Consider the linear standard Brownian motion $(B_t)_{t \ge 0}$. We define the maximum process $(M_t)_{t \ge 0}$ of $(B_t)_{t \ge 0}$ to be such that $M_t = \max_{0\le s \le t} B_s$.
Prove that the process $(M_t -B_t)_{t\ge 0}$ has the same distribution as a reflected Brownian motion $(|B_t|)_{t \ge 0}$.
The hint of this exercise consists prove that all finite dimensional distributions coincide using Donsker's theorem…
Also, I have a few questions about processes in general:
1) I know that if the processes have the same finite distribution laws, then they have the same distribution. Is there another way to prove that two processes have the same law?
2) Let $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ be sequences of processes which converge in distribution to the processes $X$ and $Y$, respectively.
a) What are the conditions that the processes and a given function $g$ should satisfy in order to have $g(X_n)\stackrel{D}{\to}g(X)$?
b) What are the conditions that should be satisfied in order to have $(X_n,Y_n) \stackrel{D}{\to} (X,Y)$?
Best Answer
1) Yes, check if they have the same characteristic function $\forall t$. Else, generally not. But you may be able to do that by some different characterization theorems for specific processes (e.g. Brownian Motion). $\\$
2a) From a theorem on weak convergence, you have $$X_n\overset{D}\to X \iff \lim_{n\to \infty} \mathbb{E}[g(X_n)]=\mathbb{E}[g(X)]$$ for every bounded Lipschitz continuous functions $g$ - or for general continuous functions, use the Mann–Wald theorem.
2b) Generally $$X_n\overset{D}\to X,Y_n\overset{D}\to Y \not\Rightarrow (X_n,Y_n)\overset{D}\to (X,Y).$$ This is also the case for $(X_nY_n)\overset{D}\to (XY)$ and $(X_n+Y_n)\overset{D}\to (X+Y)$.
Regarding the main question, I would just use the reflection principle and the invariance properties of BM here.