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2 votes
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Probability that Brownian Motion takes value in an $L^2$-Ball

The answer is positive. Indeed, there exists a continuous function $g\in B_v(\delta)$. Moreover, for some $r>0$, $$B^\infty_g(r) := \{h \in C([0,1]): ||h-g||_\infty<r\}\subset B_v(\delta).$$ It ...
zhoraster's user avatar
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2 votes
Accepted

Showing bounds of Stochastic Process

Consider the equation $$ \mathrm{d}Y_t = \biggl( 2 + 8 \frac{e^{Y_t} - 1}{e^{Y_t} + 1} \biggr) \, \mathrm{d}t + 4 \, \mathrm{d}B_t. \tag{1}\label{e:1} $$ The coefficients $\mu(y, t) = 2+\operatorname{...
Sangchul Lee's user avatar
1 vote
Accepted

May the sum of Wiener processes be a Wiener process?

Let $B_t$ and $\tilde{B}_t$ be two independent standard Wiener processes, and define $W_t$ and $\tilde{W}_t$ by \begin{align*} W_t &= \frac{1}{2}B_t + \frac{\sqrt{3}}{2} \tilde{B}_t, & \tilde{...
Sangchul Lee's user avatar

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