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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Problem with particular proof regarding infinite total variation of Brownian motion

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 $c_{1}$? ...
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14 views

2D Brownian Motion — Does this argument work?

Consider a 2D Brownian Motion $(X_1(t),X_2(t))$ starting at $(x_1,x_2) \in \mathbb{R}^2$. Let $\tau_s = \inf \left\{t \geq 0 | X_1(s) - X_1(0) > s \right\}$, and let $Y_s = X_2(\tau_s)$. The ...
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1answer
26 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
31 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
29 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
21 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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21 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
26 views

Calculate a differenciation

$$a>0,$$ $$b>0,$$ $$\sigma >0$$ $X$ 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
28 views

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

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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21 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
46 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
27 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
7 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
35 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 ...
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1answer
36 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 ...
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44 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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11 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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41 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 $ ...
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21 views

limit of sum of a brownian motion

Let $W_t$ be a wiener process and let $\pi$ be a partition of the segment $[0,T]:0\leq t_1\leq...\leq t_n=T$ I need to show without using the martingale property that the term below tends to $0$ in ...
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22 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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8 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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9 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
20 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
20 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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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 ...
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2answers
57 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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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
25 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
22 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
50 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
21 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
33 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
45 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 ...
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1answer
19 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 ...
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57 views

Ornstein-Uhlenbeck a Markov process

Consider the Ornstein-Uhlenbeck process defined by $$ X_t = e^{- \alpha t} X_0 + \sigma \int_0^t e^{ \alpha (s-t)} d W_s$$ with $\sigma,\alpha>0$. In many literature I have found they considered ...
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20 views

Brownian Motion Maximum Value Proof

Let $B(t)$ be a Brownian Motion and $$M(t) = max_{s:s \leq t} B(s)$$ and $$\tau_a = min_t{B(t) = a}$$ Then, $P(\tau_a < t) = P(B(t) - B(\tau_a) > 0 \: |\: \tau_a < t) + P(B(t) - B(\tau_a) ...
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1answer
33 views

On random rotational fluctuations in $\mathbb{R}^n$

Imagine first a disk that is mostly stationary, except for random ("thermal" if you like) "rotational fluctuations" around its axis (which is fixed). Something a bit like what's shown in the figure ...
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1answer
42 views

An equality involving the Wiener process

The equality below appears as a step in a proof in a chapter titled "Itô Stochastic Calculus" in Brzeźniak and Zatawniak's textbook "Basic Stochastic Processes", Springer 2005 (in a solution to ...
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question about brownian motion and integration

If $X(t)$ is the standard Brownian motion, $0<\alpha<\beta$, and $T$ is the first exit time of $X(t)$ from $[-\beta,\beta]$, then how can I find $E(\int_0^T \mathbb{I}_{(-\alpha,\alpha)} X(t) ...
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30 views

Generator of Wiener process and its running maximum

If we let $W$ be a standard linear Wiener process issued from zero and $M$ its running maximum $$ M_t := \sup \{ W_u: u \leq t \}, $$ then we could show that $(X,Y):=(M,M-W)$ is a Markov process on ...
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0answers
53 views

Absorbed brownian motion is a Markov process

I have been asked to prove that the Brownian motion absorbed at the origin is a Markov process. Formally, let $B_t^x$ be a Brownian motion originating from $x>0$ and let $\tau^x_0 = \inf\{t>0 : ...
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2answers
21 views

Product rule with stochastic differentials

I am encountering difficulty in seeing how this relationship holds: with $S_T$ being stock price at time $T$, I want to find the sde for $S_t e^{-rt}$ $$dS_t = rS_tdt + \sigma S_t d\hat B_t$$ Where ...
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Mutual independence of increments of Brownian motion

Brownian motion has a bunch of different definitions. My question is about showing the property in the title using a certain definition of BM and nothing else. The (partial) definition I am given is ...
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2answers
38 views

Brownian motion and covariance

Show that for $B = (B_t)$ Brownian motion, its covariance is $cov(B_s, B_t) = min(s, t)$. The solution I was given was: For $s ≤ t$, $B_t = B_s + (B_t − B_s)$, $B_sB_t = B_s^2 + Bs(Bt − Bs)$ ...
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35 views

Transition function for absorbed Brownian motion

I need an help with the following exercise. I've already seen this question Prove that Brownian Motion absorbed at the origin is Markov but I don't understand the answer. Also I would like to prove ...
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2answers
26 views

How to show stochastic differential equation is given by an equation

I I tried using substitution and I got an extra integral at the end and do not know how to proceed. Can anyone help me to break this down?
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1answer
40 views

How to solve Stochastic differential equation?

I do not have a clue on how to solve out this type of question, and how to deal with integration with a combination of brownian motion and linear function. Can anyone help me out please?
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2answers
17 views

Using Ito's formula, write down a stochastic diferential equation satiesfied by $Y_t:=X_t^2$, given both $Y_t$ and $X_t$

I am trying to solve this exercise and I am stuck in the third part of it. I checked the solution and it makes no sense to me, so I would really appreciate it if someone could explain to me how Ito's ...