All Questions
155 questions
0
votes
1
answer
51
views
Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, ...
- 5,405
2
votes
0
answers
70
views
Poisson process subordinated by a gamma process
I am working on a problem and I encountered the following situation:
$(N(t): t \ge 0)$ is a Poisson process with parameter $\lambda t $. If $T_{n} = \sum_{i=1}^n W_i$ represents the $n^\text{th}$ ...
- 21
0
votes
1
answer
78
views
Uniform concentration bound (function-valued random variable / continuous stochastic process)
I'm trying to consider a probability space $\Omega$ and
$f(x,\xi):\mathcal{X}\times\Omega\to\mathbb{R}$ (stochastic process over space? or function-valued random variable?), where $\mathcal{X}\subset\...
- 321
2
votes
1
answer
246
views
Does $X_t$ with $t>0$ admit a density?
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 835
0
votes
0
answers
87
views
Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables
Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
0
votes
0
answers
73
views
Asymptotic stochastic ordering for weighted sum of i.i.d. random variables
Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$,
\begin{equation}
a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
- 19
0
votes
0
answers
99
views
Random walks on groups
I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a ...
- 9
-1
votes
1
answer
169
views
joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
-2
votes
1
answer
152
views
Branching process with varying offspring distribution at each step
Consider a simple branching process $Z_0,Z_1,Z_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$.
...
- 109
-1
votes
1
answer
77
views
Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution
Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
- 287
0
votes
0
answers
74
views
Probability distribution for a Bayesian Update
I am struggling with a process like this:
$$X_t=\begin{cases}
\frac{\alpha\omega_t}{\alpha\omega_t+\beta(1-\omega_t)} & \text{with prob } p\\
\frac{(1-\alpha)\omega_t}{(1-\alpha)\omega_t+(1-\beta)(...
- 21
3
votes
1
answer
159
views
Are there any known results on the probability distributions of perpetuities with power law discount rates?
Currently I am working on studying stochastic integrals of the form: $$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$
where $S_t$ is a Compound-Poisson process with Exponentially-distributed ...
- 33
2
votes
1
answer
324
views
On the mean value taken by Bernoulli random variables with joint distribution constraints
We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X_1, X_2, \ldots X_n$, such that the probability $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...
- 1,677
2
votes
0
answers
201
views
Continuity of density of SDE
Consider a stochastic differential equation in $\mathbb R^m$ with a parameter $\theta\in\mathbb R$:
\begin{equation}
dX_t^{\theta,x} = v(\theta,X_t^{\theta,x})dt+\sigma(X_t^{\theta,x})\circ dW_t,~...
- 21
3
votes
1
answer
171
views
Exponential of supremum of Brownian bridge on short time frame
For each $T > 0$, let $B^T$ be a Brownian bridge on $[0, T]$, conditioned to start and end at $0$.
Question: Is it true that $\mathbb E[|\text{exp}\, (\sup_{0 \leq t \leq T} B^T_t) - 1|] \to 0$ as $...
- 6,215
0
votes
1
answer
169
views
Understanding the approximation of a random sum of random processes
I want to understand an approximation of a compound Poisson distribution in this paper.
First, let's set the environment. Consider $\mathcal{P}$ the class of distributions of real-valued and strictly ...
- 135
5
votes
1
answer
336
views
Joint distribution of drawdown time and value of geometric Brownian motion
Let $X$ be a geometric Brownian motion, satisfying the SDE
$$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$
for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Define the ...
- 6,215
3
votes
1
answer
266
views
A linearly distributed version of the balls into bins problem
Some years ago, I found a paper with all the formulas for the balls into bins problem when the "areas" (i.e., probabilities to capture a ball) of the bins are all different. However, the ...
- 1,677
0
votes
1
answer
82
views
WLLN for bootstrap means of stationary ergodic processes?
Setup:$\quad$
Suppose that $(X_n)$ is a stationary ergodic process with $E|X_1|<\infty$.
Given $X^{(n)}=(X_1, \dots, X_n)$, select a standard Efron bootstrap subsample $(X_{n,1}^*, \dots, X_{n,m(n)}...
- 127
1
vote
1
answer
215
views
Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)
Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$.
...
- 3,098
3
votes
2
answers
667
views
Is every discrete compound Poisson distribution a mixed Poisson distribution?
I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer.
The mixed Poisson distribution and compound Poisson ...
- 1,311
1
vote
0
answers
142
views
What are the Lévy processes with specific increments?
It is known that the increment of the Wiener process $W$ is drawn from a Gaussian distribution, i.e. $\Delta W \sim \mathcal{N}(0, \delta t)$.
I wonder what are the Lévy processes with increments from ...
- 11
3
votes
1
answer
251
views
Another large noise limit
Note: Here all processes take values in $[0, 1]$.
Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Let $X$ be the solution to the SDE
$$dX_t = \sigma X_t \, dW_t$$...
- 6,215
1
vote
1
answer
93
views
Probability distribution of $\sum_i^n X_i - T$ where $\sum_i^nX_i <T<\sum_i^{n+1} X_i $
Let $X_{1}, X_{2}, \ldots, X_{n}$ be IID random variables with mean $\mu$ and variance $\sigma^2$. Let $S_n=\sum_i^{n}X_i.$
Let $T\gg1$ and define $\tau=T-S_n$ where $n$ satisfies the following ...
- 117
2
votes
1
answer
150
views
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 ...
- 123
2
votes
1
answer
308
views
Maximum nearest neighbor distance for a Poisson point process
Is the maximum nearest neighbor distance between points of the process, over all the infinitely many points of a stationary Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, almost ...
- 3,098
1
vote
1
answer
2k
views
First hitting time for a drifted Brownian motion
While the solution for a first hitting time for a drifted Brownian Motion is well known, I want to post a different question.
Take a continuous-time stochastic process $X_t$ and define the the ...
- 21
1
vote
0
answers
663
views
The distribution of hitting time in 2D-lattice random walk [closed]
Assume a particle at $(0,0)$ with the same possibility of $1/4$ for moving up/down/left/right (i.e. random walk in 2D lattice). We define the stopping time 𝑇𝑐 as it hits $(a,b)$. How can we get the ...
3
votes
0
answers
516
views
The distribution of collision stopping time in 2D random walk
Assume two particles A at $(0, 0)$ and B at $(a, b)$ in 2D discrete grid, both of them have the same possibility of $\frac{1}{4}$ for moving up/down/left/right (i.e. 2D random walk). We define the ...
2
votes
2
answers
161
views
Determine the affine envelope of a random process's MGF
Suppose that a stationary random process $S(t)$ can be characterized as the figure below, which for most of the time is a straight line $S(t)=c\cdot t$, but occasionally would "stall" for a ...
- 265
1
vote
2
answers
277
views
Distribution of interarrival times for a special class of stochastic point processes
I am interested in Poisson-binomial stationary point processes (here on the real line) defined as follows. Let
$t_k=k/\lambda$, with $k\in\mathbb{Z}$ and $\lambda>0$,
$F_s(x)$ be a symmetric, ...
- 3,098
5
votes
1
answer
392
views
Uniqueness of the solution to some SDE
Consider the stochastic differential equation as follows:
$$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable and $m(t)=\mathbb P[...
- 1,334
3
votes
2
answers
297
views
Does my construction always result in a stationary Poisson point process of intensity $1$? How so?
My construction is as follows: Let $X_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F_k(x,s)$ where $k$ is the location parameter and $s$, a ...
- 3,098
5
votes
2
answers
311
views
A comparison of diffusions
Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
- 128k
2
votes
1
answer
124
views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
- 399
0
votes
1
answer
138
views
Do measure-valued dynamical systems correspond to marginals of Markov processes?
Let $(\mu_n)_{n=1}^{\infty}$ be a sequence in $\mathcal{P}_1(X)$ for some compact metric space $(X,d)$. Suppose that there is a weakly-continuous function $F:\mathcal{P}_1(X)\rightarrow \mathcal{P}_1(...
- 5,405
4
votes
2
answers
480
views
Hitting probability of a line
Consider a simple (nearest neighbor) random walk on a lattice $\Bbb Z^2$ which starts at the origin, is constrained to $x\ge 0$ halfplane, and stops when it hits the line $x=n$. Denote by $p(n,k)$ ...
- 17k
0
votes
1
answer
142
views
Covering number of the conditional distribution function
Suppose $Y$ is a random variable in $\mathbb{R}^d$, and we want to find the covering number
\begin{equation*}
\mathcal{F} = \big\{ F_{Y|W} (y | W) : y \in \mathbb{R}^d \big\}
\end{equation*}
where ...
- 331
5
votes
3
answers
601
views
Convergence speed of a random dyadic rational generator
We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$
two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...
- 1,677
1
vote
1
answer
259
views
Is the topology generated by the convergence of finite-dimensional distributions metrizable?
Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...
- 309
0
votes
0
answers
96
views
Limit of a linear discrete-time stochastic process with uniform noise
I have posted this in the math and stats sites, but I am not sure where the proper forum for this question is. If it is not here, please go on and delete it.
Suppose we have a stochastic linear ...
- 1
1
vote
1
answer
172
views
Is it always possible to determine the distribution of a random variable given all its moments? [closed]
we we're asked about it, and I know that answer is "NO", and I haven't found an good enough answer yet
and would appreciate an explanation with examples.
- 11
1
vote
0
answers
44
views
Small parameter expansion of probability density
I am trying to describe the motion of a particle that moves according to the Langevin equations
\begin{align}
\dot{x}&(t)=v_0\cos{\beta(t)},\tag{1}\\
\dot{y}&(t)=v_0\cos{\beta(t)},\tag{2}
\end{...
- 11
3
votes
0
answers
98
views
Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix
In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
- 279
2
votes
1
answer
268
views
General form for likelihood of Cox process, from Diggle–Moraga–Rowlingson–Taylor
On page 4 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at arXiv, they claim the ...
- 25
1
vote
0
answers
68
views
(Anti-)concentration of gap between largest and second largest component of multivariate random gaussian vector
Let $n$ be a large positive integer and let $Y=(Y_1,\ldots,Y_n)$ be a zero-centered random $n$-dmensional real vector with covariance matrix $\Sigma$, an $n$-by-$n$ positive definite matrix with ...
- 6,853
2
votes
1
answer
241
views
Weak continuity of law
Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and weak topology. Let $X_t$ be a strong solution to the SDE with initial ...
- 5,405
5
votes
0
answers
130
views
Random process on a sequence of rolls of an $n$-sided die
Let $\ X:=X_{k\,n}\ $ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\ $ where $\ k\in\{1, 2, \ldots,n\}\ $ and $\ n\ $ are fixed. Let $t$ be a ...
- 83
2
votes
2
answers
206
views
non-homogeneous counting process
Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$....
- 31
10
votes
4
answers
680
views
The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
- 773