$E[|X(t)|]\leq K\implies E[|X(\tau)|]\leq K $?

Question

Let $X(t)$ be a stochastic process. Assume that, for every $t\leq M$, it holds $$E[|X(t)|]\leq K, $$ for some constant $K$. Let now $\tau\in[0,M]$ be random (stopping time). Is it true that also $$E[|X(\tau)|]\leq K ?$$ Thank you in advance for every suggestion.

Answer 1

No, this is (in general) not correct. Just consider the following example:

Let $X$ be a random variable such that $\mathbb{P}(X=1) = \mathbb{P}(X=-1) = \frac{1}{2}$ and define $$X_t := 1_{\{X=1\}}$$ for $t \leq 1$ and $$X_t := 1_{\{X=-1\}}$$ for $t>1$. Then $\mathbb{E}(|X_t|) \leq \frac{1}{2}$ for all $t \geq 0$. On the other hand, if we set $$\tau := \inf\{t>0; X_t = 1\},$$ then $\tau \leq M :=1$ almost surely and $$\mathbb{E}(|X_{\tau}|) = 1 > \frac{1}{2}.$$