Moto Browniano

Il moto Browniano è un processo stocatisco a tempo continuo $(B_t{)}_{t\ge 0}$ è un elemento di $C[0,1]$ distribuito secondo la Wiener measure. Then $B_t\sim N(0,t)$ and

$$Cov(B_s,B_t)=s\wedge t$$

assume w.l.o.g. $s<t$ then

$$\mathbb{E}[B_s{B}_t]=\mathbb{E}[(B_t-B_s)B_s]+\mathbb{E}[B_s^2]$$ $$=\mathbb{E}[B_t-B_s]\mathbb{E}[B_s]+s=s$$

since $(B_t-B_s)$ is indipendent of $B_s-B_0=B_s$.

We can compute the law of the increments via the characteristic function:

$$\mathbb{E}[e^{iu{B}_t}]=\mathbb{E}[e^{iu(B_t-B_s)}]\mathbb{E}[e^{iu{B}_s}]$$ $$\varphi (B_t-B_s)=\frac{\mathbb{E}[e^{iu{B}_t}]}{\mathbb{E}[e^{iu{B}_s}]}=e^{-(t-s)u^2/2}$$

dunque $B_t-B_s\sim N(0,t-s)$. Thus increments are stationary, meaning their distribution doesn’t depend on time.

From the indipendence of the increments we can compute the probability of cylinder sets:

$$\mathbb{P}(B_{t_1}\in I_1,B_{t_2}\in I_2,\dots )=\mathbb{P}(B_{t_1}\in I_1,(B_{t_2}-B_{t_1})+B_{t_1}\in I_2,\dots )$$

and the fact that given $X,Y$ indipendent random variables:

$$f_{X,X+Y}(x,y)=f_X{f}_Y(y-x)$$

so from the properties of the Wiener measure, its value is deterined on all cylinder sets, which form a Pi-system: so the measure is uniquley determined on all measurable sets.