2
votes
1answer
38 views

Ito formula applied to $\frac{1}{t}\int_0^t W_s ds $

I got this expression and I have to calculate its differential by the Ito formula, $W_t$ denotes the Brownian motion: $$\frac{1}{t}\int_0^t W_s ds $$ I calculate the derivative of ...
0
votes
1answer
35 views

Property of Wiener process sample path

What is a mean of time, when the trajectory of wiener process $W_t$ is over the line $y=t$? We need to find $\mathbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
0
votes
1answer
14 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
0
votes
0answers
32 views

First hitting time Geometric Brownian motion

I have the following problem: My Process underlies the SDE $ d W_t = \mu W_t dt + \sigma W_t d B_t $ with $B_t$ being a standard Brownian motion, $\mu,\sigma >0$, i.e. $W_t = S_0 \exp\Big( ...
0
votes
0answers
6 views

Branching Brownian Motion and KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
0
votes
1answer
46 views

What is the distribution of this random variable? [closed]

Find the distribution of this random variable: $$X_t=\exp\left(t \int_0^t sdW_s\right)$$ knowing that $W$ is a Brownian motion in the filtered space $(\Omega, \mathcal{F},P,(\mathcal{F}_t)_{t\geq0} ...
0
votes
1answer
11 views

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 ...
1
vote
0answers
18 views

Stability of simulation of brownian noise

As I understand, Brownian noise can be simulated by the process $$x_{n+1}=x_n+R_n$$ where $R\sim U[-a,a]$. The expected value for $x_n$ is then $x_0$. But $\text{Var} x_n\to\infty$ as $n\to\infty$ ...
1
vote
0answers
34 views

Brownian motion starts fresh variant

It is a standard result that if $W_t$ is a Brownian Motion and $S$ is a stopping time of the standard filtration $F_t$ then we have that $B_t = W_{S+t} - W_S$ is a Brownian Motion. I quote the ...
0
votes
1answer
44 views

Geometric BM tends to zero but is strictly positive a.s.?

The process $\{S_t\}_{t\ge0}$ following $dS_t = \sigma S_tdW_t$ with $S_0>0$ has the solution $$S_t=S_0 e^{-\frac12\sigma^2t+\sigma W_t}$$ Now for any $\epsilon>0$ we have $$\mathbb ...
0
votes
1answer
18 views

Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
2
votes
1answer
32 views

Scaled integrated Brownian motion has limit

Let $B$ be a standard Brownian motion and put $$X(t)=\frac{1}{\sqrt{t}}\int_{0}^{t}f(B(s))ds,$$ where $f \in L_1(\mathbb{R}^{1})$ and $\int f(x)dx=1$. Show that $$ \lim_{t \rightarrow \infty} EX(t) ...
4
votes
1answer
42 views

Understanding of Brownian Motion

My background is functional analysis rather than probability, but I would like to understand what is a Brownian motion. Below I'm giving my current understanding, can anyone verify whether I'm ...
1
vote
0answers
22 views

Question regarding Notes on Strong Markov Property

I wrote the following notes from a lecture a couple of weeks ago and I don't understand a particular line. Suppose $B_t$ is a Brownian Motion Now look at $B^x_t = x + B_t$ which is a BM starting at ...
1
vote
0answers
40 views

conditional expectation of the Brownian motion [duplicate]

$(B_t)$ is a Brownian motion and i assume that $s<t<u$ we have $$E[B_t |\sigma(B_s,B_u)] = G(B_s,B_u)$$ Does anyone knows the explicit expression of $G$ ? (the calculus is easy but ...
1
vote
0answers
23 views

Expression for $B_1$

I think that it is indeed the case that $$ B_1 = \int_0^1 \frac{B_1 - B_t}{1-t} dt, $$ where $B$ is a standard one-dimensional Brownian motion. Am I right? If so, how you we prove it? Thanks a lot ...
0
votes
1answer
39 views

Distribution of Sum of Brownian Motion and Integrated BM

Let $W(t)$ be a standard Brownian motion (BM), in particular $W(t) \sim \mathcal{N}(0,t)$. Then it is easily shown that $\int_0^T W(t) dt \sim \mathcal{N}(0, T^3/3)$. Question: What is the ...
0
votes
1answer
36 views

How big a Brownian bridge can get? Confidence band.

If we know the endpoints of the Brownian path, is there any theorem telling us if it can be contained within a ball a.s. (with probability one)? For example contained in two big enough balls (call it ...
1
vote
0answers
50 views

Variance of Integrated Geometric Brownian Motion

I'm just asking for verification that my derivation is correct, as I can't seem to find this result elsewhere. I'd like to calculate $Var(\int_0^T X(t) dt)$ where $X(t) = X_0e^{(\mu - ...
0
votes
0answers
28 views

Expectation Involving Two Values of Geometric Brownian Motion

Not sure this is the best place to ask for verification, but I can't seem to find a derivation anywhere else. I want to calculate $\mathbb{E}[e^{\sigma(W_t + W_s)}]$, where $W_t$ and $W_s$ are two ...
2
votes
0answers
25 views

Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
5
votes
1answer
159 views

Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
0
votes
1answer
32 views

Brownian Motion with drift (stupid question)

How do you prove that $$ \lim_{t\to +\infty} (B_t+ct)=+\infty $$ almost surely? $(B_t)_t$ is the standard Brownian Motion starting from $0$.
1
vote
1answer
33 views

Property of Brownian Motion's paths

We are considering a Brownian Motion $(B_t)_t$ with values in $\mathbb{R} $ starting from $x$ defined on the stochastic basis: $$(\Omega,\mathcal{E},(\mathcal{F}_t)_t,\mathbb{P}^x)$$ Then, let's ...
0
votes
2answers
64 views

Is the following Itô-Integral not zero?

is the following statement true: $$\int_0^T t \, dW(t) \neq 0$$ I need it for a counter-example, that one can not change the order of integration between $dW$ and $dP(\omega)$. I thought of taking ...
4
votes
1answer
46 views

Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$ This is a clear ...
5
votes
1answer
89 views

Is $t^{-\frac{1}{2}}B_{t^2}$ a Brownian Motion?

I think the title says it all. Let $X_t = t^{-\frac{1}{2}}B_{t^2}$, with $B_t$ being a brownian motion started at $0$. I think I have proved continuity at $0$ by doing the following: $$ X_t = ...
2
votes
1answer
56 views

Hitting time for Brownian Motion Surplus Process

I'm trying to solve this question for a continuous surplus process. The surplus process is $$U_s=U_0+s-B_s$$ where $B_t$ is a Brownian motion representing payouts, $U_0$ is starting capital, $s$ is ...
2
votes
0answers
53 views

Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
1
vote
1answer
37 views

Combination of Wiener Processes

If $W_s$ and $W_t$ are wiener processes, we have that the probability that $W_s$ and $W_t$ attain maximum is (I am concluding this from "running maximum", but I am not sure) ...
4
votes
1answer
89 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...
2
votes
0answers
49 views

Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$ \tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \} $$ is a stopping time with ...
3
votes
1answer
51 views

Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$ \tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \} $$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
0
votes
2answers
33 views

Which Brownian motion property is the most important? [closed]

Which Brownian motion property is the most important? A standard Brownian motion is a stochastic process $(W_t, t\geqslant 0)$ indexed by nonnegative real numbers t with the following properties: ...
0
votes
1answer
23 views

What is the sample path of a stochastic process

Assume $\Omega $={head, tail}, let T=$\mathbb N$ and $X_t$ $t\in T$ be a collection of i.i.d random variables following Bernoulli distribution. Since a stochastic process is a function of two ...
0
votes
2answers
38 views

Merton problem: can the stock price keep rising?

I read that the stock price, $S(t)$ of the famous Merton model is given by the following differential equation $dS(t) = µS(t)dt + σS(t)dB(t).$ I gather that this is geometric Brownian motion. A path ...
1
vote
1answer
58 views

Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
0
votes
1answer
69 views

Sample continuity of Brownian motion

I wanted to know if the Brownian motion and the fractional Brownian motion are almost surely sample continuous or not? Many thanks.
2
votes
0answers
40 views

A right-inverse of Brownian motion local time at zero has stationary independent increments

Let $L_0^t$ be the local time for a standard Brownian motion at $0$ and define $$X_t=\sup\{s\ge0:L_0^s\le t\}, t\ge0. $$ I would like to show that $(X_t)$ has stationary independent increments. That ...
1
vote
1answer
88 views

Expectation and variance of correlated exponential brownian motions

What is the expectation and variance of correlated exponential Brownian motions for the random variable $F$, where $A$ is real constant, $\sigma$ is a real constant and $\rho$ is the correlation. $$F ...
1
vote
1answer
77 views

Find the distribution of the maximum of a Wiener Process with negative drift

So.. what I have now is Let $M=max\{W_t; 0\leq t <\infty\}$ since $W_0=0$, $M\geq 0$ with probability 1. So, $P(M>x)=P(T_x<\infty)$ where $T_x$ is the stopping time, so we now use the ...
0
votes
1answer
38 views

Variance of sum of Brownian Motions

Let $t_i=\frac{T\cdot i}{n}$ for $T>0$, $i=1,...,n$ and let $(W_t)_{0\le t\le T}$ be a standard Brownian motion. Now I want to evaluate $$\text{var}\left(\sum_{i=1}^n W_{t_i}\right) = \mathbb ...
2
votes
1answer
80 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
0
votes
1answer
24 views

Question about limit of Stochastic Process

Given $\mu_t$ continuous stochastic process that satisfies $\int_0^t \mu_s^2\;ds<\infty$. Define $X_t\equiv \int_0^t \mu_s\;ds$. Let $|\cdot|$ denote floor function. Then where does ...
0
votes
1answer
36 views

$dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u $

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] $. As far as I can see though, Ito's formula says ...
1
vote
2answers
56 views

Is this true about Brownian Motion?

I have the following in my notes and I'm not sure if it's true or not. Any help would be highly appreciated. If $\{W_t\}_{t\geq0}$ is a standard Brownian motion stochastic process, $\Delta>0$ and ...
4
votes
0answers
124 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
0
votes
0answers
76 views

Cubed Brownian motion

I have to do the following exercise: Let $(W_t)$ be a Brownian motion. (a) Does X given by $X_t:=W_t^3$ have constant expectation? (b) Is it a martingale? (c) Does it have independent increments? ...
1
vote
1answer
67 views

An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
2
votes
0answers
45 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 ...