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
52 views

Is this process a martingale?

Given $X_t=\int_0^t s W_s dW_s$ and the process $M_t=X_t^3-\int_0^t X_sY_s ds$. Find $Y_t$ such that $M_t$ is a martingale. I started thinking that $X_t$ can be seen as: $dX_t=tW_tdW_t$ then by ...
2
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2answers
57 views

Why is the integral $\int_0^1t\,dW_t$ a normal random variable?

Consider the random variable $X=\int_0^1t\,dW_t$, where $W_t$ is a Wiener process. The expectation and variance of $X$ are $$E[X]=E\left[\int_0^1t\,dW_t\right]=0,$$ and $$ ...
1
vote
0answers
40 views

Compute a conditional probability

Let $x\colon[0,1]\to \mathbb{R}$ be a continuous path with, and let $B_t$ be a standard Brownian motion on some probability space. I want to compute the following conditional probability: $$ ...
1
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0answers
45 views

Is the variance of an integral the same as the integral of the variance?

Consider a standard Brownian Motion $X_t$ and continuous random variable $Y_t$, where $Y_t$ is defined as $$ Y_t = \int_0^t X_t \, dt $$ My goal is to compute the variance of $Y_t$. I'd like to say ...
1
vote
1answer
39 views

What is the “distributional derivative” of a Brownian motion?

Let $\emptyset\ne I\subseteq\mathbb R$ be an open interval and $A:C_0^\infty(I)\to\mathbb R$ be a distribution. Then, $$\langle{\rm D}A,\varphi\rangle:=-\langle A,\varphi'\rangle\;\;\;\text{for ...
1
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1answer
45 views

Mean of exponential Brownian motion

I am new to stochastics and I am trying to compute the expectation of $S_t = e^{\sigma W_t}$, where $W_t$ is a standard Brownian motion and $\sigma>0$. My attempt (using the log-normal PDF here and ...
0
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0answers
47 views

Does this make sense?

Can I write this? Let $W_s$ be a Wiener process and let $x_s$ be a stochastic square integrable process adapted to the filtration generated by $W$. Is such an expectation nonsensical? And if not, how ...
0
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0answers
13 views

Find stochastic processes with given expectations of their products

Find stochastic processes Sa, Sb, Sc, Sd such that all of the following hold: E[Sa Sc]=0 E[Sb Sd]=0 E[Sa Sb] = 1 E[Sc Sd] = 1 E[Sa Sc Sb Sd] = 0 My attempt at a solution: Sa = B1(t) - B3(t) Sb ...
1
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1answer
27 views

On the independence and stationarity of supremum of increments of a Brownian Motion

In between an exercise I did, I have the following affirmation that is meant to be used without proof: given a standard Brownian Motion $B$, the rvs defined as $Z_n=\sup_{0\leq u\leq 1}|B_{u+n}-B_n|$ ...
3
votes
0answers
32 views

Clarification on the Augmented Filtration

Consider the following definition. Definition. Let $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ be a probability space and $W$ a Brownian motion. Let $\mathcal{F}^W_t=\sigma\left(\left\{W_s\mid ...
1
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0answers
15 views

Brownian Motion and Poisson's problem

Let $U\subset \mathbb{R}^d$ be a bounded domain and $g: U\to \mathbb{R}$ be continuous. A continuous function $u:\overline{U}\to \mathbb{R}$, $u\in \mathcal{C}^2(U)$ is said to be a solution of ...
1
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0answers
19 views

What is the formula for a likelihood ratio $L$ that transforms martingale Geometric BM to Geometric BM with positive growth?

What is the formula for a likelihood ratio $L$ that transforms martingale Geometric BM given by $dS = \sigma S \,dW_t$ to Geometric BM with positive growth $dS = \mu S \, dt + \sigma S \, dW_t$? ...
1
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0answers
29 views

Quadratic variation along a sequence of subpartitions

Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ with $|\pi_n|=\max\limits_{t^n_i,t^n_{i+1}\in \pi^n}|t^n_{i+1}-t^n_i|\to_{n\to +\infty} 0$ the quadratic variation of a path ...
1
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0answers
33 views

exercise 3.3.34 from Karatza and Shreve [duplicate]

In the exercise, W is a standard, one-dimensional Brownian motion and $0 \lt T \lt \infty$. We are asked to show that $$\lim_{\beta\rightarrow\infty}\sup_{0\le t\le T}|e^{-\beta t}\int_0^t e^{\beta ...
1
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0answers
29 views

Show a geometric brownian motion is a martingale

Let $\{S(t), t\geq0\}$ a geometric brownian motion with drift $\mu$ and volatility $\sigma$. Find if the process is also a martingale or not. I know that I have to prove that $$E[S(t)-S(t-1)\lvert ...
1
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0answers
26 views

When does convergence in quadratic variation imply a uniform convergence or vice versa?

Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ the quadratic variation of a path $x\colon [0,T]\to \mathbb{R}$ is defined by $$ [x]=\lim_{n\to ...
6
votes
1answer
110 views

Given Q and $X_t$ is Q-Brownian, find $\frac{dQ}{dP}$ / Uniqueness of Brownian motion or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t ...
0
votes
0answers
21 views

Ito rule for a given ratio and exponential

Helo, I have trouble performing the following differentiation following Ito calculus $$d(e^Z/B)$$ Given that $Z_t$ is a logarithm of a certain process and follows $$dZ=mu_zdt+sigma_zdW$$ $$dB=rBdt$$ ...
1
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1answer
38 views

Expectation of a function of Brownian motions

I would like to how I can compute this expectation and get the answer that is given. All terms W indicate a Wiener process. $$E_t[W_s^3]=E_t[(W_t+(W_s-W_t))^3]=W_t^3+3W_t(s-t)$$
2
votes
1answer
36 views

Expectation of product of stochastic integral and predictable process

I would like to simplify $E[X_t Y_t]$ where $X_t=\int_0^t x_sdWs$ and $Y_t=\int_0^t y_sds$ where $x_s$ and $y_s$ are square integrable (predictable??) processes adapted to the filtration generated by ...
1
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0answers
21 views

Recurrence of $\int^t_0W_s ds$, where $W$ is a Brownian Motion

Is there any easy way of showing this integral is recurrent? i.e. it visits every point infinitely many times?
2
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1answer
38 views

modulus of continuity of Brownian Motion

Is there any estimate for the following quantity $$ E\left(\sup_{\substack{0 \leq s,t \leq1 \\ |t-s| < \delta}} \left|W_t - W_s\right|\right) $$ for some small $\delta > 0$, where $W$ is a ...
1
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1answer
34 views

Simulation of Brownian motion and white noise.

Let $\{W(t)\}$, $t≥0$ be a Wiener process with $ σ^2 = \operatorname{Var}\{W(1)\} = 1$. For a real constant $ε > 0$, consider the differential ratio process $∆ε = \{∆ε(t)\}$, $t>0$ given by ...
4
votes
1answer
34 views

Suppose $X_t$ is a brownian motion with $X_0 \sim u_0$. What is the probability density of $X_t$? (heat equation)

Suppose $u_0(x) = 2x$ for $0 \leq x \leq 1$ and $u_0(x)=0$ otherwise. Suppose $X_t$ is a brownian motion with $X_0 \sim u_0$. What is the probability density of $X_t$? Since $X_t$ is a brownian ...
0
votes
1answer
26 views

is this processes a martingale

Let $X_t$ be a brownian motion define: $Y(t) = t^2X_t - 2 \int_0^t sX_s \ ds$ Is $Y$ a martingale? I am trying to use Ito's lemma, and show that the drift is 0, however I am having troubles ...
3
votes
2answers
57 views

Ito integral for Brownian motion

I know that because $W_t$ is a martingale, $$E\left[\int_{0}^{T} W_t dW_t\right] = 0$$ then what should the value for this equation be: $$E\left[\int_{0}^{T} W_t^{n}dW_t\right]?$$ $n$ is the power of ...
1
vote
1answer
35 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 ...
0
votes
0answers
45 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 ...
1
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0answers
32 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 ...
1
vote
1answer
22 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 ...
2
votes
1answer
16 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, ...
1
vote
0answers
31 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 ...
0
votes
1answer
26 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 ...
1
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0answers
30 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 ...
1
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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: ...
2
votes
1answer
18 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 ...
1
vote
1answer
19 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 ...
1
vote
1answer
56 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
17 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
46 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 ...
2
votes
1answer
36 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 ...
1
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0answers
34 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 ...
1
vote
1answer
43 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
23 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
61 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 ...
8
votes
3answers
158 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 ...
1
vote
0answers
17 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 ...
1
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0answers
25 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 ...
0
votes
0answers
12 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 ...
3
votes
1answer
52 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 ...