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prove $P\{\int_{0}^{\infty}f(W_s)ds=\infty\}=1$

We need to show that $$ \int_0^T f(B_t) dt\to\infty, T\to\infty. $$ I will use the occupation density formula (UPD see a simpler argument below): $$ \int_0^T f(B_t) dt = \int_{\mathbb{R}} f(x) L^x_T(B)...
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3 votes
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Lévy's characterization of Brownian motion: right-continuous processes

Cool question. Proposition Let $(X_u)_{u}$ be right-continuous martingale with $X_0=0$, such that $(X^2_u-u)_u,(X_u^3-3uX_u)_u,(X_u^4-6uX_u^2+3u^2)_u$ are martingales. Then for every integer $M \ge 1$,...
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Compute the probability of an event regarding a stopping time

We have that $dX_t=bdt+\sigma dW_t,\,X_0=x$. I suppose $x \in (a,b)$. To find $f$ we must find a solution to the second order ODE described in OP. Consider $$f(y)-f(z)=\int_{z}^ye^{-\frac{2b}{\sigma^2}...
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Issues with solving a Stochastic Differential Equation

To complete the solution you need to find what $F^{-1}(x)$ is. We start by finding the form of $b(X_t)$. We see from by comparing the form of $dX_t=\frac{1}{2}b(X_t)b'(X_t) + b(X_t) dW_t$ with $dX_t=-\...
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Brownian motion and martingale

Yes, this can be proved in the same way. As noted in the comments, when $f$ is harmonic the second term vanishes.
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2 votes

Lie algebras and the Hörmander condition for SDEs

After e-mailing an expert about the questions listed for the bounty, I now have some answers! Firstly, yes it is true that $[[\sigma_1,\sigma_2],[\sigma_3,\sigma_4]] \in \mathcal{V}_1$, by the Jacobi ...
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What is the integral of a Gaussian white noise

To long for a comment The Wikipedia article you found mentions under the section Continuous-time white noise that "However, a precise definition of these concepts is not trivial, because some ...
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What is the integral of a Gaussian white noise

Not exactly what you are asking for but if you use the framework developed by Takeyuki Hida, then the White noise process $[0,\infty)\ni t\mapsto w(t)$ can be seen as a $(S)^*$-valued process (i.e. a ...
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