3 votes
Accepted

How to approach a second order ODE with Dirac Delta and?

First of all, this equation makes no sense as the Brownian paths are barely continuous. So different interpretations of this equation to some regular structure might yield different results. One can ...
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1 vote

Symmetric Random Walk on a 2D grid

This is the sort of result you should have got ...
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  • 143k
1 vote
Accepted

Symmetric Random Walk on a 2D grid

the simulation is not capturing the right problem. to get the temp. at a point $(x,y)$, start many random walks at that point, then average the results. Run a loop over the starting points $(x,y)$. ...
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  • 15.5k
1 vote

What is a continuous stochastic process?

The set $\Omega$ is the sample space, which can be any set you want. Since $\Omega$ is a set, though, it doesn't make sense to say that $\omega \in \Omega$ "appears just once at time $k$". ...
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  • 9,047
1 vote

Formal (mathematical) proof of $\mathbb{P}(W_{t}>b)=\int_{0}^{t} \mathbb{P}\left( W_{t} > b \mid \tau_{b} =s \right) dF(s)$

I will write $Y$ for $\tau_b$. Let $\phi (Y)=P(W_t>b|Y)$. Then $\int_0^{\infty} P(W_t>b|Y=s)dF(s)$ is just a notation for $\int_0^{\infty} \phi (s)dF(s)$. Taking expectation on both sides of $\...
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  • 5,298
1 vote
Accepted

$\mathbb{P}(\tau_a\leq \tau_{-a}\leq t)=\mathbb{P}(\tau_{3a}\leq t)$ for a Brownian motion?

The claimed identity is not true. Rather than trying to use an identity derived from the reflection principle, it is better to use the reflection principle directly. Let $\gamma_{-a}:=\min\{t>\...
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  • 15.5k
1 vote

Why is the set $\{ W_\cdot(\omega)$ is locally $\alpha$-Hölder at some $s\in[0,1]\}$ contained in ths messy intersection/union of measurable sets?

Suppose that $f \in C[0,1]$ is locally Holder continuous of order $\alpha$ at $s \in [0,1/2]$ (the case where $s>1/2$ is similar). Then there exists $\delta>0$ and $C_1<\infty$ such that $|f(...
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