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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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15 views

Brownian motion and Running Maximum

Take B$_t$ as a standard Brownian motion such that B$_0$ = 0. And M$_t$ is the corresponding running maximum. i.e. M$_t$ = max$_{0\leq s \leq t}$ B$_s$. My goal is to compute: (i) Quadratic ...
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Expectation of stochastic process continuous in starting point.

Let $(B_t)$ be a Brownian motion started in $x$ and $f_1,\dots,f_n$ be $\mathbb{R}$-valued continuous bounded functions. Consider the map $x \mapsto \mathbb{E}^x \big( f_1(B_{t_1}) \cdots f_n(B_{t_n}) ...
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If $T_{U,V}=\inf\{t:B_t \notin (U,V)\}$ does $\int E[B_{T_{U,V}}|U,V]dP=\int E[B_{T_{u,v}}]dP_{U,V}$

I have the following question: if $T_{U,V}=\inf\{t:B_t \notin (U,V)\}$ does $$\int E[B_{T_{U,V}}|U,V]dP=\int E[B_{T_{u,v}}]dP_{U,V}$$ - or if it makes the difference does $\int E[1_{\{-\infty ,x]} ...
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Deriving Ito process with a drift from geometric Brownian motion.

Please help me solve this question. Thank you. Let the Geometric Brownian motion be: $$ \frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t} $$ $\Delta S$ = change in stock price (s) ...
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22 views

Expectation of the exponent of a constant times exiting time of a Brownian motion (i.e. $\mathbb{E}_x[e^{n\sigma}]$)

Suppose $a, n>0$ and $B_t$ is a Brownian Motion, define $$\sigma=\inf\{t:B_t\in\{-a,a\}\}.$$ I want to find $\mathbb{E}_x[e^{n\sigma}]$. (Notice since $n>0$ it is $\textbf{not}$ Laplace ...
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Is there anything known about $\sup_{s\leq t} \vert B_s \vert - f(s)$, where $B$ is a Brownian motion and $f$ measurable.

If we consider the process $$ Y^f_t := \sup_{s\leq t} \vert B_s \vert - f(s)$$ is there anything known about the distribution or at least the probability $\Bbb P (Y^f_t \leq 0 )$ ? Of particular ...
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Book-recommendation: Numerical method for stochastic differential equations

Speaking of numerical stochastic differential equations, the book of Peter Kloeden 1992 Numerical Solution of Stochastic Differential Equations is a quite famous and standard reference. But when I ...
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1answer
29 views

What is the significance of $[t/ \Delta t]$ in Ross' definition of Brownian motion?

Picture is from Ross' Introduction to Probability Models, 11th ed. I understand the definition of $[t/\Delta t]$, I just don't see how it connects to the position at time $t$ (eq. 10.1). For $\Delta ...
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1answer
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reflection principle brownian motion

In this proof they use the strong markov property but i don't understand why we need it. Could anyone explain it to me? Thank you
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Characterization of Brownian Motion (Problem Karatzas/Shreve)

In the book "Brownian Motion and Stochastic Calculus" by Karatzas/Shreve, they state the following problem (chapter 5, problem 4.4): A continuous, adapted process $W= \{W_t,\mathcal{F}_t;0\leq t < ...
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Calculate expected value of $H_t=exp(W_t/(1+t))$

If $W_t$ is a Wiener Process/Brownian motion standard, what is the expectation of: $H_t=exp(W_t/(1+t))$ I know the answer is: $E(H_t)=exp(t/(2(1+t)^2))$ But the solution I find is $1/2*exp(t/(2(1+...
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Brownian motion expected value of $W(1)$.

I know that if $W(t)$ is a Brownian motion on the interval $[0,t]$, then $$\mathbb{E}[W(t)]=0$$ but why $$\mathbb{E}[W(1)]=0?$$ Lets say that $W(1)=a$, then why we do not have $$\mathbb{E}[W(1)]=\...
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The value of dBtdWt [on hold]

Let $W =(W_1, W_2)$ be a $2-$dimensional Brownian motion. Given $ρ \in (-1,1)$ we define the $R-$valued process $B_t$ by $Bt= ρ Wt1 + \sqrt(1-ρ2)Wt2$. What is the value of $\frac{dBt}{dWti}$ for $i =...
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$E[(\int_{0}^{\infty}f(t)dW_t)^2]$ for $f(t)=(W_2-W_1)1_{[2,3)}(t)+(W_3-W_1)1_{[3,5)}(t)$

Let $f(t)=(W_2-W_1)1_{[2,3)}(t)+(W_3-W_1)1_{[3,5)}(t), t \ge 0$. $(W_t)_{t\ge0}$ is Brownian motion. What's the 'best' method to calculate $\mathbb E[(\int_{0}^{\infty}f(t)dW_t)^2]$? I would ...
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1answer
18 views

Product of two independent brownian motion is a martingale

Given $X, Y$ independent Brownian motions, I'd like to show that $XY$ is a martingale. This seems to be a fairly easy result, but I can't work it out, nor find anywhere that gives a proper proof. I ...
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1answer
26 views

First variation on Brownian motion

The first variation of function $f(t)$ on interval $[0,T]$ is defined as $$ FV(f) = \lim_{\|\pi\| \rightarrow 0} \sum_{i=0}^{n-1} |f( t_{i+1} ) - f( t_i)|.$$ How can we estimate the first variation ...
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Probability of $B_t < 0$ if $B$ is Brownian Motion

Let's consider Brownian motion $B_t$. We assume that $B_0 = 0$. I am to show that $$P(\inf{ \{t>0: B_t <0 \}} = 0) = 1.$$ It seems pretty obvious. I don't know how to start the proof however. ...
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Understanding where the Mandelbrot Van Ness representation of fractional brownian motion comes from

I would like to know more about this way of representing the fBm process. Define: $$K_H(t,u) = (t-u)^\kappa_+ - (-u)^\kappa_+,$$ where $\kappa = H - 1/2$. The Mandelbrot Van Ness representation of ...
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Showing Ito's formula gives a semimartingale decomposition for brownian motion

I'm trying to show that for a standard Brownian motion and some twice continuously differentiable function $f$ that $f(B_t)$ is a local martingale iff $f'' = 0$. Applying Ito's formula gives, and ...
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31 views

W_t^3 martingale or not? two arguments puzzle me.

I want to study whether $W_t^3$ is a martingale or not? where $W_t$ is the standard Brownian motion. I have method 1 argument, but I also got second argument which implies different conclusion. ...
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1answer
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Question on Construction of Brownian motion with Kolmogorov-Centsov Theorem

I am studying the construction of Brownian motion with Kolmogorovs Extension Theorem. In the attached section from Karatzas and Shrive (Brownian Motion and Stochastic Calculus) it says that $C([0,\...
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1answer
21 views

$\mathbb E[W_s^2W_t^2]$ for Brownian motion

Consider a Brownian motion $(W_t)_{t\ge0}$ on $(\Omega ,\mathcal F, \mathbb P)$. How can I calculate $\mathbb E[W_s^2W_t^2]$? I know that $\mathbb E[W_sW_t^2]=0$ but I don't know if that helps here.
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1answer
40 views

Markov property for 2 dimensions and example

As I try to study Markov processes I struggle in understanding how to extend the Markov property for one dimension \begin{align*} \mathbb{P}\left[X_{t} \in A | \mathcal{F}_{s}\right]=\mathbb{P}\left[...
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21 views

Distribution of $(X,Y,Z)$, $X=W(3)-W(1), Y=W(5)-W(2), Z=W(7)-W(4)$. (Brownian motion)

Let $W(t),t \ge 0$ be a Brownian motion on $(\Omega,\mathcal F,P)$ and let $(\mathcal F(t),t\ge0)$ be the natural filtration of $W$. Let $X=W(3)-W(1), Y=W(5)-W(2), Z=W(7)-W(4)$. I should ...
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1answer
21 views

Show a stochastic process is a Martingale

B$_t$ is a Brownian motion starting from 0. For any fixed constant $\sigma$ $>$ 0, X$_t$ = e $^{\sigma B_t - \sigma^2t/2}$, t$>$0 is a martingale w.r.t. the filtration generated by Brownian ...
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1answer
53 views

Conditional expectation of Brownian motion's first hitting time

Let $T_x$ be the first hitting time of $x$. Let $B_t$ be a Brownian motion started at $x\in [0,R]$. Show that $$E[T_R \mid T_R < T_0]=\frac{R^2-x^2}{3}.$$ By using the fact that $B_t^2 - t$ is a ...
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1answer
17 views

brownian translation

I have a bit of struggle with understanding what it means to have the same law as a brownian. For instance, how can i prove that $$\sup_{t\in[k-1,k]} |W_t - W_{k-1}| \stackrel{\mathcal{L}}{=} \sup_{t\...
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1answer
48 views

Calculating the expecation of the supremum of absolute value of a Brownian motion

I got a Brownian motion $B(t)$ that starts in $0$ and want to calculate the expectated value of the supremum on the interval $[0,1]$ of the absolute value of it, i.e. $E \left (\sup \limits_{t \in [0,...
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1answer
58 views

Solution to Dirichlet is necessarily given by Brownian Motion (problem with stopping times)

Let $D \subset \mathbb{R}^n$ be a bounded open set and $f: \partial D \rightarrow \mathbb{R}$ a continuous function. The Dirichlet problem consists of finding a continuous function $U: \overline{D} \...
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1answer
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Hitting times by Brownian motion

[Edited] Suppose that $A$ is a (Borel) measurable set and $X$ is an Ito diffusion, i.e., $dX_{t}=\mu(X_t)dt+\sigma(X_t)dB_t$. Consider a hitting time $\tau_A$ of the given set $A$ by the process $X$:...
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Quadratic variation of $w^2(t)$

What is the quadratic variation of ($w(t))^2$, where $w(t)$ denotes a Wiener- process and what is the expected value of it? What is $E([(w(t))^2])$? My result is: $2(t^2)$, but knowing myself, I am ...
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1answer
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Revuz and Yor's Book “Continuous Martingales and Brownian Motion” - Chapter 1 - Exercise 1.19

Context : This post is the second of a series of posts taking their origins from the exercises in the Revuz and Yor's Book "Continuous Martingales ans Brownian Motion". The reason for doing so is ...
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1answer
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Proof of Blumenthal's 0-1 law for Brownian Motion

I am currently reading the book "Brownian Motion, Martingales, and Stochastic calculus" by Jean-François Le Gall and am stuck at understanding the proof of Blumenthal's 0-1 law for Brownian Motion. ...
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Is isotopy type of brownian loops deterministic?

Let $B_1, B_2$ be two independants Brownian loops on $R^3$. Does it almost surely exists a continuous isomorphism $\phi$ of $\mathbb{R}^3$ onto itself such that $\phi(Im(B_1))=Im(B_2)$? Can we chose $...
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24 views

HJB when optimal control responds partially to Brownian motion

I cannot formulate the HJB $V$ in a simple problem where the state $a$ follows a Brownian motion, and the control $\ell$ is (for some states) such that the derivative $V_\ell$ stays constant. In ...
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Brownian Motion and Complex Analysis

I was recently taught in a lecture on complex analysis based off of this paper that much of complex analysis can be rephrased in the language of Brownian Motion. The paper gives some simple proofs of ...
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Supremum of Brownian motion almost surely not at endpoint

Let $(B_t)$ be a standard $\mathbb R$-valued Brownian motion starting from 0. Let $$T = \inf\{t \in [0, 1] : B_t = \sup_{s \in [0, 1]} B_s\}. $$ I wish to show that $T < 1$ almost surely. ...
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Proving that $\int^\infty_0 e^{2B_s} ds=\infty$ [duplicate]

I am struggling to prove that $$\int^\infty_0 e^{2B_s} ds=\infty$$ where $B_t$ is a regular Brownian motion. We know that $\limsup_{t\rightarrow\infty}B_t=\infty$, so I figured this would be a quick ...
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Brownian motion time scaling

If $B_t$ is a standard n-dimensional standard Brownian motion, with is then $B_t\sim \sqrt{t}B_1$, i.e. why are they equivalent in distribution?
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Version of Tanaka's Formula outside Brownian Motion

Given a Brownian Motion $B = \lbrace B_t, t \geq 0\rbrace$, Tanaka's Formula provides a decomposition of the submartingale process $|B|$ given by $$|B_t| = \int_0^t sgn(B_s) dB_s +L_0^B(t)$$ where $...
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Is an integrated Wiener process recurrent or transient?

Like the title says, if I take an integrated Wiener process / Brownian motion $\int ^t _0 W_s ds$, will it be recurrent or transient? Or, under what conditions will it be one or the other? I know ...
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1answer
44 views

Ito integral related proofs

Let $W = (Wt)_{t\geq0}$ be a standard one dimensional Brownian motion. Prove that $$\int_{0}^{t} W_s^2 dW_s= \frac{1} {3} W_t^3-\int_{0}^{t}W_sds$$ $$\int_{0}^{t} sdW_s=tW_t-\int_{0}^{t}W_sds$$ I ...
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2answers
33 views

$\xi$ has modification $\eta$ with continuous paths $\iff \exists c \in \mathbb R$ s.t. $P(\xi_{0}=c)=1$

Let $\xi:=(\xi_{n})_{n\geq 0}$ be IID and $\eta:=(\eta_{n})_{n \geq 0}$: A path is defined as a map for fixed $\omega$ that $[0,\infty[\ni t\mapsto\xi_{t}(\omega)$ Show that: $\xi$ has modification ...
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33 views

Banach valued stochastic differential equation

Does anybody know a reference to a proof of the existence of a local solution to a Banach space-valued stochastic differential equation of the form $$d X_t = a(t,X_t) dt + b(t,X_t) dW_t,\;X(0)=X_0>...
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24 views

Lévy processes, Brownian motion and Lie groups

We know that Brownian motion describes any (non-deterministic) Lévy process with continuous sample paths. The above statement is true in Euclidean space. My question is: does it stand for other ...
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51 views

Gaussian process is a Brownian motion.

I am referencing from the book Probability from Dava Khoshnevisan. Thats my definiton of a brownian motion: 1) $W(0) =0$ and for all $t > 0$ is W(t) normal distributed with mean 0 and variance t. ...
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1answer
23 views

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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1answer
32 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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0answers
26 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
37 views

Proof that $\frac{t \xi_0}{\sqrt{\pi}} + \frac{1}{\sqrt{2\pi}} \sum \frac{\sin(nt)}{n}\xi_n$ is Wiener process.

Consider $\xi_i$ are i.r.v with standart normal distribution. Let $$X_t = \frac{t}{\sqrt{\pi}}\xi_0 + \frac{1}{\sqrt{2\pi}} \sum_{n=1}^{\infty}\frac{\sin(nt)}{n}\xi_n.$$ We want to prove that $...