The finite-dimensional distributions for a Wiener process are given by this formula?

Question

A stochastic process $X = {X_t}$ on is Wiener Process if the following properties hold

$X_0 = 0$
$X$ has independent increments: for any $ninmathbb{N}$ and any $0 < t_0<ldots < t_m$ we have that $(X_{t_1}-X_{t_0}), (X_{t_2}-X_{t_1}), ldots (X_{t_n}-X_{t_n-1})$ are independent
$X_t - X_s sim mathcal{N}(0, t-s)$ where $sleq t$
$tto X_t(omega)$ is continuous for almost all sample paths $omega$

I am looking for a proof that these properties imply that the finite-dimensional distributions are given by the following formula. Letting $0leq t_1 < ldots < t_n$ and any finite collection of Borel sets $F_1,ldots F_n$ that
$$P(X_1in F_1, ldots X_nin F_n) = int_{F_1timesldots times F_n}p(t_1, 0, x_1)p(t_2 - t_2, x_1, x_2)ldots p(t_n-t_{n-1}, x_{n-1}, x_n) dx_1ldots dx_n $$
where $p(t, x, y) = frac{1}{sqrt{2pi t}}exp(-frac{(x-y)^2}{2t})$ when $t > 0$ and $p(0,x,y) = delta_x(y)$? We know from Kolmogorov's Extension Theorem that the family of all these finite-dimensional distributions will give us a Wiener process, I am curious if the properties go the other way and for a proof.

julian · Answer

The easiest way to prove this `Chapman-Kolmogorov' relation is by noting that it suffices to prove that it is equivalent to the fact that $(X_{t_1},dots,X_{t_n})$ is multinomial normal distributed with mean $0$ and convariance matrix $C_{i,j}=t_{iwedge j}$. For this observe that by 2. and 3., $(X_{t_1},X_{t_2}-X_{t_1},dots,X_{t_n}-X_{t_{n-1}})$ is multinomial normal distributed with mean $0$ and covariance matrix
$$
bar{C}=mathrm{diag}(t_1,t_2-t_1,dots,t_n-t_{n-1}).
$$
Now
$$
begin{pmatrix}
X_{t_1}\vdots\X_{t_n}
end{pmatrix}=begin{pmatrix}
1 & 0 &cdots & 0\ 
1 & 1 & cdots & 0\ vdots&  & ddots & vdots\
1 & 1 & cdots & 1
end{pmatrix}begin{pmatrix}
X_{t_1}\X_{t_2}-X_{t_1}\vdots\X_{t_n}-X_{t_{n-1}}
end{pmatrix}
$$
and therefore
$$
C=begin{pmatrix}
1 & 0 &cdots & 0\ 
1 & 1 & cdots & 0\ vdots&  & ddots & vdots\
1 & 1 & cdots & 1
end{pmatrix}bar{C}begin{pmatrix}
1 & 1 &cdots & 1\ 
0 & 1 & cdots & 1\ vdots&  & ddots & vdots\
0 & 0 & cdots & 1
end{pmatrix}
$$
and you can easily check that this indeed gives you the required covariance matrix.

The finite-dimensional distributions for a Wiener process are given by this formula?

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