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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1answer
27 views

Girsanov's theorem corollary

Trying to understand the proof of the corollary on the page http://en.wikipedia.org/wiki/Girsanov_theorem It remains for me the show the equality of the quadratic variations $[W, X]_t = 2[[W, X], ...
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2answers
48 views

$\mathbb{E}[e^{2B_t -t}] = e^{2t-t}$ - why?

Let $B_t$ be brownian motion at time t. I know we can use properties of MGF, but I get a different answer: in general, for $X \sim N(\mu, \sigma^2)$ $$\mathbb{E}[\exp(\alpha X)] = \exp(\mu \alpha + ...
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0answers
52 views

Brownian Motion Hitting Times

I am reading through Walsh's Knowing the Odds book and came across this problem. Let $B_t$ be Brownian motion. Find the probability that $B_t$ hits plus one and then minus one before time one. I am ...
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1answer
47 views

Deriving a closed form expression for stochastic integral (to show its a martingale)

I have $B_s = $ brownian motion at time $s$. $$ \int_0 ^t B_s \, dB_s$$ $$0 \leq t \leq T$$ And want to check if it is a martingale, first from its closed form expression, and then via conditions on ...
3
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1answer
28 views

Stochastic calculus rules $d(B_t^2) = 2B_t\,dB_t + dt$ - why?

Let $B_t$ = Brownian motion at time $t$ I know that $(dB_t)^2 = dt$ and $d(f(x)) = f'(x)\,dx$ for some differentiable function. Now, I have that $$M_t = B_t^2 - t$$ $$dM_t = d(B_t^2) - d(t)$$ ...
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2answers
42 views

Why is this wrong - conditional expectation of brownian motion: $\mathbb{E}[B_1 | B_2]$

Trying to find $$ \mathbb{E}[B_1 | B_2]$$ $$\mathbb{E}[(B_1 - B_2 + B_2) | B_1] = \mathbb{E}[(B_1-B_2)|B_2] + \mathbb{E}[B_2 | B_2] $$ $$\mathbb{E}[ -(B_2 - B_1)| B_2] + B_2$$ Since $$ -(B_2 - ...
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1answer
38 views

How to check if integral wrt Brownian motion is a martingale

As in title, I have a process $$X_{t}=\int_{0}^{t}s^{2}dB_{s}$$ I found here a sufficient condition for such integral to be a martingale on the interval. But I am asked if it is a martingale, not ...
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1answer
48 views

Prove this expectation of Brownian motion?

Prove $E[(\Delta B_j)^4]=3(\Delta t_j)^2$ where the Delta stands for the change of something i.e $B_j-B_{j-1}=\Delta B_j$ and the $B_j$ stand for the standard Brownian motion I won't show my step ...
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0answers
12 views

Sample variance matlab geometric brownion motion

I have a question about the geometric Brownian motion. I want to sample many paths and then showing that the sample variance equals the exact variance: $$\mathrm{Var}\left[S(t)\right]=S_{0}^2 e^{2 \mu ...
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0answers
48 views

Brownian motion under Girsanov change of measure

i am struggeling with the following exercise Let $r,\mu,\sigma,T>0$ and consider the market model with a money-market account $B$ and one risky asset $S$ such that \begin{eqnarray*} ...
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0answers
12 views

Brownian motion, why $p\{B_{T_{x+h,x-h}}=x\pm h\}=\frac{1}{2}$?

Let $(B_t)$ a Brownien motion. Let $\tau_a=\inf\{t\geq 0\mid B_t=a\}$ (with $a\neq 0$) and $T_{a,b}=\tau_a\wedge \tau_b$. Suppose $x\in[a,b]$ and $B_0=x$. Let $h>0$ very small. Why do me have ...
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1answer
23 views

Absolute continuity and sample paths of Brownian motion

An offhand remark in Morters and Peres' book on Brownian motion says that Brownian motion is a.s. absolutely continuous on compact intervals (see page 147, immediately preceding the statement of ...
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1answer
30 views

Brownian motion: Why $p\{\max_{0\leq u\leq t} B_u\geq a\}=2p\{B_t\geq a\}$?

Let $(B_t)$ a standard Brownian motion (i.e. $B_t\sim\mathcal N(0,t)$). Let $a\geq 0$. Prove that $$p\left\{\max_{0\leq u\leq t} B_u\geq a\right\}=2p\{B_t\geq a\}.$$ The proof goes like this : ...
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0answers
19 views

jump-diffusion hitting time

Suppose I have a stochastic process $dS_t= rS_t dt + \sigma S_t dW_t + dJ_t$ where $W_t$ is a brownian motion and $J_t$ a compound poisson process of parameter $\lambda$ with lognormal jump size, ...
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0answers
89 views

Brownian motion stopped at the hitting time of an independent Brownian Motion

While I was working on the exit time of planar BM out of a square I came across the following observation, which I cannot grasp. I define this exit time as $$\tau = \inf\{t \geq 0: \lvert B(t)\rvert ...
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1answer
25 views

Problem with particular proof regarding infinite total variation of Brownian motion [closed]

I have some problems with a proof from the last page of this pdf: Brownian motion has infinite total variation. Could we say that variance is exactly $\frac{c_{1}}{n}$ for some constant $c_{1}$? ...
2
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0answers
32 views

2D Brownian Motion — Does this argument work?

Consider a 2D Brownian Motion $(X_1(t),X_2(t))$ starting at $(x_1,x_2) \in \mathbb{R}^2$. For every $s\geq0$, let $$\tau_s = \inf \left\{t \geq 0 \mid X_1(t) - x_1 > s \right\}\qquad Y_s = ...
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1answer
41 views

Finding $E[W]$ and $E[W^2]$, where $W = \int_{t=0}^T B_s$ $ds$

I'm trying to find a)$E[W]$ and b) $E[W^2]$, where $W_t = \int_{t=0}^T B_s$ $ds$ ($B_s$ denotes a Brownian motion). In addition, I'd like to find $E[Z_sZ_t]$, where $Z_t = \int_0^t B_s^2$$ ds$ ...
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1answer
49 views

Expectation of the product of Brownian motions

I'm new to Stack Exchange. I'd like to find the expectation of the product of three Brownian motions: $E(B(t_1)B(t_2)B(t_3))$ I know from a previous post here that ...
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1answer
152 views

Independence of the components of a multidimensional Brownian motion

Let $B = (B^1, \dots, B^n)$ be an $n$-dimensional ($n \in \{1, 2, \dots\}$) Brownian motion (i.e. $B = (B_t)_{t \geq 0} \in \Omega \rightarrow (\mathbb{R}^n)^{[0,\infty)}$ has continuous paths, $B_0 = ...
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1answer
25 views

Density Function of Random Variable Related to Brownian Motion

Above is my question. I've done the first two parts, that's no problem. I'm stuck on finding the density of the rv $R = W_1 / M$. I have got as far as $$g(x,y) = \frac{\partial^2}{\partial x ...
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0answers
29 views

Covariance of two integrated Brownian motions

I have a question that is similar to the one here: covariance of integral of Brownian, but the answer that I come up with does not match what the book claims the answer is. Given that $$X_t = ...
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1answer
43 views

Calculate a differenciation [closed]

$$a>0,$$ $$b>0,$$ $$\sigma >0$$ $X$ is the solution of : $$dX_t=aX_t(b-X_t)\,dt+\sigma X_t \, dB_t,\quad X_{0}=1 $$ I have also shown before that $$L_t=e^{(ab-\sigma^2/2)t+\sigma B_t}$$ Now ...
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1answer
33 views

Does Brownian Motion return to the origin infinitely soon? [closed]

Consider a standard unidimensional Brownian Motion $B_t$ (Wiener process). Fact: This process returns to the origin infinite number of times with probability one. Consider a stopping time $\tau = ...
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0answers
25 views

Girsanov theorem [duplicate]

I work on an exercice and I have to calculate: $$E(W_{t}^2e^{(\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds)})$$ $$\theta$$ is deterministic function I don't know how to ...
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2answers
60 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
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1answer
37 views

Measurability of a set

This question is from Karatzas's Brownian Motion and Stochastic Calculus page 108. Let $W_t$ be standard one-dimensional Brownian motion then it concludes that the set ...
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0answers
41 views

Integral of Brownian Motion with respect to an independent Brownian motion

I have this seemingly simple problem which I haven't been able to solve. I have two standard Brownian motions, $B$ and $W$, on the same probability space and under the same filtration (I am not so ...
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0answers
8 views

Reflected brownian motion at arbitrary lower barrier

What is the kernel for reflected Brownian motion at some lower barrier $p_b$? The best I could come up with is: $(e^{-((x-u)/a)^2/2}+e^{-((x+u-2b)/a)^2/2)})/(2\pi)^{1/2}$ Which is equal to zero ...
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1answer
38 views

Is $X_t := W_t^2$ a Wiener process for a Wiener process $(W_t)_{t \geq 0}$?

I'm studying for exam and found this exercise which I don't really understand: Suppose $W_t$ is standard Wiener process. Is process $X_t=W_t^2, t\geq0$ a Wiener process? So I need to show that ...
2
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1answer
42 views

Independence of linear combinations of Brownian motions

Let $0<s\leq t\leq u\leq v$ and $W_x$ be a Brownian motion. Show that $aW_s+bW_t$ and $\frac{1}{v}W_v-\frac{1}{u}W_u$ are independent for $a,b$ satisfying $as+bt=0$. The question seems easy but ...
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1answer
16 views

Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration.

Let $\left(B_{t}\right)_{t\geq0}$ a Brownian motion. Show that $M_{t}=\max_{0\leq s\leq t}B_{s}$ is adapted to the natural Brownian filtration. Remark: I try the following: It suffices to show ...
3
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0answers
55 views

No drift brownian motion problem

Given two same brownian motion with no drift and different variances: $$dG_1= \sigma_1 G_1 dW $$ $$dG_2= \sigma_2 G_2 dW $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > \sigma_2 $ ...
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0answers
17 views

Covariance of the normalized brownian excursion

This question may be very simple, but... Let $X_t$ be the normalized Brownian excursion. We would like to compute the expectation $E[X_tX_u]$ for any times $0\leq t,u\leq1$. We found in the paper [1] ...
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0answers
51 views

Why is the black-scholes model arbitrage free when $\sigma >0$?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
2
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0answers
28 views

Proving that a local martingale given by a stochastic integral is not a martingale

Let $X_t=\int_0^t e^{W_s^2}dW_s$ for $0\leq t\leq 1$ and show that is not a martingale. I guess the reason is that the expectation is not finite, but I'm not sure how to show it precisely. In fact ...
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0answers
14 views

Logarithm of Brownian motion which is a local martingale but not a martingale

Let $W_1(t)$ and $W_2(t)$ be independent Brownian motions starting at positive points (not necessarily at the same point). Let $X_t=\log(W_1^2+W_2^2)$ and show that it is a local martingale but not a ...
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1answer
35 views

Density of the Absorbed Process

The curiosity arose while reading the Ch.18 of Arbitrage Theory in Continuous Time 3/ed, dedicated to pricing Barrier Options. Definition 18.1 For any $y\in R$, the hitting time of y, $\tau(X,y)$, ...
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0answers
33 views

Wiener measure on continuous function space

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. I have following problem: Given is the map $W:\Omega\rightarrow C[0,1]$ (it is not given but I think it is implicit a Wiener process). ...
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1answer
39 views

Slight generalisation of the distribution of Brownian integral

I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that $$ E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] ...
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1answer
20 views

The name of invariance principle of Donsker

I have seen the invariance principle of Donsker for the Wiener measure in Karatzas' Brownian Motion and Stochastic Calculus. I am wondering why this theorem have this name, e.g. where does the ...
4
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2answers
190 views

Implementing Ornstein–Uhlenbeck in Matlab

I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in ...
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0answers
10 views

is daily return with general stochastic volatility model stationary?

In order to estimate the parameter, we need to know whether this model will result a stationary daily return or not. And yes, actually there is an estimator for estimating the variance of this daily ...
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0answers
33 views

Survival probability (1D Brownian Particle)

Here is an interesting article from Wikipedia: First-hitting-time model I am particularly interested in how the following density is derived: $$p\left(x,t;x_0,x_c\right)=\frac{1}{\sqrt{4 \pi D ...
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1answer
30 views

Brownian Motion Hitting Time Distribution

Define $\tau_a = \inf \left\lbrace t \geq 0 | B(t) \geq a \right\rbrace $ for some $a>0$. The problem is to show that $ \tau_a \stackrel{d}{=} \sqrt a\tau_1 $. What I've done so far: $$P(\tau_a ...
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1answer
55 views

Brownian motion at infinity

This is probably a standard exercise in stochastic calculus but I haven't been able to come up with a proof that relies only on a given set of results. So my question is about proving the following ...
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1answer
22 views

does brownian motion and poisson random measure have to be independent? [closed]

Suppose a brownian motion $W$ and a poisson random measure $\mu$ are defined on the same filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), P)$, where both $W$ and $\mu$ are ...
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1answer
40 views

Local maximum of brownian motions

Let $B=(B_t)_{t\geq 0}$ be the standard Brownian motion. I want to show that for every $t_0 \geq 0$ $\mathbb{P}$($B$ has a local maximum in $t_0$)=0. I've already shown that for every ...
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0answers
57 views

How to do integration by parts with brownian motion?

I am not sure how to perform integration by parts in the following expression: $$ \left(1-t\right)\left(B_t - B_s + \int_s^t \frac{r}{1-r} \mathrm{d} B_r \right) $$ Can anyone help me to solve this ...
0
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1answer
21 views

Expected Stopping Time for BM

I'm working on this homework problem for Brownian Motion. Suppose we define a stopping time $\tau_a = inf \left\lbrace t \geq 0 : B(t) = a \right\rbrace$ for some $a>0$. I already showed in a ...