1. Wald’s equation
Let ( X n ) n ∈ N (X_n)_{n∈ℕ} (Xn)n∈N be a sequence of real-valued, independent and identically distributed (i.i.d.) random variables and let N N N be a nonnegative integer-value random variable that is independent of the sequence ( X n ) n ∈ N (X_n)_{n∈ℕ} (Xn)n∈N. Suppose that N N N and the X n X_n Xn have finite expectations. Then
2. Wald’s martingale
In probability theory Wald’s martingale, named after Abraham Wald and more commonly known as the geometric Brownian motion, is a stochastic process of the form
for any real value λ where Wt is a Wiener process. The process is a martingale.
2.1 Wiener process
In mathematics, the Wiener process is a real valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion.[1] It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.
The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and disturbances in control theory.
The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model.
3. Walt's theorem
Wald theorem:
Let {X n, n ≥ 1} \{X_n,n\geq 1\}{
Xn,n≥1 } is the iid random variable sequence,E [X 1] <+ ∞ E[X_1]<+∞And [ X1]<+ ∞ , r is a random variable with a positive integer value,E r <+ ∞ Er<+∞E r<+ ∞ , and for everythingn ≥ 1, r = nn\geq 1,{r=n}n≥1,r=n and{X n + 1, X n + 2……} \{X_{n+1},X_{n+2}……\}{
Xn+1,Xn+2…… } Aremutually independent, thenE ∑ rn = 1 X nl <+ ∞ E\sum_r n=1 Xnl<+∞E∑rn=1Xnl<+∞,且 E [ X 1 + X 2 + … + X r l ] = E r E x 1 E[X_1+X_2+…+Xrl]=ErEx1 And [ X1+X2+…+X r l ]=ErEx1
https://en.wikipedia.org/wiki/Wald%27s_equation
https://en.wikipedia.org/wiki/Wiener_process#Brownian_martingales
https://zh.wikipedia.org/wiki/%E7%BB%B4%E7%BA%B3%E8%BF%87%E7%A8%8B