A stochastic process is the sum of a set of random variables that depend on parameters (usually time). Random variables are quantitative manifestations of random phenomena, and their values change with the influence of accidental factors. For example, the number of customers received by the store from time t0 to time tK is a set of random variables depending on time t, that is, a random process.

Stochastic process theory came into being at the beginning of the 20th century [1], and developed gradually to meet the needs of physics, biology and management science. It is widely used in automatic control, public utilities, management science and other fields. One is probability method, which uses orbital properties, stopping time and stochastic differential equations. The other is analytical method, in which measure theory, [4] differential equation, semigroup theory, function heap and Hilbert space are used.

In practical research, the two methods are often used in combination. In addition, combinatorial method and algebraic method also play a certain role in the study of some special stochastic processes.

research contents

The main contents include: multi-index stochastic processes, infinite particles and Markov processes, probability and potential, and special discussions on various special processes.

Chinese scholars have done well in [5] stationary processes, Markov processes, [6] martingale theory, limit theorems, stochastic differential equations and so on.

Mathematical stochastic process is a mathematical structure caused by the concept of actual stochastic process. People study this process, either because it is a mathematical model of actual stochastic process or because of its inherent mathematical significance and its application outside the field of probability theory.

Mathematical stochastic process can be simply defined as a set of random variables, that is, a parameter set is specified, and a random variable x(t) is specified for each parameter point T. If we recall that the random variable itself is a function, a point in the domain of the random variable x(t) is represented by ω, and the value of the random variable in ω is represented by x(t, ω), then the stochastic process is completely defined by the point pair (t). If t is fixed, this binary function defines a function of ω, that is, a random variable represented by x(t). If ω is fixed, this binary function defines a function of t, which is the sample function of the process.