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Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion. Denote $Y := \int_0^1 f(t) \, dW_t$. For each $\varepsilon > 0$, consider the conditioned random va …

Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely. Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event $$ E_T := \{|B_s| \geq \lambda s\ \text{ …

Note: This question is closely related to an earlier question: A large noise limit. Let $W$ be a standard one dimensional Brownian motion. For every $\varepsilon > 0$, let $A_\varepsilon$ denote the e …

Let $W$ be a standard $d$-dimensional Brownian motion. Suppose $b: \mathbb R^d \to \mathbb R^d$ is measurable and bounded. Consider, for every $\varepsilon > 0$, the solution $X^\varepsilon$ on $[0, T …

The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\v …