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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181 views

Show that $O_t$ is a Gaussian Process

Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process. I think ...
3
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155 views

A question regarding the strong Markov property

In our lecture on Brownian motion & stochastic calculus we proved: If $ X $ is a canonical RCLL process having the strong Markov property and $ \tau $ is a stopping time with $ \tau < + \infty, ...
3
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217 views

Expected time spent in the set

An exercise 2.14 from Bernt Øksendal's "Stochastic Differential Equations": Let $B_t$ be $n$-dimensional Brownian motion and let $K\subset \mathbb R^n$ have zero $n$-dimensional Lebesgue measure. ...
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495 views

Quadratic variation of a Brownian motion up to time $T$ converges to $T$ in $L^2$?

In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve, Theorem 3.4.3. Let $W$ be a Brownian motion. Then $[W, W](T) = T$ for all $T > 0$ almost surely. where $[W, ...
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24 views

Brownian Motion with Levy's Characterization 2

Let W be a $\mathbb{R}$-valued Brownian motion. To prove that $(B_t)_{t\geq 0}$, where: $B_t:=W_t-\int_0^t\frac{W_u}{u}du$, is a Brownian Motion with respect to $\mathcal{F}^B$, I showed $[B]_t=t$ and ...
2
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51 views

Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $(exp(\lambda X_t-\frac{\lambda ^2}{2}t))_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{R}$-valued process X ...
2
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23 views

Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
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16 views

bromnian motion and use of Lebesgue's differentiation theorem

Let $M$ be a Brownian motion with $M_0=0$ and $V\in L(M)$. Use Lebesgue's differentiation theorem to prove that there exists a predictable process $H\in L(M)$ such that $V\cdot M$ and $H\cdot M$ are ...
2
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21 views

Intuition behind “stochastic orthogonality”

Whilst doing an exercise on the Brownian Motion on a sphere I came across this identity: $$ \langle Z\times B,Z\times B\rangle = 2|Z|^2dt $$ where $\times$ denotes the cross product and $Z$ is a ...
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22 views

Long term behavior of Brownian Motion

Let $(B_t)_{t \geq 0}$ be a Brownian motion. The objective is to prove that \begin{align*} \limsup_{t \to \infty} \frac{B_t}{\sqrt{t}} = \infty. \end{align*} By the scaling property of Brownian ...
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25 views

Laplace transform of survival probability for stochastic diffusion

Let $Y_t$ be a killed process defined by \begin{eqnarray} Y_t = X_t \quad \mbox{if } t<\xi,\\ Y_t = 0 \quad \mbox{if } t\geq\xi. \end{eqnarray} where $\xi$ is a random time such that $$ \xi=\inf ...
2
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36 views

Can a time-shifted Brownian motion be also a Brownian motion

Let $T<\infty$, $(\Omega,\mathcal F,P)$ a probability space carrying a standard $d$-dimensional Brownian motion $(B_t)_{t\geq 0}$ and $(\mathcal F_t)_{t\geq 0}$ the natural $\sigma$-algebra ...
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32 views

Independence of Brownian motions

Let $W_t$ be a Brownian motion at time $t$ and let $t<t_1<T$. I'm trying to find the variance of $$ k\sigma W_{t_1} + \sigma \left(W_T - W_t\right).$$ I started by letting $$ k\sigma W_{t_1} + ...
2
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23 views

If a Stochastic Process has Variance linear with t, how to prove it is not Wide Sense Stationary?

For my study, as a part of a Matlab exercise, the following question is asked: Using the results of the estimated standard deviations of the random variable $x(k)$ for $k = 10^3; 10^4; 10^5$ ...
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38 views

Brownian motion, harmonic functions and the Dirichlet problem

I am having trouble understanding one detail of the standard use of Brownian motion to solve the Dirichlet problem, I will write the statement and proof and then point to the detail I don't ...
2
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57 views

Brownian motion in $2$ dimensions on the plane

Consider a $1$ dimensional Brownian motion of a particle starting at $0$. Then, we know that the probability that the particle reaches the point $x$ at a time $\geq t_0$ is given by ...
2
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29 views

Stopping time distribution and transforms with 1-dimension B-motion.

Let $W_t$ be a 1-dimensional Brownian Motion. For $x>0$, we define: $$\tau_{x} = inf \{ t \geq 0; |W_t| = x\}$$ Compute $E[e^{-s\tau_x}]$ and prove that $\tau_x$ is equal in distribution to ...
2
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23 views

Verify argument about random time being measurable w.r.t. a reversed BM increment process.

Let $B$ be a brownian motion (assume for convenient notation that it is two-sided). Let $T:=\sup\{t<2 :\vert B_t-B_1\vert\geq 1\}.$ Let $Y$ be the process $s\mapsto (B_T-B_{T+s})$. I want to argue ...
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20 views

$2$ d Brownian motion move from one point to another

Suppose $W_t= (X_t,Y_t)$ is a $2$d standard Brownian motion starting at $(-1,0)$. How do I show that there is a positive probability that $W_t$ moves from $(-1,0)$, to a neighborhood of $(1,0)$, say ...
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31 views

Brownian motion : disjoint intervals with the same maximum

In this article, at the end of page 11, there is a proof that for a Brownian motion, almost surely, there exist disjoint intervals with the same maximum. The proof uses the random variables ...
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68 views

Definition of Standard Brownian Filtration

I am trying to learn about stochastic calculus for my research, so self study, and I came across the notion of a Standard Brownian Filtration. I cannot find a good definition of what the Standard ...
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39 views

Tail field versus germ field of Brownian motion

Continuing my foray into Brownian motion (apologies for the bombardment...), I'm trying to verify the details of a proof of Durrett of the following 0-1 property of the tail $\sigma$-algebra of ...
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40 views

Reflection principle in the proof of the distribution of $M_t - W_t$ (Brownian motion)

Let $W_t$ be the Brownian motion starting at $0$. Consider the following random variables. $M_t = \sup_{0\leq s \leq t} W_s$ and $|W_t|$. We first calculate $$\Bbb{P}(|W_t|>a ) = \Bbb{P}(W_t ...
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37 views

Extension of Cameron-Martin formula via monotone class theorem

my question revolves around the Cameron-Martin theorem: Let $(\mathcal{C}_{(0)}[0,1],\mathcal{B}(\mathcal{C}_{(0)}),\mu)$ be the Wiener space (i.e. continuous functions starting in $0$, equipped ...
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29 views

$\limsup$ of Brownian Motion Time Integral

The following are well-known: $\limsup_{t\rightarrow \infty} \frac{B(t)}{t} = 0$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt t} = \infty$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt ...
2
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93 views

Moment generating function of $(W_T, \max W_t)$

Does there exist an explicit formula for the moment generating function $\psi(u, v) = E e^{u W_T + v M_T}$ of the pair $(W_T, M_T)$ where $M_T = \max_{0\leq t\leq T} W_t$? Using the well-known pdf of ...
2
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0answers
22 views

Mean and variance regime-switching model

Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal ...
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70 views

Electrostatic capacity of two spheres with changing radii

Although I have read a lot of questions and answers here, this is my first time actually posting. Feel free to suggest needed edits. My question is the following (in a simplified setting). All this ...
2
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0answers
60 views

Cross Variation of two stochastic processes

I am currently working on a stochastic calculus exercise at the moment and I am slightly confused when it comes to finding cross variation. We are given that the process $X_t = W_t^3$ ($W_t$ is ...
2
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0answers
30 views

Law of a supremum of random variables

Let $(B_t)_{t\geq 0}$ the standard brownian motion (with $B_0=0$), $p$ be a real number greater than $1$ and $q$ its conjugate number. Prove that $X_p=\sup _{t\geq 0}(|B_t|-t^{p/2})$ is a.s. strictly ...
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30 views

Can we integrate brownian motion with respect to a deterministic function

Let $B_t$ be brownian motion at time $t$, and $x$ be some random variable. For instance, I know that $$\int_0^T 1 dB_t = 1(B_T-B_0)$$ And that $$\int_0^T \cos(B_t) dB_t$$ cannot be directly ...
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31 views

Construction of Brownian motion - differentiability

I'm working on the the construction of BM given by Lévy-Ciesielski. The author begin to prove another result and for this reason he assume that BM exists and that it is also differentiable. For this ...
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71 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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124 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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39 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 = ...
2
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101 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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46 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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79 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). ...
2
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0answers
90 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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88 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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39 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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48 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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47 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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200 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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0answers
75 views

Stochastic Integral and Ito Isometry

I am right now studying stochastic integral, and facing the following dilemma! I just want to check whether my understanding is right! The stochastic integral is defined by following: $I(t) ...
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57 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 ...
2
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0answers
66 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 ...
2
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0answers
49 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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0answers
22 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 ...
2
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0answers
78 views

Deriving mean and variance of a function of Gaussian process

Suppose $\mathbb{G}$ is a tight zero mean Gaussian process and $F$ is an absolutely continuous CDF $$Y=\int_a^b\frac{d\mathbb{G}}{1-F}+\int_a^b\frac{\mathbb{G} \, dF}{(1-F)^2}$$ I know that $Y$ is a ...