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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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 ...
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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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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69 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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46 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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36 views

If two Brownian motion starts and end at the same points, can we say something about there difference?

Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say ...
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37 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
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34 views

Question about zero set of Brownian motion

I was reading the posted to solutions to one of the questions on a probability midterm and couldn't figure out how to justify one of the steps. Let $\{B_t\}_{t\geq 0}$ be a Brownian motion and ...
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57 views

Variance of absolute value of brownian motion

Im wondering if anyone has this calculated, I cant seem to find it anywhere online. I am trying to find the variance of absolute value of BM. Here is my attempt: First, $f_{\lvert W_t \rvert} ...
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47 views

Stcochastic Integral and Ito Isometry

I am right now studying stochastic integral, and facing the following dilemma! I wjust want to check whether my understanding is right! The stochastic integral is defined by following: $I(t) ...
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51 views

Probabilities related to Brownian excursion

I am reading a paper that uses a fact about Brownian excursion which I don't understand. Let $(E_t)$ be a standard Brownian excursion, i.e. $E_t = X_t + i R_t$, where $X$ is a standard real Brownian ...
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59 views

Pathwise integral of $W^{-a}$

Denote by $\tau(x) := \inf \{t \ge 0, W_t=x\},$ where $W_t$ is a Wiener process started at $W_0 = w_0 > 0$ and I would like to show that for any $a>1$ it almost surely holds that ...
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46 views

Lebesgue Measure of “excursions” of Brownian Motion

I know that the set $S$ where a standard Brownian motion $M:=B[\mathbb{R}]$ attains a strict local minimum is a.s. dense in $\mathbb{R}$. For every point $s \in S$, consider the interval $(s,t)$ such ...
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19 views

Sufficient condition of boundedness of diffusion process

I came across the following statement in Sebastian Bossu's book "Advanced Equity Derivatives", page 27. He says that the time-homogeneous diffusion process $dX_t=a(X_t)dt+b(X_t)dW_t$ (coefficients ...
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50 views

Distribution of $(\sup_{0\leq s\leq t} W_s -W_t)$

I am interest in the law of the $(\sup_{0\leq s\leq t} W_s -W_t)$ where $W$ is a standard brownian motion. I know that $M_t:=\sup_{0\leq s\leq t} W_s \overset{\mathcal L}{=} |W_t |$ so its density ...
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27 views

Intersection of two independent 1-d Brownian motions.

I am interested in the first intersection of two independent 1-d Brownian motions. More precisely, what is the joint distribution of the intersection point and intersection time? Any help is ...
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35 views

Strong markov property in two dimensional Brownian motion

I don't understand the following claim from my book: Let $(B_t)$ be a standard Brownian motion. Let $u:\Omega \rightarrow \mathbb{R}$ be a continuous function, where $\Omega$ is a domain and $B(x, ...
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57 views

Conditional expectation involving Brownian Bridge

I have no ideas on this problem: Let $(B_t, 0 \leq t \leq 1)$ be a standard Brownian motion in $1$ dimension. Let $Z^y_t = yt+ (B_t -tB_1)$. We call $\{Z^y_t\}_{0 \leq t \leq 1}$ a Brownian Bridge ...
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72 views

Why is the pathwise integral of $\alpha_s$ w.r.t the Lebesgue measure continuous?

My class notes on stochastic calculus say that the if $(\alpha_s(\omega))_{s\in \mathbb{R_+}}$ is progressive then $\int_0^t \alpha_s ds$ is a pathwise continuous process? How does the joint ...
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41 views

How to calculate probability of an event in a stochastic setting?

Let $\left(\, B_{t}\,\right)_{t\ \geq\ 0}$ be a Brownian motion. Calculate the probability of the event: $$ E\equiv\left\{\,\exists\ \epsilon > 0 : \forall\ 0 < h < \epsilon, \max_{t\ \in\ ...
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222 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf ...
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72 views

What is the distribution of the area between a Brownian Bridge and the x-axis?

Lets say that we have a Standard Brownian Bridge ($\sigma=1$) with endpoints $(0,0),(1,0)$ Is there a way to derive the distribution of the area between a sample path of this bridge and the x-axis?? ...
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205 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
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36 views

Separation of variables and Fourier transformation

I know there's another question very similar to this argument. In the book Probabilità e Modelli Aleatori of Enzo Orsingher, at page 134, it shows that the transition function of an absorbing Brownian ...
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49 views

Independence of two processes

Suppose $X_t$ is the solution of the SDE $$dX=a(X)dt+b_1(X)dW_1+b_2(X)dW_2$$ $Y_t$ is the solution of the following SDE $$dY=p(Y)dt+q_1(Y)dW_1+q_2(Y)dW_2$$ Here, $W_1$ and $W_2$ are independent ...
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46 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 ...
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105 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 ...
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91 views

Hitting time of a maximum of random walk converges to that of Brownian motion

Suppose $S_n$ is a simple random walk; formally, $S_n=\sum_{i=1}^n X_i$ for $X_i\sim\mathcal{U}(-1,1)$, i.i.d.. Denote by $M_n$ the maximum of the random walk on $n$ steps; formally, $M_n=\max_{0\le ...
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99 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 ...
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88 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 ...
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48 views

Eigenfunctions of a 2D fractional Brownian motion covariance

The fractional Brownian motion is a centered Gaussian process with the following covariance function (covariogram): $E[B(t)B(s)]=C(\Vert t \Vert ^{2H}+\Vert s\Vert^{2H}-\Vert t-s\Vert^{2H})$ ...
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59 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 ...
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171 views

Why is a brownian motion conditioned to stay positive a Bessel-3

I am told this result long ago but I still don't know how to prove it. Is it because that this conditioning can be turned into a Girsanov probability change? Or is there any simpler ways to see it?
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123 views

Geometric Brownian motion - Volatility Interpretation

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
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135 views

Expectation of the infimum of a GBM

does somebody know a reference, where I can find the value of the expectation of the running infimum of a geometric Brownian motion, namely: Given a filtered probability space ...
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57 views

Local time of fractional Brownian motion

For BM, there is a downcrossing representation of the local time at 0. Namely, $L_t(0)=\lim_2 (b_i-a_i)D(a_i,b_i,t)$, where $D$ is the number of downcrossing between level $b_i$ and $a_i$. I am ...
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252 views

Brownian motion conditional probability

If $B$ is the standard brownian motion and $a,b >0$ I want to show, using the reflection principle $$\mathbb{P}\left(B_t\geq a-b | \inf_{s\leq t} B_s \geq -b\right) = \frac{\mathbb P(|B_t+x|\leq ...
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301 views

Is the absolute value of Brownian motion a super martingale?Is it a sub martingale? Is it a Markov process?

I've just started to study random processes and I'm trying to solve the following problem: Let $W(t)$ be a Brownian motion with filtration $F(t)$ generated by $ W(t)$ (i.e., $F(t)=\sigma \left( ...
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0answers
194 views

Almost sure non differentiability of Brownian Motion

Problem: Let $t>0$, show that the standard Brownian motion is almost surely not differentiable a $t$ Now, through a Borel Cantelli argument I proved that, almost surely $$\limsup_{\epsilon ...
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115 views

Verifying a standard Brownian Motion?

Let $\{X_t, t\ge 0\}$ be a standard Brownian motion process. For a fixed positive number s and all $t\ge 0$, we define $Y_t = X_{t+s} - X_s$. Is $\{Y_t, t\ge0\}$ a standard Brownian motion? Attempt: ...
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53 views

Can anyone explain me this proof about a Brownian Motion?

Prove that the process $W_t=(1+t)U_{t/(1+t)}$ on $[0,\infty)$ is a Brownian motion. $\text{(b)}$ Clearly $Y_0=U_0=0$, and inherits continuity of sample paths from $U_t$ (and hence from $W_t$). ...
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64 views

Bounded Brownian Bridge

I am trying to calculate the following expectation value of $$ E[exp(-\int_{t_0}^{t_1} X_s ds)] $$ in which Xs is a bounded Brownian bridge, which means $X(t_0)=a$, $X(t_1)=b$ and $A<X(t)<B$. ...
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1k views

Integrating deterministic function with respect to Brownian motion

I have looked everywhere for a satisfactory answer to this, including Shreve's textbooks, but I can't find one. If I want to integrate a some deterministic function f(t) with respect to brownian ...
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74 views

first hitting time probability for a Brownian motion with variable diffusion

I am looking for the first hitting time probability of the following Brownian motion: $dX=\mu X dt+ \sigma (X) X dW$ assuming $X(0)=X_0$ and $\sigma(X)= \sigma_1$ if $X>X_1$ and ...
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61 views

Exercise in brownian motion

Consider a system of n particles moving in three dimensional space under the action of an external force with $C^1$ potential V and coupled to a heat bath causing an external random effect. Then we ...
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132 views

Quadratic variation process of $G$–Brownian motion

I would like to prove the inequality $$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$ where $\langle B ...
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167 views

Independence of Brownian motion-related stopping times

Let $(B_t,\mathcal{F}_t)_{t \geq 0}$ a Brownian motion on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. For $a \in \mathbb{R}$ define a stopping time $\tau_a$ by $$\tau_a := \tau(a) := ...
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152 views

Independence of Brownian Motion with respect to a stopping time

Let $B_t$ be a brownian motion, $B_0=0$, and $\gamma \in \mathbb{R}$. Now, let's build the following stopping time: \begin{equation} T = \inf \{ t \geq 0 : |B_t + \gamma t| = 1 \}. \end{equation} If ...
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101 views

A problem with regard to Wiener process

Let $W$ be a Wiener process and $U_x$ is the amount of time spent below $x$ during time interval $(0,1)$. Hence $U_x=\int\limits_0^1I_{\{W(t)<x\}}dt$. My question is: what is the probability ...
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237 views

Correlated diffusion processes and covariance matrix

I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct. I would like to simulate $n$ dimensional diffusion processes with $n$ noises. Each process has its ...