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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45 views

Distance between Brownian Motion and scaled Gaussian random walk

I'm currently reading this paper: http://user.math.uzh.ch/barbour/pub/Barbour/SteinDiffusion.pdf and in equation (2.26) the author uses the following fact: If $Z(t)$ is a standard Brownian Motion and ...
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1answer
24 views

how to show that definition for stochastic process in continuous time applies to stock prices

I know that the formal definition of a stochastic process is: {$X(t,\omega)\,\,t\ge0$} is a stochastic process if: For any fixed $t\ge0$, $X(t,\omega)$ is a random variable For any fixed $\omega$ ...
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38 views

Why is $E(X_t|B_t)=\frac{E(X_tB_t)}{E(B_t^2)}B_t$?

Why is $E(X_t|B_t)=\frac{E(X_tB_t)}{E(B_t^2)}B_t$ ? Does this always hold In an exercise I have to show that $E(X_t|B_t)\neq X_t$, where $X_t=\int_0^t B_s ds$, I think the definition of $X_t$ ...
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1answer
35 views

Compute $ \mathbb{E} [W(t_1)W(t_1 + t_2)W(t_1 + t_2 + t_3)] $ when $W$ is a Brownian motion

Let $(W(t))_{t \geq 0}$ be standard Brownian motion, and let $t_1, t_2, t_3 \in \mathbb{R}_{> 0}$ with $t_1 < t_2 < t_3$ be arbitrary. Compute: $$ \mathbb{E} [W(t_1) * W(t_1 + t_2) * ...
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1answer
49 views

Computing expectation of brownian motion

I need to compute the following: $E\left[ B_t \int_0^tB_s^2 \, ds \right]$ for $t≥0$ Where $B_t$ is a standard Brownian motion. I'm thinking this is really obvious, But I cannot get my head round ...
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24 views

Wiener's construction of the Wiener Measure

I am writing an essay about Norbert Wiener and I already have sufficient info about him in general and his history, but now I would like to know how he constructed the Wiener measure. I found some ...
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42 views

Brownian motion, harmonic functions and the Dirichlet problem

I am having trouble understanding one detail of the standard use of Brownian motion to solve the Dirichlet problem, I will write the statement and proof and then point to the detail I don't ...
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1answer
29 views

An issue of dependent and independent random variables involving geometric Brownian motion.

Let $X(t)=X(0)e^{\mu t + \sigma Z(t)}$ be a geometric Brownian motion (GBM) where $Z(t)$ is the standard Brownian motion with drift $0$ and the variance rate per unit of time is $1$. Now, let $s<t$ ...
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1answer
51 views

Iterated logarithm law for difference (supremum(W) - infimum(W) ) is it 2srt(2/pi) sqrt(t loglog(t))?

Law of iterated logarithm says that $$\sup(W(t)) \sim \sqrt{2 t \log(\log(t))}.$$ Consider $\sup(W(t)) - \inf(W(t))$ my guess based on numerics that it should be $$2\sqrt{\dfrac 2\pi} \sqrt{t ...
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49 views

Simulation of brownian motion and fractional brownian motion

It's easy to simulate a path of a brownian motion with the method explained in Wiener process as a limit of random walk: ...
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1answer
78 views

Exercise 8.12 Introduction to stochastic processes Gregory Lawler [closed]

Let $X_t$ be a standard Brownian motion starting at 0 and let $T=min \{t:|X_t|=1\}$ and $\hat{T}=min \{t:X_t=1\}$ (a) Show that there exist positive constants $c$, $\beta$ such that for all ...
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0answers
30 views

Construction of Wiener Process using integral of covariance multiplied by a function

I read in the notes of Stochastic Processes that there is a construction of Wiener Process (knowing that $Cov(W_s, W_t)=min(s,t)$ ) which going like this: consider operator $Q$ on $C([0,1])$ $$Qf(t)= ...
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27 views

How to evaluate the expectation of the exponential of reflected brownian motion

How do you compute this expectation $\mathbb{E} \left [ e^{\varepsilon|W_t|} \right] $ where $W_t$ is a Brownian Motion Do I need to expand the absolute value? Can I use the standard Taylor series ...
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61 views

Showing that $P(W_{t}/\sqrt{t \log(t)}>1+\epsilon)\to0$ when $t\to\infty$, where $(W_t)$ is a Wiener process

I have a question about the martingales $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$. With use of this martingale I want to show that $P(\dfrac{W_{t}}{\sqrt{t log(t)}}>1+\epsilon)$ goes to $0$ if $t$ ...
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1answer
23 views

Mean time for the trajectory. Find mean

What is the mean of time when the trajectory of the wiener process, $W_t$, is over the line $y=t$? We need to find $\Bbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
3
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1answer
86 views

Quadratic variation of semi-martingale

$X_t = e^{B_t-\frac{1}{2}t^2}$ I need to find $[X]_t$, the quadratic variation process. I have tried to solve the problem and my main question is whether this approach is correct or not. ...
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0answers
27 views

Brownian motion and sup of a Brownian motion

I am stuck with the following problem: let $B_t$ be a standard Brownian motion and let $S_{t}:=\sup_{0 \leq s \leq t} B_s$. Prove that for every $\lambda \geq 0$ and $\mu \leq \lambda$, ...
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37 views

Proof finite stopping time and Wiener process bounded

Let $T_{-a,b}=\inf\{t\geq 0: W_{t} \notin [-a,b]\}, a,b>0$. I want to show that this is a finite stopping time ($P(T_{-a,b}<\infty)=1$) and that $|W_{\min(T_{-a,b},t)}|$ is bounded by a ...
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36 views

modulus of continuity of Ito process

We know from Levy's (uniform) modulus of continuity that for Brownian Motion, almost surely any sample path is locally Holder continuous for any $\rho <\frac{1}{2}$, i.e. $$ |W_t - W_s | \leq ...
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0answers
31 views

Why is Wiener measure on $C[0,1]$ strictly positive?

The question is as stated. I have thought about this for a while and can't really get anywhere. Here strictly positive means non-zero on non-empty open sets (in this case with a finite interval we are ...
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20 views

Why does $1 \leq \sup \limits_{0\leq t \leq 1}( C|B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold?

Why does $1 \leq C\sup \limits_{0\leq t \leq 1}( |B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold ? I am trying to show by contradiction that the Burkholder-Gundy ...
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1answer
33 views

Proving a simple equality involving integrals and a brownian motion

I'm trying to prove the following equality $$ \int_0^T W(t) dt = \int_0^T (T-t) dW(t) $$ where $W(t)$ is a standard brownian motion. I'm been trying to make use of the fact, that $dt = dW(t) dW(t)$ ...
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1answer
34 views

Why does $(W_t)^2$ have mean $t$?

This is in the context of Ito Calculus. Here, $W_t$ is a $P$-Brownian Motion. My book says that "... $(W_t)^2$ has mean $t$, because of the variance structure of Brownian motion.." I understood that ...
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1answer
43 views

Solving equation with Wiener process

I want to show that if $E(f(X_{t}))=E(f(W_{t})e^{\lambda W_{t}-0.5*\lambda^2*t})$, where $W_{t}$ is a Wiener Process, then $X_{t}\sim N(\lambda t,t)$. Does anyone have a clue how to solve this ...
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1answer
79 views

Showing martingale for a Brownian motion $(W_t)_{t \geq 0}$

I want to show that $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$ is a martingale with respect to $F_{t}$. We can use that $$E(e^{\alpha ...
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5answers
113 views

Solve the integral $\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx$

To find the Variance of a Wiener Process, $Var[W(t)]$, I have to compute the integral $$ Var[W(t)]=\dots=\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx=\dots=t. $$ I've ...
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0answers
45 views

Show that the solution to a stochastic differential equation is satisfied by the following

I am confused on how to get from the first statement to the second. Getting from the second statement to the third would just a simple case of substituting s=0. The solution sheet says to use ...
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1answer
98 views

What is a “continuous modification”? And can we always modify an almost surely continuous process, such that every path is continuous?

Let's motivate the question by a classical result: Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ which ...
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1answer
76 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 ...
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2answers
59 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 $$ ...
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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: $$ ...
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50 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 ...
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1answer
51 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 ...
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1answer
57 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 ...
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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 ...
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0answers
16 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 ...
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1answer
32 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 $Z_n=\sup_{0\leq u\leq 1}|B_{u+n}-B_n|$ ...
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37 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 ...
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23 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 ...
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29 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 $dS = \mu S \, dt + \sigma S \, dW_t$? ...
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0answers
32 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 ...
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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 ...
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0answers
72 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 ...
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0answers
32 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 ...
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1answer
130 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 ...
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0answers
29 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 $$dZ=mu_zdt+sigma_zdW$$ $$dB=rBdt$$ ...
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1answer
39 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
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1answer
53 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 ...
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0answers
23 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
44 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 ...