Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If a stochastic process $X_{t}$ $\sim N(0,t^3/3)$ and $Y_{t}$ is defined as follows:

$Y_{t} = X_{t}/t$, if $t>0$ and $ Y_{t} = 0$, if $t=0$, then how can I show that $Y_{t}$ is continuous in $t>=0$ almost everywhere?

I started as follows: $X_{t}$ is continuous in $t$ and since the probability measure of $w$ such that $Y_{0}(w)=0$ is $0$, $Y_{t}$ is continuous in $t$ almost everywhere. Any help would be great.

Thanks Trambak

Later. (Copied from an answer where Trambak mistakenly added it —Mariano)

Well $X_t$ is actually $\int_{0}^{t}(t-s)dW_{s}$, where $W_{s}$ is a Brownian Motion. One can show that $X_{t}$ will have a distribution as given in the question above. From this point onwards, how can I prove that $Y_{t}$ is continuous in $t$ with probability 1?

share|improve this question
    
Is $X_t$ supposed to be continuous? –  Ilya Jan 29 '13 at 12:18

1 Answer 1

Hint: Rewrite $\int_{0}^{t}(t-s)dW_{s}$ as $\int^t_0 W_s ds$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.