All Questions
141 questions
1
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215
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Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)
Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$.
...
- 3,098
1
vote
1
answer
139
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Characterization of Brownian motion: processes with right-continuous paths
I am looking for a reference with a proof for the following fact:
If a right-continuous martingale $(X_r)_{ r \geq 0}$ is such that $X_0=0,(X^2_r-r)_r,(X_r^3-3rX_r)_r,(X_r^4-6rX_r^2+3r^2)_r$ are ...
- 573
1
vote
2
answers
88
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Lower-bound on zero-crossing probability of the nonstationary gaussian process $X(t) = tU+(1-t^2)^{1/2}V$, with $(U,V) \sim N(0,I_2)$
Let $(X(t))_{t \in [-1,1]}$ be a centered non-stationary smooth gaussian process with covariation function $\rho(t,s) = \mathbb E[X(t)X(s)]$. For $t_0 \in (-1,1)$ and $\epsilon \in (-1-t_0,1-t_0)$, ...
- 6,853
1
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1
answer
141
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Differentiable approximation of Brownian diffusion with bounded volatility
Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
- 335
1
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1
answer
739
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Joint law of a standard Brownian motion and its local time at a nonzero level
Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\...
- 1,649
1
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0
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133
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A question about one Malliavin derivative calculation
Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
- 620
1
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0
answers
99
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Expectation of $B_u \operatorname{argmax}_t B_t$
This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here.
Yesterday I asked a question about the joint law of ...
- 620
1
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0
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134
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Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1
Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
- 21
1
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0
answers
100
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Ito formula for fractional BM + drift and supremum bound
Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
- 111
1
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1
answer
207
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How to prove the coupling version of the Donsker's Invariance Principle?
Donsker's invariance principle:
Let $X_1,X_2,...$ be i.i.d. real-valued random variables with mean 0 and variance 1. We define $S_0=0$ and $S_n= X_1+ ... + X_n$ for $n \geq 1$. To get a process in ...
- 288
1
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0
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160
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Laplace Equation for Brownian Motion [closed]
So, I know that there is this theorem (taken from here):
For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have
$$
u(x) = \...
- 121
1
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0
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68
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Differentiable approximation of Brownian diffusion with unbounded volatility
Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
- 335
1
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0
answers
745
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Local martingale but not martingale
For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process
$Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...
1
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0
answers
243
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Intersection of a Poisson bridge and a Brownian bridge
Take a Poisson process $N_t$, a Brownian motion $W_t$ and constants $T > 0$ and $a > 0$. Suppose $N$ and $W$ are independent. I'm interested in the probability that $W$ does not cross over $a + ...
- 222
1
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0
answers
77
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"Return map" for Brownian motion
Consider a Brownian motion $W$ reflected at the boundary of a domain $D$ in Euclidean space. I want to look at the process obtained by "restricting" it to the boundary.
I was thinking of ...
- 2,772
1
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0
answers
239
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2-d geometric Brownian motion hitting time distribution
I am trying to solve following problem: Given two independent geometric Brownian motions
$\frac{d x_t}{x_t}=\mu_x dt + \sigma_x dw_t^x$
and
$\frac{d y_t}{y_t}=\mu_y dt + \sigma_y dW_t^y$
and ...
- 119
1
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0
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78
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Showing an "obviously-optimal" control is optimal (without smoothness assumptions)
Let $\mathcal{A}\subseteq\mathbb R$ be a compact interval, $T\in\mathbb R_+$ be a finite horizon, and $g:\mathbb R\to\mathbb R_+$ be a continuous function with $g\leq 1+|\cdot|$. Consider an optimal ...
- 537
1
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0
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72
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Local time as a measurable map from Wiener space
Let $B$ be a Brownian motion on $[0,1]$. The local time of $B$, which I will denote by $L$, is defined as the process on $\mathbb R$ such that
$$\int_0^1 F(B_t)~dt=\int_\mathbb R F(x)L(x)~dx,\qquad\...
- 891
1
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1
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250
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Azema's martingale and quadratic covariation
Given a filtered space $(\Omega,\mathcal F,\mathbb F,\mathbb P)$ supporting a Brownian Motion $B$, where the filtration $\mathcal F$ is the augmented Brownian filtration, the Azema's martingale is ...
- 325
1
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0
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57
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Matching Numbers in Ito McKean
Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as
$e_1 = \lim_{b \...
- 163
1
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0
answers
66
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$X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]
What are the SDE's satisfied by the following processes?
$X_t = B_t^q$
$X_t = (\sin B_t)^q$
$X_t = B_t^q (\sin B_t)^r$
Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the equations ...
- 11
1
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0
answers
160
views
number of times Brownian motion hits boundaries
Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...
- 19
1
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0
answers
1k
views
What is the characteristic functional for Brownian motion on a sphere?
I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...
- 111
1
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1
answer
971
views
Integration of independent Brownian motions
I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where $\tilde{...
- 11
0
votes
2
answers
6k
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Quadratic covariation of two not independent Brownian motions
Given two not independent Brownian motions, $X$ and $Y$. I was wondering if we can say anything about the quadratic covariation of $X$ and $Y$, $\langle X,Y \rangle_t$. I know that for two independent ...
- 103
0
votes
1
answer
163
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Stability of SDE fBM
Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s),
$$
where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian ...
0
votes
1
answer
211
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Step in proof of Itô formula
I am reading a book on stochastic processes. The author proved Itô formula for $f(t,w(t))$ where $w(t)$ is brownian motion with filtration $F_t$. Then he wants to prove Itô formula for $x(t)=a(t)+b(t)...
0
votes
1
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74
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$\lim_{r \to +\infty}\frac{1}{\sqrt{2r \ln(\ln(r))}}(B_r-B_{\left \lfloor{\sqrt{2r \ln(\ln(r))}}\right \rfloor})= 0$ a.s.?
Consider a Brownian motion $B$ and let $f(r)=\sqrt{2r \ln(\ln(r))}.$
Is it true that $\lim_{r \to +\infty}\frac{1}{f(r)}(B_r-B_{\left \lfloor{f(r)}\right \rfloor})= 0$ a.s. ?
If so, how to prove it? ...
- 249
0
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1
answer
160
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Probability to cross an envelopp for 1D random walk?
Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence.
I can make an analogy with random walk: let ...
- 5
0
votes
1
answer
183
views
Probability to cross dynamic boundary for 1D-random walk?
context: Imagine we have an evolving bit sequence (ex: 001011...) where the probability to get 0 or 1 is 1/2. n is the lengh of my sequence (the number of bits)
I can make an analogy with random walk: ...
- 5
0
votes
3
answers
480
views
How to prove that a Brownian bridge $\mathbb{P}(M[0, 1/2]\geq s)\leq 2\mathbb{P}(B(1/2)\geq s/2)?$
Consider a Brownian bridge $B: [0,1]\to \mathbb{R}$ with $B(0)=B(1)=0$. Let $M[0, 1/2]=\max_{x\in[0,1/2]}B(x)$. How to prove that
$$\mathbb{P}(M[0, 1/2]\geq s)\leq 2\mathbb{P}(B(1/2)\geq s/2)?$$
...
- 288
0
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0
answers
95
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Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$
I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion.
I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
- 11
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0
answers
117
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Estimate of cumulative probability of geometric Brownian motion
Let $B_\tau$ be the standard BM, $t$ be the initial time, $s$ be the time variable, $r$ and $\theta$ are positive constants. We also assume that $x$ is the initial position of the below geometric ...
- 54
0
votes
0
answers
185
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Probability that a $d$-dimensional Brownian bridge is greater than a given parameter
Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known :
$$ \mathbb{...
- 186
0
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0
answers
88
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Independent increments for the Brownian motion on a Riemannian manifold
In am not a probabilist, but I must do some stochastic-flavoured work on a connected Riemannian manifold $M$. A nice thing about the Brownian motion on $\mathbb R^n$ is that we may talk about its ...
- 5,407
0
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0
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77
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Construct a random time such that the strong Markov property of Brownian motion fails
Let $\{B_t, \mathcal{F}_t; t\ge 0\}$ be a standard, one-dimensional Brownian motion. Can we construct a random time $S$ such that $P[0\le S < \infty] = 1$ and $W_t = B_{S+t} - B_S$ is not a ...
- 903
0
votes
0
answers
111
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Wiener measure in the space of functions of two or more variables
Wiener measure in the space of continuous functions of two variables had been introduced by J. Yeh in the 1960 paper "Wiener Measure in a Space of Functions of Two Variables" (AMS free access)
(p. ...
- 105
0
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0
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69
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Convergence of a stochastic process in probability
I came across the following. For any fixed $n$, let $\{X_{n}(s) \}_{s\geq0}$ be a stochastic process and let $\{B_n(s) \}_{s\geq0}$ be a Brownian motion. We wish to study the behaviour of $\{X_{n}(s) \...
- 101
0
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1
answer
279
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Expected properties for a PDE whose solution is supposed to be something that doesn't exist
My understanding of Lecture #33, 34: The Characteristic Function for a Diffusion:
As an alternative to directly computing the characteristic function of a random variable $X_t$ in a stochastic ...
- 247
0
votes
0
answers
92
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Movement of a random walk in the limit (a particle in diffusion)
I asked this question in Math Exchange and obtained no answer.
Let $X(t)$ be a stochastic process in time such that $X(0)=0$ and, at each increment of time $\Delta t$, it can move $h$ units in space ...
- 141
-2
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1
answer
121
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Brownian motion and Durret book [closed]
I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B_t \in A\}$. It seems to ...
- 719