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Results tagged with brownian-motion
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user 127334
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answers
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Intersection of a Poisson bridge and a Brownian bridge
Take a Poisson process $N_t$, a Brownian motion $W_t$ and constants $T > 0$ and $a > 0$. Suppose $N$ and $W$ are independent. I'm interested in the probability that $W$ does not cross over $a + N_t$ d …
- 222
1
vote
Accepted
Limit of an integral / Boundary behaviour of a Gaussian convolution / single layer potential
First note that
$$ \partial_x k(s, x) = -\frac{x}{s}k(s,x),
\quad s > 0, x \in {\mathbb R},$$
so your integral becomes
$$
q_a(t,x) := \int^t_0 \partial_x k(s,x-a)k(t-s,a)ds.
$$
Now suddenly your …
- 222