Possible values of x(t)

[Assassinos]

Suppose that we have a function of the form [MATH]x(t)=e^{-b(W(t))^2}[/MATH] where [MATH]W(t)[/MATH] is a Wiener process. Since the values that the Wiener process can take belong in [MATH] [0,T] [/MATH], i assumed that the values that x can take belong in [MATH][1,e^{-bT^2}][/MATH], but is that right? Thank you in advance.

 

[Dr.Peterson]

Suppose that we have a function of the form [MATH]x(t)=e^{-b(W(t))^2}[/MATH] where [MATH]W(t)[/MATH] is a Wiener process. Since the values that the Wiener process can take belong in [MATH] [0,T] [/MATH], i assumed that the values that x can take belong in [MATH][1,e^{-bT^2}][/MATH], but is that right? Thank you in advance.

Almost. As long as [MATH]W>0[/MATH], the function [MATH]e^{-bW^2}[/MATH] is monotonic, so the answer to your question is relatively easy. But if [MATH]b>0[/MATH], as I would guess, then it's monotonically decreasing.

Did you observe that [MATH]e^{-bT^2}<1[/MATH]? So what needs to be changed in your answer?