Tagged Questions

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Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
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Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
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Brownian Motion first hitting time distribution

I have a question concerning the distribution of the first hitting time of Brownian Motion $\tau_x = \inf_{t\geq 0}\{W_t=x\}$, where $W_t$ is Brownian motion. Using some calculus, I found out that the ...
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A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
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Limit of stationary increment of Brownian Motion [closed]

Does the following limit $$\lim_{s \to \infty}(B_{t+s}-B_{s})$$ have the same distribution with $B_t$?
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Distribution of stochastic integral

Assume that $\mathrm{d}S = \sigma \, \mathrm{d}W$ with initial level $S(0)$ and where $\mathrm{d}W$ is usual Brownian motion. Now $$A(T) = \frac{1}{T} \int_0^T S(t) \, \mathrm{d}t.$$ ...
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Strong Markov Property Brownian Motion Question

If $\tau$ is a stopping time and $\omega(t)$ is Brownian Motion then the Strong Markov Theorem states that $Z(t)=\omega(t+\tau) -\omega(\tau)$ conditioned on $\{\tau <\infty\}$ is distributed as ...
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I wonder if it is possible to estimate $\mathbb{P}(X<t)$ with the moment generating function? This question popped up when I tried to proof this estimate $\mathbb{P}(T_a<t)\leq ... 1answer 78 views Brownian motion at exponential time I want to find the law of$B_T$, where$B_t$is a brownian motion,$T$is exp-distributed with parameter 1, with$B_t$and$T$being independent. My idea is to say that the density of$B_T$is given ... 1answer 243 views Joint Distribution of two correlated ito integral I have a question regarding finding the joint distribution of two process$$dX_{t}=a_{t}dB_{t}$$$$dY_{t}=b_{t}dW_{t}$$where$B_{t}$and$W_{t}$are two Brownian motions with correlated increments, in ... 1answer 121 views Assistance with a BM exercise A friend and I are attempting to answer part 3) of the exercise quoted below (from Continuous Martingales and Brownian Motion) regarding Brownian Motion (BM). We have some questions apropos thereof. ... 2answers 127 views That Brownian Motion's increments are gaussian is “not surprising”? In section 1 of chapter 1 of Continuous Martingales and Brownian Motion, the authors claim that the fact that the increments of of Brownian motion are gaussian random variables "is not ... 2answers 106 views why is the expected value of a Wiener Process = 0? This section of wikipedia says that the expected value of a Wiener Process is equal to 0. Why is that? 2answers 459 views Conditional distribution in Brownian motion I need to prove the following: Let$X$be a Brownian motion with drift$\mu$and volatility$\sigma$. Pick three time points$s < u < t$. Then, the conditional distribution of$X_u$given ... 1answer 173 views Distribution of the integral of a diffusion process Suppose$X(t)$is a diffusion process with$E[X(t)]=0$and variances$\sigma^2_t$concave in time. If$X$is also a Brownian motion, then the distribution of$\int_0^T X(t) dt$is known to be ... 1answer 161 views What is the conditional distribution of$B(s)\mid B(t_1)=x_1,B(t_2)=x_2$for$0<t_1<s<t_2$? Given that$\{B_t,t\ge0\}$is a standard Brownian process. What is the conditional distribution of$B(s)$given$B(t_1)=x_1$and$B(t_2)=x_2$, for$0<t_1<s<t_2$? My try: First i tried to ... 2answers 2k views Integral of Brownian motion is Gaussian? Let$(W_t)$be a standard Brownian motion, so that$W_t \sim N(0,t)$. I'm trying to show that the random variable defined by$Z_t = \int_0^t W_s \ ds$is a Gaussian random variable, but have not ... 3answers 106 views Derivation of Wiener process first passage times using probability generating function? I would like to find the distribution of first passage times in a simple Wiener process using the idea of probability generating functions. Thus there will be, at certain point, a limiting step to go ... 1answer 205 views The joint distribution of zeros of brownian motion Let$\gamma_t$be the last zero of brownian motion before$t$and$\beta_t$be the first zero after$t$. I need to calculate the joint distribution of$\gamma_t$and$\beta_t$, i.e.$P(\gamma_t<x, ...
Let $X_1, X_2, \dots,X_n, X_{n+1}$ be independent random variable of exponential distribution, and the mean is 1. Let $S_i = X_1 + \dots + X_i$ I want to know ...
How would I go about finding the distribution of $B(u) + B(u+v)$ where $u+v > u$? I know that both $B(u)$ and $B(u+v)$ are normal random variables. The sum of two normal random variables is also ...