The brownian-motion tag has no usage guidance.
-1
votes
1answer
67 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
65 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
78 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
335 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
93 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
222 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 ...
0
votes
0answers
49 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
54 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
74 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
108 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
84 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" ...
3
votes
0answers
79 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)
$$ \tilde W^\alpha_t = \int_0^...
2
votes
1answer
101 views
Generator of Wiener process and its running maximum
This was originally posted on Math StackExchange a long time ago, but got no answer (even after a bounty).
See https://math.stackexchange.com/questions/1274775/generator-of-wiener-process-and-its-...
3
votes
2answers
169 views
Supremum of difference of Brownian bridges: strictly positive wp 1?
EDIT: the original $\ge$ is now $>$ (sorry for the typo!)
Let $B_1(\cdot)$ and $B_2(\cdot)$ denote independent, standard Brownian bridges, i.e., they are mean-zero Gaussian processes on $[0,1]$ ...
0
votes
0answers
67 views
Movement of a random walk in the limit (a particle in diffusion)
I asked this question in Math Exchange and obtained no answer.
Let $X(t)$ be a stochastic process in time such that $X(0)=0$ and, at each increment of time $\Delta t$, it can move $h$ units in space ...
0
votes
0answers
99 views
Quadratic variation and DDS theorem
DDS theorem tells us, that for every continuous martingale M such that $⟨M⟩_{\infty} = \infty ~~a.s.$ then there exists a Brownian motion $B$ such that $M_t = B_{⟨M⟩_t}$.
I have a question whether I ...
2
votes
1answer
141 views
Brownian sausage surgery of Poisson point process
Fix some $r >0$ and let $\mathcal P$ be a unit intensity Poisson point process on $\mathbb R^d - \mathbb B(0,r)$. Let $W_t = \cup_{s \leq t} \mathbb B(B_t,r)$ be the Brownian sausage around a ...
1
vote
0answers
52 views
Dynamics for sets related to Brownian motion: zero set, fast points
For sets like the Cantor set, we have preserving maps (eg. the shift-maps and conjugates to it) that allows us to study dynamical quantities such as invariant measure and entropy. I am wondering if we ...
1
vote
1answer
103 views
Conditional stochastic integration
Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$
What is the ...
1
vote
0answers
84 views
Unique EMM & completeness in the Black-Scholes model
Consider the Black-Scholes model
$$ dS(t) = \mu(t) S(t) dt + \sigma(t) S(t) dW^{\mathbb{P}}(t) $$
$$ dB(t) = r(t) B(t) dt$$
Steele shows now in "Stochastic Calculus & Financial Applications" (Ch. ...
1
vote
2answers
329 views
2 dimensional brownian motion hitting time
If we have two independent brownian motion in $x$ and $y$ direction. At time zero we sit at $(a,b)$ with $a>0, b>0$.
What is the probability that we will hit positive $x$ axis before hitting ...
0
votes
0answers
81 views
reflected Wiener process and Doob's h-transform
Is there a way to see a real valued Wiener process $W^R$ starting from $x \geq 0$ and reflected in 0 as the Doob's transform of a standard real valued Wiener process $W$?
Recall that
$$
W_t^R = x + ...
0
votes
0answers
94 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
274 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
299 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 ...
1
vote
2answers
118 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
82 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, ...
5
votes
1answer
488 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
190 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
143 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
86 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
123 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
$$
p(x,t;y,T) = \frac{1}{\...
2
votes
1answer
279 views
Exercise on a hitting time for a Brownian Motion
I'm following Chapter 3 of "Brownian Motion", by Peres and Mörters, about The Dirichlet Problem(DP). As it is known, in order to obtain existence and uniqueness of a solution for DP it is necessary to ...
2
votes
0answers
150 views
Weighted sum of Standard Brownian Bridges
Suppose, $\{B_j\}_{j=1}^k$ be a sequence of Brownian Bridges.
Let us consider, $$X(t)=\sum_{j=1}^m w_j(t)B_j(t),$$ where $w_j$ are positive weight functions.
Then what can we say about (...
3
votes
1answer
100 views
Covariation of the stochastic integral and the Wiener process
Let$^1$
$T>0$
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint operator with finite trace $\operatorname{tr}Q$
$(e^n)_{n\in\mathbb N}$ be an ...
7
votes
1answer
169 views
A Converse of the Skorokhod Embedding Theorem
I am wondering whether the following "sort of converse" of Skorokhod's embedding theorem holds:
Suppose that $\{D_t\}_{t \geq 0}$ is a stochastic process with continuous paths, $D_0 = 0$, and suppose ...
6
votes
1answer
283 views
Moment bounds on exponential martingale
Consider the exponential martingale used in the Girsanov transformation of
measure:
$$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
83 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
182 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 ...
1
vote
1answer
128 views
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 ...
1
vote
0answers
49 views
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 \...
2
votes
0answers
90 views
constructing brownian motion adapted to a given brownian motion
Suppose we are given a Brownian motion $\{B_t,\mathcal F_t^B\}_{t\in[0,1]}$. I would like to know if it is possible to construct another Brownian motion $\{W_t,\mathcal F_t^W\}_{t\in[0,1]}$ from $B$, ...
2
votes
1answer
101 views
Neat little proof using local time from Ito McKean
This is a cool little result, the proof of which uses the machinery of local time. On p. 72, Prob 1 asks to show that $\int_0^1 dt/x(t)$ exists, where $x(t)$ is a continuous time brownian motion.
In ...
2
votes
1answer
210 views
Any modern/recent version of Ito & McKean?
This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their ...
4
votes
0answers
176 views
Explicit martingale representation for a Brownian bridge
Let $W$ denote a Wiener process, $\displaystyle M_t = \max_{0 \le s \le t} W_s$ its running maximum. The martingale representation of $M$ is known explicitly:
$$M_T = \sqrt{\frac{2T} \pi} + \int_0^T ...
1
vote
1answer
136 views
Deterministic function in the support of Brownian motion
Is there an explicit example of a function in the (topological) support of the law of Brownian motion (with respect to the topology of uniform convergence of continuous functions)?
(You can take "...
5
votes
2answers
281 views
Reference for Feynman-Kac
I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...
2
votes
1answer
74 views
Equivalence of tail sigma-fields between two triangular arrays of Brownian motions
Let
$$\{B^{(i)}_t : 0 \leq t \leq T, i \geq 1\}$$
be a sequence of i.i.d. standard Brownian motions. For each $n,$ let $V^{(n)}_t, \,0 \leq t \leq T$ be a continuous bounded process adapted to the ...
8
votes
0answers
167 views
Basis for $L^2(\mathbb{R})$ that Solves the Heat Equation
This is a less-than-serious question that I asked on math.SE, but I suspect it is slightly more appropriate to ask it here. Consider the heat equation $$
u_t = \frac12 u_{xx}
$$ On $\mathbb{T}$ with ...
2
votes
0answers
107 views
Feynman-Kac formula for *general* Sturm-Liouville operator
One way to state (omitting technical requirements) the Feynman-Kac formula that I am familiar with is as follows.
Let $u$ be a solution to the pde
$$u_t(x,t)=-\frac{\sigma^2(x,t)}2u_{xx}(x,t)-V(x,t)u(...
