Questions tagged [brownian-motion]

Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

2,583 questions
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Evaluate $\int_0^r e^{ku}\; dW_u$.

I want to know how to find $$\int_0^r e^{ku}\; dW_u$$ where $W_u$ is a Wiener process with mean 0 and variance $u$.
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Simplifying a Stochastic Equation

Let $$dG_t=\mu(t)dt+\sigma(t)dW_t$$ where $W_t$ is standard Brownian motion and define $$f(t,u)=E[e^{-r(u-t)}(\mu(u)-rG_u)|F_t]$$ where $(F_t)_{t\in [0,T]}$ is a filtration, $T\ge u\ge t$ and $r$ is a ...
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Skorokhod Embedding theorem proof

It occurs in Durrett's proof of Skorokhod embedding that he needs the following. Suppose that we have a Brownian motion $B_t$ in $1$-d that starts at $0$. (To be clear, it is not necessarily ...
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Simple Exploitation of Markov Property for Brownian Motion

The elementary Markov Property as I learned it reads: For a positive real number $a$ the stochastic processes $(B_t)_{0 \leq t \leq a}$ and $(W_t)_{t \geq 0}:=(B_{t+a}-B_a)_{t \geq 0}$ are ...
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Revuz and Yor's Book “Continuous Martingales and Brownian Motion” - Chapter 1 - Exercise 1.19

Context : This post is the second of a series of posts taking their origins from the exercises in the Revuz and Yor's Book "Continuous Martingales ans Brownian Motion". The reason for doing so is ...
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