All Questions
Tagged with stochastic-processes brownian-motion
219 questions
0
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1
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323
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Integrated square difference of Brownian bridges
I am doing some work with measuring the distance between distributions, and someone pointed out to me that I should look into calculating the integrated squared difference of two brownian bridge ...
CommunityBot
- 1
2
votes
1
answer
186
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Upper left Dini derivative of Brownian motion at a hitting time
Let $W$ be a standard Brownian motion. Define the upper left Dini derivative $D^-W$ by
$$D^-W_t := \limsup_{h \to 0^-} \frac{W_{t+h} - W_t}{h}.$$
Fix $a > 0$, and define the stopping time $\tau$ by
...
- 3,669
3
votes
1
answer
655
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Forgery theorem: the Brownian motion stays close to any curve with positive probability
In a paper I am reading the authors claim that, if $B$ is a standard BM in $\mathbb{R}$ and $f\in C([0,1],\mathbb{R})$, then for any $\epsilon>0$
$$
\mathbb{P}(\sup_{t\in [0,1]}|B_t-f(t)|<\...
- 14.2k
2
votes
1
answer
392
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Full version of Cameron Martin theorem for Brownian motion
I’m looking for a version of the Cameron Martin theorem for the Brownian motion under random shifts. Here is the precise statement:
Let $\mathbb P$ be Wiener measure on $\Omega := C[0, 1]$. Given a $C[...
- 8,992
2
votes
1
answer
2k
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Alternate proof of Levy’s characterisation of Brownian motion
Levy’s characterisation theorem for Brownian motion states that for a local martingale $X$ with $X_0 = 0$, $X$ is a Brownian motion if and only if it has quadratic variation $\langle X, X \rangle_t = ...
- 8,992
1
vote
1
answer
739
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Joint law of a standard Brownian motion and its local time at a nonzero level
Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\...
CommunityBot
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4
votes
2
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455
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Converse of Itô's formula
Let $f,h,g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\forall u \in \mathbb{R}_+,f(B_u)=f(B_0)+\int_0^ug(B_r)dB_r+\frac{1}{2}\int_0^uh(B_r)dr.$$
Prove that $f$ is ...
- 6,155
1
vote
0
answers
103
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Continuity of Wiener measure on open balls
Let $\mu$ be the Wiener measure on $C_0 [0, T]$, the space of continuous functions starting at $0$, under the sup norm.
Question: Is it true that the function $r \mapsto \mu(B_r(x))$ is continuous in $...
- 6,155
4
votes
1
answer
350
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Lebesgue differentiation theorem at a stopping time
Let $W$ be a standard Brownian motion, and $\mathcal F_t$ it’s natural filtration. Let $H$ be a continuous process, adapted to $\mathcal F_t$ and integrable with respect to $W$.
Question: Is it true ...
- 6,045
1
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0
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89
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Comparison of the numbers of particles surviving forever
Consider two $N\text{-}$particle systems as follows : for $1\le i\le N$,
$$X^i_t=1+\int_0^t(b+\phi^i_s) \, ds+W^i_t \quad\mbox{and} \quad Y^i_t=1+ct+W^i_t,\quad \forall t\ge 0,$$
where $c>b>0$ ...
1
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1
answer
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
0
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1
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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)...
- 63.9k
2
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1
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182
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Mean of log-normal variable when exponent is replaced by runnung maximum of Ito-integral
Let $W=\{W_t\}_{t\in[0;1]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;1]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{\sigma_t\}_{t\in[0;1]}$ be a continuous and ...
- 7,514
1
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1
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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
5
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2
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688
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Endpoint of Brownian motion conditional on high maxima
Note: This question is closely related to an earlier question: A large noise limit.
Let $W$ be a standard one dimensional Brownian motion.
For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...
- 6,155
2
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1
answer
264
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Chung's law of the iterated logarithm for Brownian motion
I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...
- 188k
5
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1
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283
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Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
- 5,474
1
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1
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118
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For some $\alpha>0$, $ e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\frac{|B_s-B_t|^2}{|s-t|})<\infty\right) $?
I am reading one lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it says that
for some $\alpha>0$,
$$
e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\...
- 7,514
6
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0
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292
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Running maximum/supremum of Brownian motion: add information to make it a Markov process?
Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it ...
- 398
1
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0
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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 ...
CommunityBot
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8
votes
2
answers
1k
views
The Wiener measure of an open set
There is so much written about the Brownian motion and I suspect the answers to the questions below are hidden in somewhere in the literature but I cannot find them
Denote by $E$ the Banach space ...
- 14.2k
6
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1
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579
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Is this a Brownian motion?
I am building a 2D stochastic process as follows. I start with a point $P_0=(0,0)$. Then $P_k=(X_k,Y_k)$ is defined as follows, for $k>0$:
\begin{align}
X_k & =X_{k-1}+R_k \cos(2\pi\theta_k) \\
...
- 52.3k
1
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1
answer
1k
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The joint distribution of the min and max of a Brownian [closed]
The joint distributions of the brownian and both the minimum and the maximum respectively are known. What could be said about the joint distribution of the maximum and the minimum of a Brownian ...
- 63.9k
2
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1
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150
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Existence of a process on $\mathbb{R}^2$ that looks like two 'independent' brownian bridges $B_1(x)$ and $B_2(x)$ conditioned on $B_1(x)+B_2(x) > 0$
Consider any probability density function $f(x)$ that has mean zero variance one and say all finite moments. You may assume standard normal density if you like.
Given $a_1,a_2>0$, I consider two ...
- 471
2
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1
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960
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On the range of Holder continuity of Brownian motion
It is known that Brownian motion is almost surely locally Holder continuous, on a range that is random, i.e. depends on the particular path. This question explores the maximal range on which Brownian ...
- 14.2k
0
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0
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176
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A convergence question in $L^2$ construction of Brownian motion
I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...
- 227
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1
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74
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$\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? ...
- 1,172
1
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2
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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
0
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1
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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
2
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1
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203
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Comparison of probabilities that drifted Brownian motion never hits barriers
Let $k , h: \mathbb R_+\to [0,1]$ be non-decreasing and right continuous s.t. $k(t)\le h(t)$ for all $t\ge 0$. Define $\tau_{k}$ (resp. $\tau_h$) by
$$\tau_k : = \inf\{t\ge 0:2+\beta t+ W_t \le k(t)\}\...
- 128k
3
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1
answer
229
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How to prove excursion process is a Poisson point process?
This question comes from book Ju-Yi Yen and Marc Yor P59 and P60,
On page 59, "Define $\mathcal{Z}_\omega=\{t:B_t(\omega)=0\},$ and $\tau_l$ is the inverse local time. The complement of $\mathcal{...
- 14.2k
2
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0
answers
53
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Continuity of translation operator in fractional white noise analysis
Fix $H\in(\frac{1}{2},1)$, and let $\Omega:=C_0([0,T],\mathbb R^d)$ be the space of $\mathbb R^d$-valued continuous functions. There is a probability measure $P^H$ on $(\Omega,\mathcal B(\Omega))$, ...
- 515
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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 ...
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1
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1
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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 ...
CommunityBot
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1
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210
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Existence of two stochastic processes
I am wondering if I can show that
For given $x,y\in \mathbb{R}$ there are two stochastic processes $S_t$ and $B_t$ such that $S_t$ and $B_t$ are two one dimensional Brownian motions starting at $x$ ...
CommunityBot
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1
vote
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 ...
- 14.2k
2
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2
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240
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Reference request (Brownian local time): for fixed $t$, $a\mapsto L_a(t)$ is a.s. continuous and with compact support
So the title is quite self explanatory.
In the book "Continuous Martingales and Brownian Motion" by Rebuz and Yor, in the proof of Proposition $(2.1)$ of chapter XIII it's stated that:
For ...
- 1,989
4
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0
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166
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Occupation time of SDE
Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
- 6,155
1
vote
1
answer
215
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The long run average amount of time the deviation of Brownian motion spends above its expected value
Let $B_t$ be a standard one dimensional Brownian motion. Is it true that
$$\lim_{s \to \infty} \frac{\int_{[0, s]} \mathbf 1_{ \{|B_t| \geq \sqrt{2t/\pi} \} } \ dt}{s}$$
exists almost surely?
- 14.2k
5
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1
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548
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Largeness of the set of zeroes of a Brownian motion
Definitions:
A measurable subset $S$ of $\mathbb R$ is said to be mesoscopic if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero ...
- 14.2k
0
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0
answers
87
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Why the distribution of M(t) is the same as X(t)?
Let $ B(t)(t\geq 0) $ be the standard Brownian motion and $ M(t)=\max_{0\leq s\leq t}{B(s)} $. If we define $ X(t)=M(t)-B(t) $ as a new stochastic process, how can I show that $ X(t) $ has the same ...
- 1,219
1
vote
0
answers
124
views
L2-closure of absolutely continuous stochastic processes?
Assume we have a possibly multidimensional Brownian motion on a probability space $(\Omega,\mathcal F,\mathbb P)$ where $(\mathcal F_t)_{t\in[0;T]}$ is the Brownian standard filtration. Let $\Vert X\...
- 335
5
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2
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289
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Bounding Brownian motion and an Ito process simultaneously
Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
- 1,172
0
votes
0
answers
185
views
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 :
$$ \mathbb{...
- 186
2
votes
1
answer
538
views
Generalized Fokker-Planck equation
Consider the diffusion process
$$
d X = \mu(X, t) dt + \sigma(X, t) dY.
$$
When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...
- 188k
1
vote
0
answers
160
views
Laplace Equation for Brownian Motion [closed]
So, I know that there is this theorem (taken from here):
For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have
$$
u(x) = \...
- 121
1
vote
1
answer
57
views
Bound moments wrt. known initial and final moments
Let $X$ be an $L^p$ random variable, where $p\in (0,1)$ and $W_t$ usual Brownian motion (with $W_t$ independent from $X$). I'd like to bound
$$\mathbb E|X+W_t|^p$$
purely in terms of $\mathbb E|X|^p$ ...
- 128k
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 ...
- 128k
1
vote
0
answers
68
views
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 ...
- 3,065
1
vote
0
answers
222
views
Is my quadratic variation derivative bounded?
Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
- 335