# Questions tagged [brownian-motion]

In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.

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### How can there be Brownian motions under different measures?

According to the definition, for a Brownian motion it holds that $W_0 = 0$, and $W_t - W_s \in N(0, t-s), \quad t > s$. This implies that $W_t \in N(0, t)$, for all $t \geq 0$. Hence, the ...
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### How to generate normalized factor scores for beta exposure

I'm working on building a time series momentum model (TSMOM) based on price alone for currency pairs. I'm implementing a paper that produces a buy/sell signal based on geometric brownian motion and a ...
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### Hermite polynomials as martingales [closed]

Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
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### Mutual variation of Brownian motions

Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
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### What does it mean to "compute" an Itô integral?

I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
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### What is the expectation of a change in Brownian motion? [closed]

I know $E[W_T-W_t]=0$ but I have a solution which implies this is wrong. Question Answer
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### How to show the following conditional expectation relation holds for a Brownian motion?

Suppose that $B_k$ stands for a standard Brownian motion process. \begin{equation} \mathbb{E}\Big(e^{-w\int_{t}^{S}B_k dk\, -uB_T}\Big| B_t = x\Big) \end{equation} where $w$ and $u$ are constants, and ...
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### How to prove that the following is still a Brownian motion [closed]

Given a Brownian motion $B_t$ on a filtered probability space, how can I prove that $W_t=B_t+\alpha t$ is still a Brownian motion, with $\alpha \in \mathbb{R}$? Is it always true? Do I need necessarly ...
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### Future price in continous time

I am in the following continuous time market: $S_t^0 = rS_t^0dt$ $S_t^1 = (\mu - \delta) S_t^1dt + \sigma S_t^1 dB_t$ where $r, \mu, \delta$ and $\sigma$ are constant values in $\mathbb{R}$. $\delta$...
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### Conditional probability of Brownian motion (with drift and scaling) hitting barrier

I am trying to understand the pricing of barrier options, and am considering the Brownian motion $\mathrm{d}X_t=a\mathrm{d}t+b\mathrm{d}W_t$, $a$ and $b$ constant. I am trying to: derive the ...
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### Calculate Ito integral $\int_0^t W_s^2\text dW_s$ from first principles

I am stuck on the 1st equation of the solution where the Wiener process $W_{t_i}^2$ is expanded so that the Itô integral (in terms of infinite sums) looks like the RHS of the first equation of the ...
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### Do we model stock prices using non-Markovian processes in continuous setting?

In a continuous setting, is it common to model stock prices using non-Markovian processes ? If so, do you have some examples of models ? Or is Markovianity something "embedded" in the ...