Probability Density of ##x## (Wiener Process)

[lelouch_v1]

Homework Statement

Given $x(t)=\exp{-b(W(t))^2}$, where $W(t)$ is a Wiener process, solve the below questions:
i)What values can x take?
ii)What is the probability density for x?

Relevant Equations

$P(W)=\frac{\exp{-W^2/(2t)}}{\sqrt{2\pi t}}$

Suppose that W(t) is just a Wiener process (i.e. a Gaussian in general). I want to know what the probability density for x, P(x), is. I started off by just assuming that I want to measure the expectation value of an observable f(x), so $<f(x)>=\int_{W=0}^{W=t}{P(W)f(g(W))dW} \ \ ,\ \ x=g(W)$ Then I transformed variables from W to t and I got $$<f(x)>=\int_{x=g(0)}^{x=g(t)}P(g^{-1}(x))f(x)(\frac{dW}{dx})dx=\int_{x=g(0)}^{x=g(t)}\frac{P(g^{-1}(x))}{g'(g^{-1}(x))}f(x)dx$$ so I just assume that the probability for x is $$P(x)=\frac{P(g^{-1}(x))}{g'(g^{-1}(x))}$$ Since x=g(W), then $$W=g^{-1}(x)$$, but from (1) I get $$\frac{dW}{dx}=\frac{1}{2x\sqrt{blnx}}$$ and assuming that $$\sigma^2=V=t$$, I get $$P(x)=\frac{e^{lnx/{2bt}}2\sqrt{blnx}}{x\sqrt{2{\pi}t}}$$ Is this right? Shouldn't I get a Gaussian? Also, was I right to take the values of x to be [0,t] and not $(-\infty,+\infty)$ ? Thank you in advance :)

 

[andrewkirk]

I get $$P(x)=\frac{e^{lnx/{2bt}}2\sqrt{blnx}}{x\sqrt{2{\pi}t}}$$ Is this right? Shouldn't I get a Gaussian?

There's no reason to expect a Gaussian. The random variable $W_t$ is Gaussian, but $X_t$, being a non-linear transformation of $W_t$, is not.

Rather than wrestling with all those integrals, a simpler and more intuitive way to calculate the probability density is to first calculate the cumulative probability function, then differentiate.

Start with:

$$Pr(X_t < x) = Pr \left(\exp\left(-bW_t{}^2\right) < x\right)$$
then manipulate that until you get a probability that $W_t$ lies in a certain region (which will consist of the real line excluding an interval centred on zero). Then express that probability in terms of $\Phi$, the cumulative probability function of the standard normal distribution.

Then differentiate that wrt $x$ and you'll have the probability density function.

There may be an alternative approach using Ito's Lemma, but you may not have studied that lemma yet, and I think it may take longer than the above anyway.