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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Independence of two events in a Brownian motion

Let $\{X_k:k\geq0\}$ a Standard Brownian motion. Compute the following propability $$P(X_2>0|X_1>0).$$ The question is: Are $\{X_2\}$ and $\{X_1\}$ independent? I know: ...
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1answer
20 views

Variance of a time dependant gaussian

I'm trying to find the variance of the following: $$ \int_{0}^{t} N\Bigl(0,\sigma^2e^{-C(t-\tau)}\sin^2\bigl(B(t-\tau)\bigr)\Bigr)d\tau $$ where $N$ is a Gaussian distribution with zero mean and ...
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1answer
21 views

Discontinuous expression cannot be a Brownian motion?

Let $W_t$ and $\tilde{W}_t$ be two standard independent Brownian motions and for a constant $-1 \leq \rho \leq 1$, define $X_t := \rho W_t + \sqrt{1-\rho^2} \tilde{W}_t$. Is $X := (X_t)_{\{t ...
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1answer
88 views

Use Ito's Formula to prove following identity

Again, I am not sure how the following works; Could someone please give me an almost stupidly detailed explanation of why/what is happening in the part below. First, the question itself; Q. $B_t$ ...
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19 views

Is there a difference between Brownian motion and Standard Brownian motion?

I find the two very confusing as some seem to use them interchangeably and some don't seem to. Wiki says they're both the same "...is often called the standard Brownian motion" it says in the "Wiener ...
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4answers
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Find $\mathbb{E}[W_t^3]$ and $\mathbb{E}[W_t^4]$

I am very stuck on this past paper question. $W_t$ is a brownian motion and find $\mathbb{E}[W_t^3]$ and $\mathbb{E}[W_t^4]$ I thought, since $W_t$ is normally distributed with density function ...
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1answer
43 views

How to integrate the following geometric brownian motion in Black-Scholes framework

As my previous questions make it obvious, I am very new to this field of mathematics and wondering if I am doing things right in the following question. Let $T \in (0, \infty)$ and consider a ...
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40 views

What is $Var(X_t-X_s)$ if $X_t = \sqrt{t} Z$

What is $Var(X_t-X_s)$ if $X_t = \sqrt{t} Z$ where $Z \sim N(0,1)$ The answer is given by $(\sqrt{t}-\sqrt{s})^2$. How do they get this? My thoughts: $X_t\sim N(0,t)$ and $X_s\sim N(0,s)$ I ...
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1answer
47 views

random walk and calculating the probability of paths

Consider a random walk $(X_n)_{n≥0}$ with $p = 0.7$, starting from $X_0 = 3$. Find the probability that $X_{10} = 5$, but $X_n ≥ 1$ for $n = 0, . . . , 10.$. Essentially what I got from the ...
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34 views

What is the variance of a Brownian Motion?

In my attempt to digest the answers to my previous question about stochastic integrals, I have stumbled upon yet another question that I need some clarification on... Simply, what is the variance of ...
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75 views

Asymptotics for the probability a discrete Brownian bridge remains below a logarithmic barrier

Let $(\mathcal{Z}(i))_{1\leq{i}\leq{\text{N}}}$ be a discrete Brownian bridge of lifespan $\text{N}$ conditioned to start and end at $0$, i.e. $\mathcal{Z}(1)=0$ and $\mathcal{Z}(\text{N})=0$. I would ...
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3answers
199 views

Stochastic Integrals are confusing me; Please explain how to compute $\int W_sdW_s$ for example

I have been trying hard to understand this topic, but only failing.Reading through my lecture notes and online videos about stochastic integration but I just can't wrap my head around it. The main ...
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0answers
29 views

Interchange intersection and union in proof of Blumenthal’s zero-one law

I am trying to prove Blumenthal's zero-one law using Kolmogorov's zero-one law. I use that $B_t$ Brownian $\iff$ $tB_{1/t}$ Brownian. Can I change the intersection and union as follows? Start of my ...
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28 views

differential stochastic equation and solutions

Let E(K):$Z_0=0$ and $dZ_t=K_t(B_t-Z_t)dt+\alpha K_td \beta_t$ with B and $\beta$ two brownian motions, $\alpha>0$ and K a continue function. 1)Show that E(K) has a solution Z with $E(\int_0^1 ...
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13 views

Compute the conditional distribution for $R(t):=[X(t)]^2$

Good evening, I can't solve the following exercise: Let be $R(t):=[X(t)]^2$, while $X(t)$ a Brownian motion with $X(0)=0$ and drift $\mu=0$. Compute the distribution of $R(t)$. I don't have any ...
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1answer
79 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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25 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 ...
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1answer
124 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
77 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 ...
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1answer
42 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} ...
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1answer
45 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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19 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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32 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
25 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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23 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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42 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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0answers
31 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
33 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
118 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 $e^{-s-t}$. ...
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1answer
93 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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33 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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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
112 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
47 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
42 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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21 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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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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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
69 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, ...
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1answer
32 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 ...
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1answer
58 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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34 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
49 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 ...
3
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1answer
67 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 ...
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1answer
69 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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22 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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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} ...