What is the Stochastic Differential Equation associated with $X_t = \frac{1}{t+W_t}$ where $W_t$ is a Wiener Process? I have tried to apply Ito's Formula here, but there is a discontinuity at $0$. Does anyone have any ideas?
What is the Stochastic Differential Equation associated with $X_t = \frac{1}{t+W_t}$ where $W_t$ is a Wiener Process?
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stochastic-processes
stochastic-calculus
brownian-motion
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0Thinking that $W_t$ returns to $0$ infinitely many times near $t = 0$ and $W_t$ grows like $\sqrt{t}$, I am worried that zeros of $t+W_t$ may accumulate near $0$. Do you really want that our Wiener process begins at $0$? – 2017-01-17
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1@SangchulLee: Indeed by the Cameron-Martin theorem the zeros do accumulate near 0. – 2017-01-17
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0@SangchulLee Hi, yes I was given this problem for a class I taught. – 2017-01-17
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0So is this problem undefined? – 2017-01-18
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1I would go back to whoever posed this problem and ask them just what they think is going on near $t=0$. – 2017-01-18