# Tagged Questions

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 $t-s$.

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### Correlated brownian motions and Lévy's theorem

$W^{(1)}_t$ and $W^{(2)}_t$ are two independent Brownian motions. How can I use Lévy's Theorem to show that $$W_t:=\rho W^{(1)}_t+\sqrt{(1-\rho^2)} W^{(2)}_t,$$ is also a Brownian motion for a given ...
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### Can Fluctuation-Dissipation Theorem Apply to Magnetic Forces in Multi-Spin Systems [closed]

Let's say I have multiple spin systems (atoms in a protein) in a solution of water and the spin systems are all producing a magnetic field $\mathrm{B_{loc}}$ that affects nearby spin systems. Will the ...
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### Distribution of marginal Wiener process.

Let $(W^1,W^2)$ be a two-dimensional Wiener process with correlation $\rho$. Let $\mathcal{F}^1_t$ be the filtration generated by $W^1$ up to $t$. I would intuitively think that for $h$ measurable ...
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### $(B_t)_{t\ge 0}$ be Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t}$ is an analytic function. [on hold]

Let $(B_t)_{t\ge 0}$ be a one-dimensional Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t} \; \text{for all} \; t\ge 0, \xi \in \mathbb{R}$ is an analytic function. A more general question ...
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Let $W_1(t)$, $W_2(t)$, $W_3(t)$ be independent Brownian motions and $f$, $g$ smooth functions. I want to know if the following is true: $$\mathbb{E}\left[ \left( \int\limits_0^t f(... 0answers 46 views +100 ### Brownian Motion in Confined space, any results? I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ... 2answers 37 views ### Why is a continuous Lévy process twice integrable? In his textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008), Jochen Wengenroth shows (p. 144) that if (X_t)_{t\in[0,\infty)} is a continuous, real-valued Lévy process with X_t\in \mathcal{L}_2 ... 1answer 57 views ### Confusion about the average distance traveled on a 1D random walk The average absolute distance on a one dimensional random walk is supposed to be \sqrt{n}. Where n steps are taken from the origin or n is the time. I don't have an intuitive understanding or ... 0answers 6 views ### What is the rate of convergence of Brownian motions Increments? Would like to know what the rate of convergence of brownian motion is? I know each brownian motion increment is distributed with N(0,t) so do i need to apply a CLT? 0answers 23 views ### Integrated Brownian Bridge is a Gaussian Process Let W(t),t \in [0,1] be a (Standard) Wiener Process. The Brownian Bridge B(t), t \in [0,1] can be constructed via B(t):=W(t) - t \cdot W(1) and is a Gaussian process with zero mean and ... 0answers 18 views ### What does it mean that a stochastic process is independant of a filtration? Let (\Omega ,\mathcal F,P) a proba space and (\mathcal F_t)_{t\geq 0} a filtration. Let (B_t)_t a Brownian motion adapted to \mathcal F_t. We know that (B_{t+s}-B_t)_{s} is independent of \... 0answers 17 views ### Expected value of the exponential of a Geometric Brownian motion I am trying to compute the following expectation:$$ E[ \exp (A_T)], $$where A_T = - C \int_{0}^{T} \exp( 2 \alpha W_t - \alpha^2 t) dt , with C and \alpha positive constants, W_t a standard ... 0answers 27 views ### Black scholes model for down and out European call option using Monte Carlo I tried to implement Matlab program computing the price of the European down and out call option using Monte Carlo and Euler discretization scheme. I have initial price S0=50, strike K=50, barrier ... 0answers 19 views ### What is the distribution of the maximum of scaled Gaussian random walk? A Gaussian random walk is the sum of standard normal variables$$Z(n) = \sum_{i=1}^n X_i,$$where X_i\sim N(0,1). What I mean by a scaled Gaussian random walk is the following:$$ U(n) = \frac{Z(n)}{...
What I mean by a normalized Brownian motion is the following: $$U(t) = \frac{W(t)}{\sqrt{t}},$$ where W(t) is a standard Brownian motion (with $\mu = 0$ and $\sigma = 1$). Is there a name for such a ...