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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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Let $\{B_t\}_{t \ge 0}$ a Brownian Motion, prove that for all $a \gt 0$: $P(B_t \ge a$ for some $t \ge 0 )=1$

Let $\{B_t\}_{t \ge 0}$ a standard Brownian Motion, prove that for all $a \gt 0$: $P(B_t \ge a$ for some $t \ge 0 )=1$ Use the principle of reflection Thoughts: The reflection principle is: ...
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17 views

Density and Probabilities of Geometric Brownian Motion

Let $\{ X_t \}_{t \ge 0}$ defined by $ X_t := 10+3t+3B_t$ where $\{ B_t: t \ge 0 \}$ is a standard Brownian Motion. Let $\{ S_t \}_{t \ge 0}$ defined by $S_t := e^{X_t}$ a) Write explicitly the ...
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18 views

SDE with absolute value

Find the unambiguous strong solution of the equation: $dX_{t} = dt+ |X_{t} - t|^{\frac{1}{2}}dW{t}, X_{0} = 0$. It is easy to tell that the solution exists but I don't know how to find it. Any help ...
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1answer
68 views

How to I use Ito's formula to compute quadratic variation?

Let B be Brownian motion. Use Itos formula to compute the quadratic variation of $\left[X_t^i\right]$ for $\left[X_t^1\right]=e^{B_t}$, $\left[X_t^2\right]=\ln(B_t^2+1)$ and $\left[X_t^1\right]=\sin^...
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14 views

Prove that $X_t = 1/c B_{c^2 t}$ with $c \gt 0$ is a Brownian Motion

Let $\{ B_t \}$ a standard Brownian Motion, prove that the next process are Brownian Motions: a) $\{ X_t \}$ where $X_t = -B_t$ b) $\{ X_t \}$ where $X_t = 1/c B_{c^2 t}$ with $c \gt 0$ c) $\{ ...
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1answer
17 views

Difference of normal random variables

I have two random variables $$X_{s+t} \sim N(0, s+t)$$ $$X_s \sim N(0, s)$$ where $s \leq t$. How do I show that... $$X_{s+t} - X_s \sim N\left(0,s + t + s -2\sqrt{s(s+t)} \right)??$$ I understand ...
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1answer
29 views

Quadratic Variation and Brownian Motion

Let $(X_n,F_n)$ be a martingale with $X_n \in L^2(\Omega,F,\mathbb{P})$. The quadratic Variation $(<X>_n)_n$ of the process $(X_n)_n$ is defined as $$ <X>_n := \sum\nolimits_{i=1}^{n}(\...
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0answers
53 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 function is convex and it ...
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0answers
10 views

Expectation of the product of Brownian processes (higher powers)

I have recently sat an exam that had elements of stochastic calculus, but I am now feeling like I might have gone wrong in some questions of it like the following. I am trying to evaluate $\mathbb{E}(...
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36 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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16 views

Wiener process density

Let $(W_{t},V_{t})$ be two independent Wiener processes. We define $\tau_{a} = \inf\{t>0 : |W_{t}|=a\}$. Find density of random variable $V_{\tau_{a}}$ for any $a>0$. How do I do that?
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1answer
689 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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2answers
23 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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Find a distribution. Wiener proccess [closed]

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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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
12 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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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
42 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
27 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
28 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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0answers
39 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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0answers
14 views

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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0answers
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
45 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
74 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
44 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
121 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
550 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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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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0answers
14 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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0answers
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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0answers
17 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
53 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
27 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
27 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
52 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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0answers
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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0answers
57 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 ...