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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51 views

lim sup and lim infs of Brownian Motion: $B_t/\sqrt{t}$ as $t \to \infty$ or as $t \to 0$.

Below is my question. Q7.9 is what I'm stuck on. I've done Q7.8; I included it in the picture because I'll use it in Q7.9, and it gives a definition that I'll use. Update: This question is now ...
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22 views

Guess quadratic variation of 2 processes, with same/different BM.

I have 2 processes with stochastic parts R, S. $$dR = \mu_1 dt + \sigma_1 dW_{t1}$$ $$dS = \mu_2 dt + \sigma_2 dW_{t2}$$ I am trying to show what precisely quadratic variation $[R,S]$ means for 2 ...
5
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1answer
98 views

Help integrating the transition probability of the Brownian Motion density function.

1. Problem: Given the Brownian Motion with Drift: $$ dx = \mu \, dt+\sigma \, dW $$ It can be shown that the transition density function is the following: $$ p(x, t) = \frac{e^{-\frac{(x-\mu ...
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1answer
31 views

Proof of Ito's Identity using Ito's formula

Prove $\int_{0}^{t} W_s dW_s=1/2 {W_t}^2-1/2t$ using Ito's formulas. I don't really know how to approach this problem since I'm not given a function to find it's derivatives and plug into the Ito's ...
3
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1answer
41 views

Estimate the probability $P(X > C\frac{(n-1)^p}{\sqrt{n}})$ for $X\sim N(0,1), C>0, p>1/2$

Let $(B_t)_{t\geq 0}$ be brownian motion, let $p>1/2$. I want to show that $$\lim_{t\to\infty} \frac{B_t}{t^p} \to 0 \quad a.s.$$ Atm I'm trying to show that $$\limsup_{t\to \infty} ...
5
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1answer
39 views

Prove that $\tilde{W}_t := W_{t+r}-W_r$ is a Brownian motion.

I am to prove that, given a Brownian Motion(Wiener Process) $\{W_t\}$, a newly defined $\tilde{W}_t=W_{t+r}-W_r$ where $r \geq 0$ is a Brownian motion. I am stuck with showing it is a Gaussian ...
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17 views

Solution of heat equation with cross terms on rectangle.

I would like to find the fundamental solution of the following PDE $$ u_t = \frac12 u_{xx} + \rho u_{xy} + \frac12 u_{yy} $$ on the rectangle $[-a,a]\times[-b,b]$, with $a,b>0$ and with homogeneous ...
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22 views

Laplace transform of the square of Brownian motion hitting time

Let $B_{\mu}(t)$ be a one-dimensional Brownian motion with drift $\mu \geq 0.$ For $a > 0,$ let $$T_a = \inf\{t > 0: B_{\mu}(t) = a\}$$ denote the first hitting time of $B.$ The Laplace ...
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1answer
19 views

Solve and prove uniqueness of the SDE $dY_t = tY_tdt + e^{t^2/2}dB_t$ without using the general linear SDE formula

Let $(B_t)_{t \in [0,T]}$ be standard brownian motion, and let $(Y_t)_{t \in [0,T]}$ be a stochastic process in $(\Omega, \mathscr F, \mathbb P)$. Without using the general linear SDE formula, solve ...
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2answers
51 views

Compute $\mathbf E[B_s^4B_t^2-2B_sB_t^5+B^6_s]$

Let $\mathbf B=\{B_t\}_{t\ge0}$ a continuous Brownian motion, what is then $\large\mathbf E[B_s^4B_t^2-2B_sB_t^5+B^6_s]$, for $t\ge s$ ? How can I factorize the expression in the parenthesis, If ...
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21 views

What does $W_{1}$ and $(W_{1},W_{2})$ mean under the context of Brownian motion $W_{t}$?

As part of some practice questions for a course I'm taking, I was given the definition of a Brownian motion $W_{t}$ as a unique continous-time stochastic process satisfying: $W_{0}=0$ The function ...
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39 views

Weak convergence $B\circ f_n\to B\circ f$

Let $B$ be a brownian motion and consider a sequence of continuous functions $f_n$ defined on $[0,1]$, such that $f_n(x)\to f(x)$ for each $x$. Is it true that $B\circ f_n$ converges to $B\circ f$ as ...
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17 views

First passage time of the Brownian motion

In an exercise (4.1 Krapinsky, "A kinetic view of statistical physics") I am asked to show that: The probability that a brownian motion on a 1D discrete lattice never reaches the site $n$ scales as ...
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2answers
54 views

Continuity of probability measures for a process

Let $(B_t)_t$ be a Brownian motion, then I am given a stopping time $\tau_s:=\min(\inf\{t \ge 0; B_t=a\}, \inf\{t \ge s; B_t=b\}; \inf \{t \ge 0;B_t=c\}),$ where $a<0<b<c.$ Now, I want to ...
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1answer
32 views

Estimate the typical numeber of times a brownian motion on ℤ starting from $0$ does a particular transition

Consider an 1D infinite lattice. The lattice is fully occupied except from a vacancy in the origin which undergoes simple diffusion (in countinuous time). At position $n>0$ in the lattice there is ...
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1answer
81 views

Covariance function for a Brownian motion

Let $B(t)$ be a standard Brownian motion. For $t\geq 0$, define $$U(t) = e^{-t}B(e^{2t}).$$ The problem is to determine the covariance function of the process. Supposedly, the answer is $e^{-s-t}$. ...
5
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1answer
70 views

To confirm the Novikov's condition

I have a question about Novikov's condition. Let $L$ be a local martingale such that either $\exp \left(\frac{1}{2}L \right)$ is a submartingale or $E[\exp\left(\frac{1}{2} \langle L,L \rangle_{t} ...
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0answers
28 views

Prove $X_t$ is a Gaussian process?

Let $X_t = \int_0^t K(t-s)\,dBs$, where $K$ is the kernel and $B_s$ is a Brownian motion. Is $(X_t)_{t\in\mathbb{R}^+}$ a Gaussian process? Why? If so, how can I compute its covariance function ...
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0answers
19 views

Why is the stochastic integral $\int_0^t \nabla u(B_s)\cdot dB_s $ a local martingale?

This is from Durrett's book Stochastic calculus: a practical introduction. I don't understand the last sentence in the picture. Could anyone help explain why the first term is a local martingale? ...
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1answer
51 views

Brownian motion: hitting times for closed sets are stopping times (and more).

Let $(B_t)$ be a $d$-dimensional Brownian motion, and consider the filtrations $(\mathcal{F_t^B}) = \sigma(B_0,...,B_t)$ and $\mathcal{F_t} = \cap_{\epsilon > 0}{\mathcal{F_{t+\epsilon}^B}}$ (the ...
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1answer
35 views

Asymptotic behaviour of brownian motion

Let $(B_t)_{t\geq 0}$ be a brownian motion, i want to show that $$\frac{B_t}{t^p} \xrightarrow[t\to\infty]{a.s.} 0, $$ for all $p>\frac{1}{2}$. I was told to use that $$X_t = \frac{B_t^2 - ...
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1answer
27 views

What is the distribution of the position of the maximum of a Brownian bridge?

What is the distribution of the position of the maximum of a Brownian bridge?
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17 views

Distribution of $L(\text{inf}\{t: |B_t| \ge x\})$.

Let $B_t$ be a standard Brownian motion. Let $L_t = L(t)$ denote the Brownian local time. Can anyone supply a reference as to the distribution of $L(\text{inf}\{t: |B_t| \ge x\})$? I know that ...
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63 views

Is $X_t = tW\left(\frac{1}{t}\right)$ a Martingale?If not, how could it be a Brownian Motion?

As is proved, $X_t = tW\left(\frac{1}{t}\right)$ is a Brownian motion. For example see Theorem 4.2 in this paper http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf I'm just confused because ...
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18 views

Expectation of exponential of an additive functional of Brownian motion

I have a question about an additive functional of Brownian motion. Let $d \in \mathbb{N}$. Let $b:\mathbb{R}^{d}\to \mathbb{R}$ be a measurable function and $(X_{t})_{t \in [0,\infty[}$ be a ...
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1answer
67 views

Uniqueness in law associated to nonlinear SDEs

I do not understand the following when reading a paper on Propagation of Chaos, written by A.S.Sznitman: Consider an $n$- dimensional process $X$ satisfying the following SDE: $$ dX_t = b(t, ...
2
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1answer
31 views

Determine $P(\lim_{t \to \infty}X_0 e^{t(1-\sigma^2/2) + \sigma W_t)}=0)$

Let $W_t$ be standard brownian motion and define the process $$X_t = X_0 e^{t(1-\sigma^2/2) + \sigma W_t}$$ where $\sigma$ has exponential distribution $$ P(\sigma \leq x) = 1-e^{-x}$$ for ...
2
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1answer
46 views

Show that $\frac{N}{m}((N+m)\ln(N+m)+(N-m)\ln(N-m)-2N\ln(N)) \to 1$ when $N\gg m$

$$\frac{N}{m^2}((N+m)\ln(N+m)+(N-m)\ln(N-m)-2N\ln(N))\to 1 \text{ when } N \gg m$$ I got this expression from fiddling around with Brownian motion. From inputing values for $N$ and $m$ I can see ...
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0answers
26 views

Martingale expectation value

I want to show that for $(B_t)_t$ being the Brownian motion and a stopping time $\tau:= \text{inf}_{t \ge 0} \{B_t= a+bt\}$ where $a,b>0$ we have that the expectation value $E(e^{-\lambda \tau}, ...
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1answer
32 views

Solve SDE $dX_t = (e^{-\gamma t} - \gamma X_t) dt + 2 e^{-\gamma t/2} \sqrt{X_t} dW_t$

Solve the SDE given by: $dX_t = (e^{-\gamma t} - \gamma X_t) dt + 2 e^{-\gamma t/2} \sqrt{X_t} dW_t$. My attempt Following the hint of my professor: suppose $X_t = e^{\gamma t} g(W_t)$. Then we ...
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1answer
9 views

Symmetrical function of Brownian Motion

Let $W_t$ be Brownian motion. Using software, I can compute $E[e^{\beta t} \sin{(\gamma W_t})] = 0$. Could one void this computation with a clever symmetrical argument. That is: Since $sin(t)$ is an ...
2
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1answer
41 views

Brownian motion martingale

I have been wondering about the following equality in the textbook by Liggett. I put a red circle at the position where my question is. They use the theorem that $B_t^2-t$ is a martingale and the ...
5
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1answer
51 views

Can the integral of Brownian motion be expressed as a function of Brownian motion and time?

Let $W_t$ be standard Brownian motion, and define $$ X_t := \int_0^t W_s ~\textrm{d}s. $$ The marginal distributions of $X_t$ are easy to write down (see here), but it doesn't seem possible to express ...
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12 views

Brownian motion noise strength in discrete time step and in continuous time.

In this Langevin dynamics tutorial In the second part talking about Implementation. It says because we are using discrete time step, we need to divided the variance by time step. In Langevin ...
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15 views

Compute $E(Y_t^2)$ with $Y_t =\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} dW_{u})ds$

Consider the process, $Y_t =\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} dW_{u})ds$. To compute the variance of this process, I need to compute $E[(\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} ...
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54 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 ...
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1answer
31 views

Brownian motion a martingale

Let $B(t)$ be a Brownian motion and $\tau$ a finite stopping time. Assuming that I know that $B(\tau)$ is a martingale, does this imply that $E(B(\tau))=0$ and $E((B(\tau))^2)=\tau$ just as one would ...
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0answers
17 views

Brownian Motion $W(t)$ or Functions of BM in $dx$

Let $W(t)$ be a standard Brownian Motion. I'm considering the case where $X(t) = \mu t + W(t)$, i.e. $X(t)$ is a BM with constant drift $\mu \neq 0$. My instructor said that: $\mathbb{P}(M^X(t) \leq ...
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1answer
58 views

Conditional probability of Brownian motion

I am given the following exercise, where I don't see how to do this Here $B_t$ is a Brownian motion. The expectation value $E^x$ means that we take the expectation value w.r.t. a Brownian motion ...
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1answer
28 views

Finding Moments of Brownian Motion

I am trying to calculate the K-th moment for Standard Brownian Motion: $Z(t) \sim N(0,t)$ I'm given that the second moment is $t$, but I'm having trouble seeing how that was arrived at. I thought ...
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1answer
45 views

Augmented filtration martingale proof.

Part a: Consider a Wiener process, $W_t$ and denote by ${\mathscr{F}_t}_{(t \geq 0)}$ the natural filtration generated by W. Let $\mathbb{R}_{+} = \{x : x \geq 0\}$ and $\mathscr{B}$ be a sigma ...
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1answer
49 views

Mistake in proof on Brownian motion?

I am currently reading these notes and in particular try to understand the proof of theorem 5.12. Actually, my only questions are about the last paragraph. He claims that $T_n$ is a uniformly ...
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27 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 ...
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1answer
26 views

Expected value related to Brownian motion

Suppose $\{W_t:t \geq 0\}$ is a Brownian motion, let $0 \leq s<t$, what's the following expected value: $\mathbb{E}[W_t^3-W_s^3-3(tW_t-sW_s)]$ $\mathbb{E}[W_t^4-W_s^4-6(t-s)W_s^2-3(t-s)^2]$. ...
2
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1answer
48 views

If $B(t)$ is Brownian motion then prove $W(t)$ defined as follows is also Brownian motion

Let $B(t)$ be standard Brownian motion on $[0.1]$ Define $W(t)$ as follows $W(t) = B(t) - \int_0^t \frac{B(1)-B(s)}{1-s} \, ds$ Prove $W(t)$ is also Brownian motion So I'm not sure how to deal ...
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0answers
34 views

The conditional expected value of Brownian motion $B(t)$ given $B(s)$, $s\leq t$, as well as $B(1)$

I am trying to calculate the expected value of the increment of Brownian motion $B(t+h)-B(t)$ given the enriched filtration generated by both $B(s)$,$s\leq t$, as well as $B(1)$. Now as $B(t)$ is ...
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0answers
28 views

Construction of the Itō integral with (local) martingales as integrators

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$. $\xi_i$ be a real-valued random variable on ...
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20 views

Prove Poisson kernel of the upper half-plane is the Cauchy distribution

Let $d=2$, and consider the domain $D=H$, the upper half-plane. Show that the Poisson kernel is the Cauchy distribution by following the steps below. (A) Let $W_t=(X_t,Y_t)$. Show that for any ...
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1answer
32 views

Is $(W_{2t}-W_{t})_{t \geqslant0}$ a brownian motion?

$W_{t}$ - Stochastic process ( Brownian motion). I need to check if $ (W_{2t}-W_{t})_{t \geqslant0}$ is a Brownian motion. I look at the independent increment property. I want to find contradiction ...
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21 views

How to use the modulus of continuity of the Brownian motion.

I have a deterministic sequence $r_n$ such that $r_n\rightarrow 0$ and $r_n\,\sqrt{n}\rightarrow 0$ when $n\rightarrow\infty$. Then I have the following random variable $$ A_n = ...