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Let $Y_t$ be a stochastic process defined by $Y_t := M_t - W_t$, where $W_t$ is the Wiener process and $M_t := \max_{0 \leq u \leq t} W_u$.

Does $Y_t$ have independent increments?

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No. Sketch:

Fix $\epsilon>0$. Since $Y$ is non-negative we have: $\mathbb{P}[Y_1<\epsilon, Y_2-Y_1<-\epsilon]=0$, but $\mathbb P[Y_1<\epsilon]>0$ and $\mathbb{P}[Y_2-Y_1<-\epsilon]>0$.

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+1, Nice proof. – Sasha Nov 17 '11 at 20:10

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