Suppose $(X_n)_{n\ge 1}$ and $(Y_n)_{n\ge 1}$ are two stochastic processes which are equivalent in law. Then the joint distributions $(\sum_{k=1}^nX_k)$ and $(\sum_{k=1}^nY_k)$ also coincide. Can we say that the law of the running maximums $X_n^*=\max_{1\le j\le n}\sum_{k=1}^jX_k$ and $Y_n^*=\max_{1\le j\le n}\sum_{k=1}^jY_k$ coincide?
Sorry is this is trivial. I do not have much intuition regarding probability!
Yes. For any measurable function $f$ on $\mathbb R^{n}$ the random variables $f(X_1,X_2,...,X_n)$ and $f(Y_1,Y_2,...,Y_n)$ have the same law.
Take $f(x_1,x_2,...x_n)=\max_{1\leq j\leq n}\sum\limits_{k=1}^{j}x_k$.