2
votes
0answers
31 views

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 ...
-2
votes
0answers
40 views

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 $B_t$?
1
vote
1answer
53 views

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 ...
0
votes
1answer
32 views

Finding a tail-probability with the momentgenerating function

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 ...
0
votes
1answer
64 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 ...
0
votes
1answer
178 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 ...
1
vote
1answer
113 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. ...
3
votes
2answers
117 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 ...
1
vote
2answers
92 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?
3
votes
2answers
282 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 ...
2
votes
1answer
159 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 ...
2
votes
1answer
143 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 ...
4
votes
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 ...
2
votes
3answers
99 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 ...
2
votes
1answer
189 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, ...
10
votes
2answers
247 views

Problem about partial sum of exponential random variable

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 ...
2
votes
1answer
716 views

Distribution of Brownian motion

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 ...