Normal Distribution and Strong Markov Property
BROWNIAN MOTION
1. I NTRODUCTION
1.1. Wiener Process: Definition.
Definition 1. A standard (one-dimensional) Wiener process (also called Brownian motion) is a stochastic process {Wt }t≥0+ indexed by nonnegative real numbers t with the following properties: (1)
(2)
(3)
(4)
W0 = 0.
With probability 1, the function t → Wt is continuous in t.
The process {Wt }t≥0 has stationary, independent increments.
The increment Wt+s − Ws has the N ORMAL(0, t) distribution.
A Wiener process with initial value W0 = x is gotten by adding x to a standard Wiener process. As is customary in the land of Markov processes, the initial value x is indicated
(when appropriate) by putting a superscript x on the probability and expectation operators. The term independent increments means that for every choice of nonnegative real numbers 0 ≤ s1 < t1 ≤ s2 < t2 ≤ · · · ≤ sn < tn < ∞, the increment random variables
Wt1 − Ws1 , Wt2 − Ws2 , . . . , Wtn − Wsn are jointly independent; the term stationary increments means that for any 0 < s, t < ∞ the distribution of the increment Wt+s − Ws has the same distribution as Wt − W0 = Wt .
In general, a stochastic process with stationary, independent increments is called a L´vy e process; more on these later. The Wiener process is the intersection of the class of Gaussian processes with the L´vy processes. e It should not be obvious that properties (1)–(4) in the definition of a standard Brownian motion are mutually consistent, so it is not a priori clear that a standard Brownian motion exists. (The main issue is to show that properties (3)–(4) do not preclude the possibility of continuous paths.) That it does exist was first proved by N. W IENER in about 1920.
´
His proof was simplified by P. L E VY; we shall outline L´ vy’s construction in section ?? e below. But notice that properties (3) and (4) are compatible. This follows from the following elementary property of the normal distributions: If X, Y are independent,
1. I NTRODUCTION
1.1. Wiener Process: Definition.
Definition 1. A standard (one-dimensional) Wiener process (also called Brownian motion) is a stochastic process {Wt }t≥0+ indexed by nonnegative real numbers t with the following properties: (1)
(2)
(3)
(4)
W0 = 0.
With probability 1, the function t → Wt is continuous in t.
The process {Wt }t≥0 has stationary, independent increments.
The increment Wt+s − Ws has the N ORMAL(0, t) distribution.
A Wiener process with initial value W0 = x is gotten by adding x to a standard Wiener process. As is customary in the land of Markov processes, the initial value x is indicated
(when appropriate) by putting a superscript x on the probability and expectation operators. The term independent increments means that for every choice of nonnegative real numbers 0 ≤ s1 < t1 ≤ s2 < t2 ≤ · · · ≤ sn < tn < ∞, the increment random variables
Wt1 − Ws1 , Wt2 − Ws2 , . . . , Wtn − Wsn are jointly independent; the term stationary increments means that for any 0 < s, t < ∞ the distribution of the increment Wt+s − Ws has the same distribution as Wt − W0 = Wt .
In general, a stochastic process with stationary, independent increments is called a L´vy e process; more on these later. The Wiener process is the intersection of the class of Gaussian processes with the L´vy processes. e It should not be obvious that properties (1)–(4) in the definition of a standard Brownian motion are mutually consistent, so it is not a priori clear that a standard Brownian motion exists. (The main issue is to show that properties (3)–(4) do not preclude the possibility of continuous paths.) That it does exist was first proved by N. W IENER in about 1920.
´
His proof was simplified by P. L E VY; we shall outline L´ vy’s construction in section ?? e below. But notice that properties (3) and (4) are compatible. This follows from the following elementary property of the normal distributions: If X, Y are independent,