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On a probability space $(\Omega,\mathscr{F},\mathbb{P})$ with filtration generated by Brownian motion, there is a progressivley process $(A_t)_{t\in[0,T]}$. If for any stopping times $0\leq \sigma\leq \tau\leq T$, $A_{\sigma}\leq A_{\tau}$, then the process $(A_t)_{t\in [0,T]}$ is an increasing process?

How to prove the proposition?

shixuejun
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1 Answers1

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Fix $0 \leq s \leq t \leq T$. Then $$\sigma := s \qquad \quad \tau := t$$ are (trivial) stopping times satisfying $\sigma \leq \tau$. By assumption, $$A_s = A_{\sigma} \leq A_{\tau} = A_t.$$

saz
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