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Questions tagged [brownian-motion]

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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17 views

Proof: the $\max{(B_t,0)}$ is submartingale without using convex function with Jensen's inequality

Proof: the $\max(B_t,0)$ is submartingale without using convex function with Jensen's inequality To prove it using convex function and jensen's inequality, we know the max funxtion is convex and it ...
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1answer
683 views

What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
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12 views

Is a vector of independent Brownian motions a multivariate Brownian motion?

Given a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P)$: If $B_1, B_2, \dots, B_m $ are all real $\mathcal{F}_t$ Brownian motions, jointly independent. Is the resulting ...
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2answers
22 views

Mean stopping time of a Brownian motion

I came across the following proof of the fact that the mean stopping time of a Brownian motion to hit $-1$ or $1$ is $1$: Let $B$ be a Brownian motion. We already know $B_t^2-t$ is a martingale. Let $...
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20 views

Find a distribution. Wiener proccess [on hold]

Find the distribution of $$\frac{1}{t-s} \left(W_t^2 + W_s \left[ \frac{t}{s} W_s - 2W_t \right] \right), \qquad 0 < s <t. $$ How do i do this? Where $$ W_t, W_s $$ - Wiener process
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12 views

Conceptual question on conditional expectations for numerically solving Backward SDEs

I am starting to study the field of Backward Stochastic Differential Equations (BSDE) and have a conceptual question on numerical techniques to solve them. BSDE are of the form: $$Y_t=F((B_s)_{0\leq s\...
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1answer
11 views

Does time changed brownian motion have no-memory property?

Let $W=(W_t)_{t \geq 0}$ be a Browniwn motion. Do the processes $$X_t = W_{e^t} \quad \text{and} \quad Y_t = \exp \left(- \frac{t^2}{2} \right) W_{e^t}$$ have the no-memory property, i.e. are the sets ...
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35 views

Is a Brownian motion bounded in probability?

I'm trying to figure out if a Brownian motion $\{B(t): t \geq 0\}$ is bounded in probability, i.e. if $B(t) = \mathcal{O}_P(1)$ holds for all $t \geq 0$. By definition I need to verify if for all $\...
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1answer
21 views

$P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$

Let $W(t)$ be a Brownian motion and $T_x=\inf\{t:W(t)=x\}$. I need to calculate $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$. I'm not sure if I understand these ...
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1answer
38 views

Calculating $E(2W(s)+W(u)|W(u)=2)$

Let $W(t)$ be standard Brownian motion and let $u<s$. I know that $W(s)\sim\mathcal{N}(0,\sqrt{s}), W(u)\sim\mathcal{N}(0,\sqrt{u})$ and $2W(s)+W(u)\sim\mathcal{N}(0,\sqrt{8s+u})$. How should I ...
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1answer
9 views

Compute the conditional expectation when brownian motion is involved

Consider the following conditional expectation $E[(x_t + \mu(s-t) + \sigma(\beta_s - \beta_t) )^2 | F_t] $ where $\beta_s$ and $\beta_t$ are brownian motion. Then, = $E[x^2_t + 2x_t\mu(s-t) + ...
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2answers
25 views

Doubt about distribution of the brownian motion

Let $B_{t}$ a brownian motion (stochastic process) then I know $B_{t} -B_{s}$ has a normal distribution with mean$=0$ and variance $=t-s$ I want to calculate the following probability: $P(3B_{2}>4)...
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1answer
24 views

Why does calculating the quadratic variation of a Brownian motion in this way not work?

This seems like a simple question, but I am stumped. I know the proofs for quadratic variation and cross variation, etc..., but for some reason can't understand why the following doesn't make sense to ...
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38 views

Fokker-Planck equation

I'm struggling to proof the Fokker-Planck equation. Let $b:[0,T]\times \mathbb{R}^N\to\mathbb{R}^N$ and $\sigma:[0,T]\times \mathbb{R}^N\to\mathbb{R}^{N\times d}$ two measurable functions. Let $X=\...
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How to prove arcsine law for amount of time Brownian motion is positive in $[0,t]$?

Ross's Introduction to Stochastic Processes states, but does not prove this result: For Brownian motion, let $A(t)$ denote the amount of time in $[0,t]$ the process is positive. Then, for $0<x&...
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20 views

How do I make sense of this expectation?

I am having some difficult in making sense of ${\rm{E}}\left[ {\int\limits_0^t {{W_u}du} } \right]$ where W is just your standard one dimensional brownian motion. Is interchanging the order of the ...
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1answer
103 views

“Conditional distribution” of Brownian sample paths

I would like to consider the "conditional distribution" of the Brownian sample paths conditional on certain sample path functionals, in a similar way that one considers the Brownian bridge. For ...
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1answer
39 views

Brownian motion, compact interval

Let $(W_t)_{t\geq0}$ denote a standard Brownian motion and $I=\left[a,b\right]$ a compact interval. Show that $P\left[\frac {W_{t+h}-W_t} {h} \in I\right] \rightarrow 0$ as $h\rightarrow 0$. What does ...
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2answers
72 views

Distribution of last exit time of Brownian motion with drift

If $X_t = \mu t + \sigma W_t$ with $W_t$ a Wiener process, I would like to know if the distribution for the last time $X_t = a$ is known - and if so, what it is. My googling has turned up a bunch of ...
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1answer
40 views

Gaussianity of a stochastic process

I am given the process $X_t = B_t -\int_0^t \frac{B_u}{u}du$ How can I show that it is gaussian, given a standard continuos Brownian motion $B$? As I know that $sB_{1/s} \rightarrow 0$ as $ s \...
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1answer
110 views

A random series has infinitely many zeros in $[0,1)$ almost surely.

These days I've been learning the properties of Brownian sample paths(Chapter 2 in Le Gall's Brownian Motion, Martingales, and Stochastic Calculus). As he mentioned in Proposition 2.14: If $B=(B_t)...
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2answers
549 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that $\operatorname{Cov}...
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1answer
162 views

Computing $E\left(\exp\left(B_t+\int_0^tB_sdB_s - \frac12\int_0^tB^2_sds\right)\right)$ if $B$ is a standard Brownian Motion

Let $B_t$ be a standard Brownian motion under probability $P$. I'm considering computing this: $$E^P\left[ e ^ { B_t + \int_{0}^{t}B_s\mathrm dB_s - \frac{1}{2}\int_{0}^{t}B^2_s\mathrm ds } \right], ...
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14 views

Variation of brownian motion [closed]

Q1. Bt is a standard Brownian motion, the second-order variation of Bt^2 between [0,T] a)0 b)T c)finite d)infinite The answer is c, but I do not know the reason. Q2 Bt is a standard Brownian ...
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15 views

Extending a Gaussian process to non-standard Brownian motion

Consider $(B_t:t\in \mathbb{Q}\cap [0,T])$. Show that it is possible to extend this to a process $(\tilde{B_t}:t\in [0,T])$ which satisfies $(\tilde{B_t})$ is a.s. continuous for every fixed $t\in [0,...
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1answer
22 views

Self similarity of fractional Brownian motion

The fractional Brownian motion with Hurst Paramter $H\in(0,1)$ is a Gaussian Process $\{X(t),t\ge0\}$ with mean $0$ and covariance function $$\gamma(t,s)=1/2(|t|^{2h}+|s|^{2H}-|t-s|^{2H}).$$ I want ...
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11 views

Interpolating process that follows geometric Brownian motion

So I have a set of time series data that follows (at least assumed) GBM but there are missing data. To interpolate, I'm thinking about simulation and create some sample paths. I have two options in ...
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37 views

Density function related to Brownian motion

I am dealing with a question listed below. I am trying to use the running maximum of Brownian motion to deal with the problem, but it does not work out. Let $ \tau_{M}=\inf\{t;W(t)=M\},M>0,$ and ...
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15 views

GBM within brownian bridge

I noticed that a brownian bridge between say a and b amy start at B0 and B1 where B1 can be greater than B0. I wonder if this path can be a GBM. If so, what’s the SDE like?
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scaling invariance of brownian local time

I am studying Brownian local time processes and several references mentioned the scaling invariance of local time. For example, page 10 of this reference (https://hal.archives-ouvertes.fr/hal-00091335/...
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Reference/suggestion needed - Hitting time of domain equal everywhere -> {domain = sphere, x0 = centre}

So, given a Brownian motion under drift $dX(t) = c dt + dB(t)$ where B(t) is a std Brownian process. Let the process start within some (convex, etc) domain $A$, i.e. $x_0\in A$. Does the following ...
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1answer
28 views

If Bt is a Brownian movement. Show that for any t and s, P (Bt> Bs) = 1/2. [closed]

My teacher said it was because of the symmetry but I don't really understand ): somebody can help me?
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1answer
48 views

Convex functions of a martingale.

Let $X_1(t)=e^{B(t)}$ and $X_2(t)=e^{-B(t)}$ where $B(t)$ is the standard brownian motion and $\{G_t\}$ is the filtration generated by the brownian motion. Determine what kind of martingale $X_1(t)$ ...
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1answer
33 views

Significance of the term $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$.

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there any significance of the term $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$. ...
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1answer
44 views

Finite sequence of random step processes such that $\lim_{n\to\infty}E(\int_{0}^{\infty}|f(t)-f_n(t)|^2dt)=0$ for $f(t)=e^{-t^2/4}$

Let $$f(t)=e^{-t^2/4}, \ \ \ t \ge 0$$ I want to show that $f$ is in $M^2$ where $M^2$ denotes the class of stochastic processes $f(t),t\ge0$ such that $$E\left(\int_0^\infty|f(t)|^2dt\right)&...
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1answer
24 views

Independence of Solution of SDE $S^{(x_0, \sigma, \mu)}_t$ of Initial Information $\mathcal{G}_0$

Question Consider the following stochastic differential equation, given as an equivalent stochastic integral equation, where the multidimensional integrals are to be read componentwise: \begin{...
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0answers
50 views

Using Ito's lemma to determine $dY(t)$ when $Y=\sin(t+B_t), \ \ \ t\ge0$

Let $(\Omega, \mathcal F, P)$ be a probability space and $\{B_t\}_{t\ge0}$ a Brownian motion. Furthermore let $\{F_t\}_{t\ge0}$ be the natural filtration of $B$. Let $$Y(t)=\sin(t+B_t), \ \ \ t\ge0$$ ...
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3answers
3k views

Prove $A_t := W_t^3-3t W_t$ a martingale

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(A_t)_{t \geq 0}$ where $A_t = W_t^3 - 3tW_t$. ...
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1answer
26 views

Poisson process on Skorokhod's space

For each $n=1,2,\ldots $, let $\ \xi_{n1},\ldots, \xi_{nn}$ be random and independent variables such that $\mathbb{P}(\xi_{ni}=1)=p_n \ \ $ and $\ \ \mathbb{P}(\xi_{ni}=0)=1-p_n$. Let consider the ...
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1answer
30 views

Time homogeneity of Ito diffusion

Consider a time homogeneous Ito diffusion satisfying a SDE, \begin{equation}\label{1} dX_t=b(X_t)dt+\sigma(X_t)dB_t, X_s=x \end{equation} $t\geq s$. The unique solution of the SDE is denoted by $...
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2answers
49 views

Find the quadratic variation process of $\int f(s) \, dB_s$

Let $f \in L^2[a,b]$ and let $\displaystyle M(t)=\int_a^tf(s)dB(s)$. Find the quadratic variation process, $[M]_t$ , of $M(t)$. Here the quadratic variation process is the limit in probability of $\...
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23 views

Basic Questions about Brownian Motion

I was hoping someone could answer a few basic Brownian Motion questions. A Brownian Motion, $B_t$, (let us assume defined on the continuous path space with the Brownian Measure starting at $0$) has ...
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54 views

How can I numerically compute a stochastic integral?

I am trying simulate a to solve a 2-dimensional stochastic process and $Y_t^1$ is a mean-reverting square root process which I simulated on a time grid using its known conditional distribution. I ...
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1answer
25 views

Derivation of Joint and Conditional density of a Brownian Motion and its Maximum

Given $W(t)$ and $M(t)=\max\limits_{0\leq s\leq t}W(s)$ where $\{W(t),\ t\geq 0\}$ is the standard brownian motion, compute their joint distribution and the conditional distribution of $M(t)$ given $W(...
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44 views

Every local martingale with respect to the Brownian filtration has its continuous version.

I would appreciate some help on the following. In class, we said that Every local martingale with respect to the Brownian filtration has its continuous version. To prove this, it is apprently enough ...
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1answer
84 views

Girsanov THM and Radon-Nikodym derivative

I've been having a hard time to applicate Girsanov theorem with Radon-Nikodym derivative in the demonstration of German-El Karoui-Rochet formule. I know that $\Pi_0:=S_0\mathbb{Q}^S(S_T\geq K)-K\...
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1answer
48 views

Showing if sum converges in $L^2$ (Brownian motion)

Consider a probability space $(\Omega, \mathcal F, P)$ and a Brownian motion $(W(t),t\ge 0)$. Let $T>0$, $t_j^n=jT/n$ and $$\xi_j^n=\frac{1}{3}t_{j+1}^n+\frac{2}{3}t_j^n, j=0,\ldots,n-1$$ ...
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0answers
17 views

Proof of strong Markov continuity of Brownian motion

Let $(B_t)$ a Brownian motion and $\sigma $ a stopping time finite a.s.. I want to prove that $W_t=B_{\sigma +t}-B_\sigma $ is a Brownian motion. The way to prove it is first to prove that for all $0\...
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0answers
12 views

Existence of $L^2$ limit of sequence of martingales

I am currently revising some martingale theory, and was trying out an old past paper question. I haven't come across anything like this before, so was wondering how one would approach it, and also in ...
2
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1answer
71 views

Is the fact that $dW_t\sim (dt)^{1/2}$ come from the $1/2-$holder property of Brownian motion?

(I offer 100 bounty because I really would like to have a constructive answer to this question) I often see that if $W_t$ is a Brownian motion, then $dW_t\sim (dt)^{1/2}$. Can it come from the fact ...