# Tagged Questions

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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### When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
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### 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 $W_t$). Now,...
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### Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
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### First hitting time for a brownian motion with two exponential boundaries

I asked a previous related question here: First hitting time for a brownian motion with a exponential boundary Now Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first ...
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### infinitesimal generator of reflecting Brownian motion

Suppose $f\in C_0^{\infty}([0,\infty))$ and $f'(0)=0$. I'm having trouble proving that $$\frac{1}{t}E_x[f(|W_t|)-f(x)]\to\frac{1}{2}f''(x)$$ uniformly on $[0,\infty)$ as $t\downarrow0$. Showing the ...
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### Negative moments of a functional of Wiener process

At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
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### 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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### 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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### Survival probability of a biased random walker

A random walker moves to $+1$ with probability $p$ and moves to $-1$ with probability $q=1-p$. If he starts at point $m$, what is the probability that he doesn't hit the point zero after $k$ steps, ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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The following are well-known: $\limsup_{t\rightarrow \infty} \frac{B(t)}{t} = 0$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt t} = \infty$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt {2t\... 0answers 94 views ### Moment generating function of$(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 ... 0answers 23 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 ... 0answers 72 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 ... 0answers 66 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 ... 0answers 31 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 ... 0answers 35 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 ... 0answers 32 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 ... 0answers 76 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 ... 0answers 129 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 $$\tau = \inf\{t \geq 0: \lvert B(t)\rvert ... 0answers 41 views ### 2D Brownian Motion — Does this argument work? Consider a 2D Brownian Motion (X_1(t),X_2(t)) starting at (x_1,x_2) \in \mathbb{R}^2. For every s\geq0, let$$\tau_s = \inf \left\{t \geq 0 \mid X_1(t) - x_1 > s \right\}\qquad Y_s = X_2(\... 0answers 50 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$...
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). ...