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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1answer
46 views

Check solution to the SDE $dX_t = - \mu X_t \, dt+ \sigma \, dW_t$

I get stuck in this problem. I just can't get the hang of how we need to "guess" a function first and almost everything along the process of solving depends on it; It's not entirely logical to me when ...
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0answers
53 views

Time to exit strip for a geometric brownian motion

I have a question about the geometric brownian motion ${\rm d}S = \mu {\rm d}t + \sigma dW$. I want to calculate $v(x)$ which is the expected time $\tau$ at which the particle first exits the strip ...
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0answers
37 views

Hitting time for Browian motion with upper reflecting boundary

I was wondering if there exist a known distribution function or a nice closed form describing the first hitting time to a given threshold $a$, $T_a$, for a Brownian motion bounded by a upper ...
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1answer
30 views

Check process is a martingale

I have such stochastic process with which I struggle all day, finally I found 2 mistakes, however answer is still unsatisfying. $$X_t = atW_t^2 - \int_0^t(W_s^2+s)ds,$$ I need to check if it is a ...
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1answer
20 views

Ito's formula but not given $\mu$ and $\sigma$

I have a little question from one of my worksheets(the solution I was given was almost not even a solution, super brief). let $f(t,x)=t\cos(x)$. Use Ito's formula to calculate $df(t,W_t)$. Well, ...
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0answers
33 views

Integration wrt BM

How do I integrate: $\int_{\mathbb{R}} (S_t - K)^+ \phi(t) dt$ where $\phi$ is a normal density and $S_t$ is a geometric brownian motion? I know my answer should be $\Phi(d_1)$, where $\Phi$ is the ...
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1answer
27 views

What is wrong with my calculation for variance?

I don't get this, I am asked if the following $X_t$ is a brownian motion or not. $Z$ is a standard normal variate. $X_t=\sqrt{t}Z$. I s$X_t$ a Brownian motion? Answer is apparently no and one of ...
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0answers
36 views

Black Scholes derivation; How and Why

A 15 mark past paper question essentially ask s me to derive the Black Scholes formula for pricing options. Let $S_t=S_0e^{(r-\frac{\sigma^2}{2})t+\sigma B_t}$ where $B_t$ is a standard Brownian ...
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0answers
25 views

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
20 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
84 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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0answers
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
51 views

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
40 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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0answers
38 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
46 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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0answers
32 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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0answers
72 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
192 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
22 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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0answers
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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0answers
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
67 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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0answers
24 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
120 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
67 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} ...
5
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1answer
44 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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0answers
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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0answers
31 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
23 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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0answers
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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0answers
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
30 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
105 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
81 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
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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0answers
20 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
92 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
40 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
40 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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0answers
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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0answers
74 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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0answers
22 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 ...
1
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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
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 ...