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If I have a stochastic process defined as usual by $dx=f(x,t)dt+g(t,x)dW$, how can I compute the Ito's formula for a process $n=\phi(t,x):=(x/t>a)$, i.e., $dn = (\ldots)dt + _\ldots$ ?

I have relaxed $n$ as $\tilde{n}:=\frac{1}{1+e^{-2k\xi}}$, where $\xi:=\frac{x}{t}-a$ and $k$ is large.

However, I find several difficulties in applying Ito's lemma numerically to such a process. Maybe I am missing something fundamental...

asciatopo
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1 Answers1

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Consider $y_t=u(t^{-1}x_t)$ where $$ u(z)=(1+\exp(-2k(z-a)))^{-1}, $$ then $$ \mathrm dy_t=t^{-1}u'(t^{-1}x_t)\mathrm dx_t-t^{-2}u'(t^{-1}x_t)\mathrm dt+\tfrac12t^{-2}u''(t^{-1}x_t)\mathrm d\langle x,x\rangle_t. $$ One knows that $$ \mathrm dx_t=g(x_t,t)\mathrm dw_t+f(x_t,t)\mathrm dt, $$ hence $$ \mathrm d\langle x,x\rangle_t=g^2(x_t,t)\mathrm dt, $$ and it remains to identify $u'$ and $u''$.

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