# 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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### Holder continuity of Brownian Motion

Can any one help me in this question please: By using the law of iterated logarithm, I have to show that $\forall t \in [0, T]$ ; where T>0, $\exists A_{t} \in \mathcal{A}$ with $P(A_{t})=1$, a random ...
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### To show that $\dim Z=1/2$, why do I have to show that $p\{\dim Z=1/2\}=1$?

Let $(B_t)_{t\geq 0}$ a standard Brownien motion. I have to show that $\dim Z=\frac{1}{2}$ where $Z=\{t\in [0,1]\mid B_t=0\}$. Why to do this, I have to show that $$\mathbb P\{\dim Z=1/2\}=1\ \ ?$$ I ...
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### Brownian Motion hitting time is finite yet has infinite expectation?

I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I ...
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### Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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### Ito's formula application

Let $\alpha, \beta \in R$ and define $$N(t)=e^{\beta t} \cos(\alpha W (t))$$ I need to use Ito formula to compute $dN(t)$ Suppose $\alpha$ is fixed. What should $\beta$ be so that $N$ is a ...
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### Yet another application of Ito's formula

Question : Let $dW^4(t)$ be the sum of an ordinary integral with respect to time and an Ito integral. Where $W^4(t)$ are standard Brownian motion. I am trying to apply Ito's formula to this, say ...
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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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### show that a function with brownian motion is a martingale

Let $B=(B^1,B^2)$ be a two-dimensional Brownian motion w.r.t. the Filtration $\mathcal{F}^B$. Show that $(M_t^2)_{t\in \mathbb{R}_{+}}:=(e^{B_t^1} \cos(B_t^2))_{t\in\mathbb{R}_{+}}$ I've tried it ...
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### Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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Let $M$ be a true real martingale adapted to some brownian motion $B$. What are the most generic conditions on $M$ to find a deterministic map $\Phi:\mathbb{R}_+\times\mathbb{R} \to \mathbb{R}_+\... 2answers 39 views ### Checking if$X(t) = \exp(t/2)\cos(W(t))$, with$W(t)$a Wiener process, is a martingale This is what I've done: Let$s < t$and$F_t$be a filtration adapted to$W(t)$$$E[e^{t/2}\cos(W(t))|F_s] = e^{t/2} E[\cos(W(t)) - \cos(W(s)) + \cos(W(s))|F_s]$$ $$= e^{t/2} [E[\cos(W(t)) - \cos(... 1answer 42 views ### How to use the Markov property of Brownian motion This is a problem from Durrett's probability with examples, exercise 8.2.1. It is not homework. The exercise states: Let T_0 = \inf\{s > 0 : B_s = 0\} and let R = \inf\{t > 1 : B_t = 0\}. ... 0answers 30 views ### Conditional expectation of a hitting time of a Brownian motion and Laplace transform I am trying to solve the following problem: Suppose B is a 1-dim Brownian motion, let \mathcal{T}_a = inf\{t: B_t = a\}, \mathcal{T}_{a,b}=min\{\mathcal{T}_a,\mathcal{T}_b\}. For a < 0 < b ... 0answers 88 views ### Brownian motion on sphere proof? proving the brownian motion on the sphere equation the stratonovich form differential equation$$\partial X=n(X)\times \partial B$$the equation in ito's form becomes$$dX=n(X)\times dB+H(X)n(X)... 1answer 32 views ### Probability that a Wiener process is negative at 2 given that it was positive at 1 Let$W_t$be a standard Wiener process, i.e., with$W_0=0$. If$W_1>0$, what is the probability that$W_2<0$? This is my attempt: we want to determine the conditional probability $$\mathbb P(... 0answers 9 views ### Upper bound involving simple Ito process Let (B(t),\{\mathcal{F}_t \}) be one-dimensional Brownian motion. Suppose that \sigma(t,ω) is a \mathcal{F}_t-adapted process satisfying |\sigma(t,ω)| ≤ R, for all t and w. I was ... 0answers 33 views ### Laplacian in spherical coordinates - brownian motion Consider the Laplacian equation on the unit sphere, for a vector f. \theta is polar angle, and \phi is azimuthal angle. The Laplacian in spherical coordinate is :$$ \Delta f = {1 \over r^2} {\... 1answer 35 views ### Brownian hitting time of a closed set I am trying to prove that the first hitting time of a closed set H by a Brownian motion is a stopping time. I have found a proof that states: $$\{\mathcal{T}\leqslant t\} = \bigcap_{n=1}^{\infty}\... 0answers 40 views ### Simple application of Donsker's theorem I am trying to do exercise 5.15 in Moerter's book "Brownian Motion". It seems quite easy, but I can't solve it somehow: Suppose S(j)_j is a SRW on the integers, started at zero. Show that:$$ \frac{... 0answers 30 views ### BM hitting times with exponential killing process Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point$p_s=(x_s,y_s,z_s)$. BM starts at point$p_0=(x_0,y_0,z_0)$and when it riches the subdomain ... 0answers 20 views ### Coupling Brownian Motions I want to simulate three freight rate indices which are naturally correlated. The freight rate dynamics ($X$) can be modeled as a geometric Brownian motion:$dX_{t} = \mu X_{t}dt + \sigma X_{t}dW_{t}$... 1answer 19 views ### Distribution of the first exit time of a one-dimensional diffusiom/ Brownian motion I have a one-dimensional diffusion on$[0,1]$and I need to calculate the distribution of the first exit time of the interval$(-\epsilon,\epsilon)$for an$\epsilon > 0$. A good first step would ... 0answers 17 views ### How to find the mean and variance of a stochastic integral? If$B(t)$is a standard Brownian motion, let$Z(t)= \int_{0}^{t} s^2 dB(s)$. I want to find the mean and variance of Z(t). It is given that$Z(t)$is Gaussian process. My approach for finding the ... 0answers 22 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 ... 1answer 34 views ### Prove that$\text{lim}_{\Delta t} \rightarrow 0$of the transition PDF of a std Weiner process is 0 The transition probability density function of the standard Wiener process is: $$f(x_2,t_2|x_1,t_1) = \frac{1}{\sqrt{2 \pi (t_2-t_1)}}e^{-\frac{(x_2-x_1)^2}{2(t_2-t_1)^2}}$$ I know that if Markov ... 1answer 29 views ### Show that$p(t_0\ \text{is a local maximum for}\ B)=0$. Let$B$a Brownian motion. Show that for all$t_0$,$$p\{t_0\text{ is a local maximum for }B\}=0$$ but a.s. local maximal are a countable dense set in$(0,\infty ). For the first part, p\{t_0\text{... 1answer 16 views ### Einstein's number of particles that experienced a certain shift explanation I am reading a Gardiner's Stochastic Methods handbook and I am wondering about the meaning of the following (this is the very beginning of the chapter): dn = n \phi(\Delta) d \Delta This is ... 0answers 32 views ### Proof of normal distribution property used in Levy's construction of the brownian motion? I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its ... 0answers 31 views ### Can we apply an Itō formula to the solution of a SPDE? Let V\subset H\subset V^\ast be a Gelfand triple (\Omega,\mathcal A,\operatorname P) be a probability space and (\mathcal F_t)_{t\ge 0} be a filtration of \mathcal A (W_t)_{t\ge 0} be a ... 0answers 26 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 ... 1answer 33 views ### Why does the Borel-Cantelli lemma finish the job? - Law of Large Numbers Brownian Motion The objective is to prove that \begin{align*} \text{\lim_{t \to \infty} \frac{B_t}{t} =0 \qquad a.s.} \end{align*} By the strong Law of Large Numbers, we have that: \begin{align*} \text{\lim_{t \... 0answers 17 views ### Brownian motion: Why \xi_n and \xi_m are independent. Let \{B_t\} a Brownian motion, n\neq m and\xi_n=\sup_{s\in [n,n+1]}|B_s-B_{\lfloor s\rfloor}|\quad \text{and}\quad \xi_m=\sup_{t\in [m,m+1]}|B_t-B_{\lfloor t\rfloor}|.$$We suppose WLOG that ... 1answer 107 views ### Discrete and continuous Girsanov I'm trying to write a proof of the Girsanov theorem based on a discrete version of it. Discrete version Suppose that I have a random vector X and two equivalent probability measures \mathbb{P}, \... 0answers 18 views ### If B_t - B_s, \ 0\leq s < t, is normally distributed, there are constants C_n, \ E|B_t - B_s|^{2n}=C_n|t-s|^n I am working on the following problem: Show that if B_t - B_s, 0 \leq s < t, is normally distributed with mean zero and variance t-s, then for each positive integer n there is a positive ... 1answer 27 views ### Quadratic Variation of Wiener's process I know I'm wrong, but I still don't understand why can't this operation be performed:$$ \sum_{j=0}^{n-1}(W(t_{j+1})-W(t_j))^2\le \max[W(t_{j+1})-W(t_j)]*(W(T)-W(0) )$$which would have a limit of ... 0answers 25 views ### Show that W(t) is almost surely non-differentiable at t=0 Show that W_t is almost surely non-differentiable at t=0. Of course, W(t) denotes a standard Wiener process. It is enough to show that$$P(\{\omega : \exists \epsilon>0 \: \forall \delta &... 0answers 24 views ### Compare moments of\int_0^t h(B_s) ds$and$\int_0^t h(\sqrt{s}Z)ds$for$(B_t)$Brownian motion and$Z$standard normal If we let$B_t$be a standard Brownian motion and$\sqrt{t}Z$, where$Z$is our standard normal random variable, we know that they have the same distribution. However, how can I show that the process ... 0answers 38 views ### How to find the variance of$\int_0^t B_s^2 ds$where$B_s$is a standard Brownian motion random variable? I am trying to find the variance of$\int_0^t B_s^2 ds$where$B_s$is a standard Brownian motion random variable. My approach is to represent the integral as a sum. However, I am not sure how this ... 0answers 25 views ### Proving if$W(t)$is a weiner process, then$W^2(t)$is also a Weiner process [duplicate] I'm trying to solve this question: For a stochastic process to be a Weiner Process it must have these properties:$W(0) = 0$so$W^2(0) = 0E(W(t)) = 0$but$E(W^2(t)) = t$I think this is enough ... 0answers 19 views ### What is meant by local time of BM on the boundary$\partial D$? I'm familiar with local time$L_t^a$at level$a$for a 1-D Brownian motion$B$. I'm reading this paper which talks about a 2D Brownian motion$B$in a bounded domain$D$that gets reflected when it ... 0answers 27 views ### Probability of hitting a barrier We have a stochastic process$ Y_t= \alpha t+ W_t$where W is a standard brownian motion. Is there a way to calculate the conditional probability with respect to$Y_1$for this process to hit a ... 1answer 47 views ### Hitting times for Brownian Motion (2) In this post there is shown that for a standard Brownian motion$\mathbb{E}[\tau^p]<\infty$for all$p \geq 1, where \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\... 1answer 50 views ### How can we identify\omega\in\Omega$with a path of Brownian motion$t\rightarrow B_t(\omega)$? In the Stochastic Differential Euqations written by Oksendal(see page 12), As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. ... 0answers 25 views ### PDE for Brownian Bridge Expectation? Let$\displaystyle Y(t)=\int_0^t v(s)ds+B(t)$, where$B(t)$is the standard Brownian motion and$v(t)$a deterministic function. Compute$m(t,y):=\mathbf E\Big[\max\limits_{s\in[0,t]} Y(s)\big|Y(t)=y\...
Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...