Questions tagged [brownian-motion]
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2
votes
1answer
77 views
How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?
How far away is
$$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$
from
$$\max_{0 \leq t \leq 1} |W(t)|$$
In other words, if you simulate a Wiener process over a finite ...
0
votes
0answers
41 views
$P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ has no closed-form expression… right?
$P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ shows up in a formula for computing $p$-values for a certain statistic, where $W(t)$ is a $d$-dimensional (standard) Wiener process. My advisor says the ...
1
vote
1answer
78 views
Is there Brownian motion on Alexandrov spaces?
It is well known that there is a notion of Brownian motion on smooth Riemannian manifolds.
I am wondering if there is a more general notion of Brownian motion on finite dimensional Alexandrov ...
1
vote
1answer
67 views
Martingale representation theorem for symmetric random walk
Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that
$$ X(t) = \int_0^t ...
0
votes
0answers
30 views
Doob's h transform
Consider $B$ a one dimensional brownian motion, a probability $\mathbb{P}$ and its filtration $\mathcal{F}_t$. Let $t>0$, and define
$$
H_0^{(t)} \triangleq \inf \{ \tau > t, \: B_\tau = 0 \}
$$
...
0
votes
0answers
79 views
Can we have Levy area for N dimensional process?
Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there a equivalent area for N dimensional Brownian motion, if so what ...
0
votes
0answers
29 views
Supremum of a general Gaussian Process
I have a stochastic integral of the form
\begin{align*}
X(t) = \int_0^t h(v) W(v) dv
\end{align*}
where $W(v)$ is the standard Brownian motion and $h(v)$ is a positive, integrable function. While $X(t)...
2
votes
0answers
51 views
Mutual dependencies of BSDE solutions with markovian drivers with different starting points
Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$.
Let ...
1
vote
1answer
105 views
conditional expectation brownian motion
$B=(B_t,t\in[0,1])$ a standard brownian motion on $[0,1]$.
For $t\in[0,1]$, we define $$\mathcal{F}_t=\sigma(B_s,s\in[0,t]),$$ $$\mathcal{G}_t=\mathcal{F}_t\,\vee\,\sigma(B_1).$$
How can we show $$\...
2
votes
1answer
126 views
Gaussian sum VS Brownian motion
Given independent Gaussian $d$ dimensional vectors $G_i$,
Let $ \sigma^2_n=\mathbb{E}(\sum_{i \le n} G_i) \cdot (\sum_{i \le n} G_i)^T$. $||\sigma_n^2||$ is norm of $\sigma_n^2$.
Is there a $d$-...
2
votes
0answers
37 views
Floquet stochastic process
Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
0
votes
0answers
46 views
quadratic variation on n-sphere
Is it true, and if so, is there an easy way to see that the quadratic variation of standard Brownian motion on n-sphere is $\leq$ t?
Note: I am a novice in stochastic analysis.
-1
votes
1answer
71 views
Expected value of $W_{t_i} W^2_{t_{i+1}}$
I stuck in determining the expected value of the following product
$E[W_{t_i}W_{t_{i+1}}^2]$ where $W_{t_i}$ and $W_{t_{i+1}}$ are Brownian with normal distribution, i.e. $W_{t_i}\sim N(0,t_i)$. I ...
1
vote
0answers
73 views
Schilder's theorem for brownian bridges
I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE.
A bit of context: usually, Schilder's theorem tells us that the ...
0
votes
0answers
84 views
Concerning some Tauberian-type asymptotics of Laplace transform involving $e^{-\sqrt{s}}$
There are some well-known Tauberian theorems concerning the asymptotics of the original function (say as $t$ tends to $0$) and that of its Laplace transform (as $s$ tends to infinity). I want to ask a ...
2
votes
1answer
409 views
Is the ito integral $\int_0^t \operatorname{sign}(W_s)\mathrm{d}W_s$ a Brownian motion?
Consider the ito integral of the sign of the Brownian motion $W_s$ from $0$ to $t$:
$$\int_0^t \operatorname{sign}(W_s)\,dW_s$$
This appears for instance in the Tanaka formula. I think this is a ...
3
votes
3answers
95 views
Moments of the Hölder norm of Brownian process
It is well known that for a brownian process $B(t),t\geq 0$, it holds
$$
\sup_{0\leq s<t\leq T}\frac{|B(t)-B(s)|}{|t-s|^\alpha}<\infty
$$
almost surely, for any $T>0$ and $\alpha<1/2$.
...
7
votes
1answer
297 views
Covariance function of Brownian motion and the second derivative operator
I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me.
Suppose $W$ is a Brownian motion, and we ...
3
votes
1answer
114 views
What is the Wiener measure of the curves with Hölder index $\frac 1 2$?
One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index ...
0
votes
0answers
55 views
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) \...
5
votes
1answer
85 views
Convolution of two Brownian motions
Suppose $B_1(t)$ and $B_2(t)$ are two independent, standard Brownian motions. What is the distribution of
\begin{align*}
G(t) = \int_0^t B_1(\tau)B_2(t-\tau)d\tau \qquad
\end{align*}
Or, at least an ...
1
vote
1answer
112 views
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 ...
4
votes
0answers
88 views
Who proved the reflection principle in random walks and Brownian motion?
I've heard Henry McKean say that the reflection principle is due to Désiré André. But the wikipedia page seems to say that André did not use a reflection principle. Does anyone know where the "modern" ...
4
votes
0answers
82 views
Normalisation in fractional integration and Brownian motion
Fractional Brownian motion comes in two forms (following Marinucci and Robinson 1998) for fraction $\alpha$ and Brownian motion $W_s$:
Type II (Levy, Volterra, Riemann)
$$ \tilde W^\alpha_t = \int_0^...
2
votes
1answer
112 views
Generator of Wiener process and its running maximum
This was originally posted on Math StackExchange a long time ago, but got no answer (even after a bounty).
See https://math.stackexchange.com/questions/1274775/generator-of-wiener-process-and-its-...
3
votes
2answers
174 views
Supremum of difference of Brownian bridges: strictly positive wp 1?
EDIT: the original $\ge$ is now $>$ (sorry for the typo!)
Let $B_1(\cdot)$ and $B_2(\cdot)$ denote independent, standard Brownian bridges, i.e., they are mean-zero Gaussian processes on $[0,1]$ ...
0
votes
0answers
68 views
Movement of a random walk in the limit (a particle in diffusion)
I asked this question in Math Exchange and obtained no answer.
Let $X(t)$ be a stochastic process in time such that $X(0)=0$ and, at each increment of time $\Delta t$, it can move $h$ units in space ...
0
votes
0answers
105 views
Quadratic variation and DDS theorem
DDS theorem tells us, that for every continuous martingale M such that $⟨M⟩_{\infty} = \infty ~~a.s.$ then there exists a Brownian motion $B$ such that $M_t = B_{⟨M⟩_t}$.
I have a question whether I ...
2
votes
1answer
148 views
Brownian sausage surgery of Poisson point process
Fix some $r >0$ and let $\mathcal P$ be a unit intensity Poisson point process on $\mathbb R^d - \mathbb B(0,r)$. Let $W_t = \cup_{s \leq t} \mathbb B(B_t,r)$ be the Brownian sausage around a ...
1
vote
0answers
55 views
Dynamics for sets related to Brownian motion: zero set, fast points
For sets like the Cantor set, we have preserving maps (eg. the shift-maps and conjugates to it) that allows us to study dynamical quantities such as invariant measure and entropy. I am wondering if we ...
3
votes
1answer
164 views
Conditional stochastic integration
Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$
What is the ...
1
vote
0answers
92 views
Unique EMM & completeness in the Black-Scholes model
Consider the Black-Scholes model
$$ dS(t) = \mu(t) S(t) dt + \sigma(t) S(t) dW^{\mathbb{P}}(t) $$
$$ dB(t) = r(t) B(t) dt$$
Steele shows now in "Stochastic Calculus & Financial Applications" (Ch. ...
1
vote
2answers
459 views
2 dimensional brownian motion hitting time
If we have two independent brownian motion in $x$ and $y$ direction. At time zero we sit at $(a,b)$ with $a>0, b>0$.
What is the probability that we will hit positive $x$ axis before hitting ...
1
vote
0answers
99 views
Brownian motion on $[0,1]$
Let $\{W'_t\}_{t\in [0,1]}$ be the Brownian motion on the real line obtained by taking the standard Brownian motion $\{W_t\}_{t\ge 0}$ and conditioning on the events $W_1 = 1$ and $0\le W_t\le 1$ for $...
3
votes
2answers
280 views
Does the hitting time of +1/-1 of a Brownian motion posess a density?
The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$.
I define $T=\inf \{t>0,|W|(t)= 1\}$. ...
6
votes
2answers
312 views
Reference for LIL for fractional Brownian motion
(Cross-posted to https://math.stackexchange.com/questions/2377810/law-of-iterated-logarithm-for-fractional-brownian-motion.)
It seems strange but, even after consulting several books, and hours ...
2
votes
2answers
134 views
Isometry for the stochastic integral wrt fractional Brownian motion for random processes
Let us fix $(\Omega,\mathscr A,\Bbb P)$ a probability space. Let then $\Bbb F:=(\mathscr F_t)_{t\ge0}$ be a complete and right continuous filtration.
Now if $B$ is an $\Bbb F$-standard Brownian motion,...
3
votes
0answers
83 views
Subordination process and Bismut's proof of asymptotic formula for the heat kernel
J. Bismut proved the asymptotic formula for the heat kernel of the Laplace-Beltrami operator $\Delta$ on a manifold $M$ in one of his well-known books. Later, in his paper on the index theorem, ...
6
votes
1answer
564 views
Brownian motion and its maximum and its minimum
Let $W_u, 0\leq u \leq t$ be Brownian motion.
Let $m_t= min_{0\leq u\leq t} W_u$ and $M_t = max_{0 \leq u \leq t} W_u$.
The fact that $(M_t , W_t)$ is absolutely continuous with respect to Lebesgue ...
4
votes
1answer
193 views
Uniqueness of a SDE with positivity constraint
We start by fixing some notation.
If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it ...
2
votes
1answer
149 views
Brownian motion and random walk
Let $M_{\Gamma}$ a Riemannian covering of a closed compact manifold $(M,g)$ with deck transformation $\Gamma$ (its neutral element will be denoted by $e$). If we denote by $p_t^{\Gamma}(x,y)$ the heat ...
1
vote
1answer
89 views
Literature on the total variation of fractal graphs/fractal Brownian motion?
I know that for standard Brownian motion, the total variation sampled at intervals of length $\Delta$ converges to $V(\Delta) = C \Delta^{-1/2}$ for some constant $C$. I wish to use this fact to study ...
1
vote
1answer
138 views
interpretation of the transition probability of a brownian motion in terms of the Wiener measure
Let $W(t)$ be a standard brownian motion in $E \triangleq \mathbb{R}^d$.
The transition probability from a state $x \in E$ at time $t$ to a state $y \in E$ at time $T$ is
$$
p(x,t;y,T) = \frac{1}{\...
2
votes
1answer
312 views
Exercise on a hitting time for a Brownian Motion
I'm following Chapter 3 of "Brownian Motion", by Peres and Mörters, about The Dirichlet Problem(DP). As it is known, in order to obtain existence and uniqueness of a solution for DP it is necessary to ...
2
votes
0answers
173 views
Weighted sum of Standard Brownian Bridges
Suppose, $\{B_j\}_{j=1}^k$ be a sequence of Brownian Bridges.
Let us consider, $$X(t)=\sum_{j=1}^m w_j(t)B_j(t),$$ where $w_j$ are positive weight functions.
Then what can we say about (...
3
votes
1answer
108 views
Covariation of the stochastic integral and the Wiener process
Let$^1$
$T>0$
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint operator with finite trace $\operatorname{tr}Q$
$(e^n)_{n\in\mathbb N}$ be an ...
7
votes
1answer
173 views
A Converse of the Skorokhod Embedding Theorem
I am wondering whether the following "sort of converse" of Skorokhod's embedding theorem holds:
Suppose that $\{D_t\}_{t \geq 0}$ is a stochastic process with continuous paths, $D_0 = 0$, and suppose ...
6
votes
1answer
324 views
Moment bounds on exponential martingale
Consider the exponential martingale used in the Girsanov transformation of
measure:
$$Z(t) = \exp\Big(\int_0^tXdW - \frac{1}{2}\int_0^t|X|^2ds\Big)$$
so that $Z$ solves the sde $dZ = ZXdW$ where $W$ ...
1
vote
0answers
89 views
forward Ito integral
Forward integral is introduced by Francesco RussoPierre Vallois as a generalization of Ito integral. For simplicity, let $B$ be a standard Brownian motion and let $\phi$ be a measurable process. The ...
2
votes
1answer
190 views
“Brownian motion” without assuming continuity of path at origin of state space
This question is inspired partly by this question Any reference on Brownian Motion continuity. In this post, the author asked if the following three axioms can define a Brownian motion without ...
