criteria for its smooth density by means of ‘the order condition’ at the center of the Lévy measure.
Then we consider stochastic processes. In Sect. 1.4, we consider Wiener processes, Poisson processes, Poisson random measures and Lévy processes. Among them, Poisson random measures are exposed in detail. Next, in Sects. 1.5 and 1.6, we discuss martingales, semi-martingales and their quadratic variations. These are standard tools for the Itô calculus. In Sect. 1.7, we define Markov processes. The strong Markov property will be discussed. In Sect. 1.8, we study Kolmogorov’s criterion for a random field with multi-dimensional parameter to have a continuous modification.
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