1
vote
0answers
34 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
2
votes
1answer
101 views

Brownian Bridge as a Gaussian Process

Let $B=\{B_t:t\geq 0\}$ be a standard Brownian motion. Define the Brownian brige $X=\{X_t:t\geq0\}$ as $$ X_t=B_t-tB_1\quad t\in[0,1] $$ Show that $X$ is (i) Gaussian and find its (ii) mean and (iii) ...
5
votes
1answer
201 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
1
vote
1answer
36 views

I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. [closed]

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
0
votes
2answers
69 views

how to prove $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$

Let $B(t)$ is Brownian Motion. I want to prove the integral $\int_{0}^{a}B(t)dt$ has normal distribution , $N(0,\frac{a^3}{3})$. means $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
1
vote
1answer
121 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
1
vote
1answer
39 views

distribution of Brownian Motion involving integral

What is the distribution of $\int_{t}^{T} W(s)ds$? Given that W(t) is brownian motion. So far, I have the following, $\int_{t}^{T} W(s)ds$ = $(T-t)W(t) + \int_{t}^{T} (T-s)dW(s)$ Also, ...
1
vote
2answers
102 views

Standard Brownian Motion

Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$. Can anyone give me some hint to start the ...
1
vote
1answer
99 views

Variable t times a Wiener Process W(1/t)

If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that Since $W(1/t)\space \sim N(0,1/t)$ that $V(t) \sim t\cdot N(0,1/t)$? And if so then is $t\cdot N(0,1/t) = ...