Suppose that we have a function of the form $$x(t)=e^{-b(W(t))^2}$$ where $W(t)$ is a Wiener process. Since the values that the Wiener process can take belong in $ [0,T] $, i assumed that the values that x can take belong in $ [1,e^{-bT^2}]$, but is that right? Thank you in advance.
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What is $T$? If $b$ is a positive constant, then $W_t^2$ can take on any value on the positive real line, which implies that $x(t)$ can take on any values in the interval $(0,1]$. – Jan Stuller Dec 01 '20 at 15:03