All Questions
11 questions
5
votes
1
answer
319
views
Spherical average of $\frac{1}{x}$
Let $X_1,...,X_n$ be points on $\mathbb S^1.$
We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$
Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
- 124
0
votes
1
answer
55
views
Looking for a family of random variables such that only the second clause is fulfilled [closed]
Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if
i) $sup_{i \in I} E(X_i) <\infty$
ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....
- 11
6
votes
1
answer
684
views
Differentiable dependence on the initial condition of the solution of a SDE
Let
$b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-...
- 167
0
votes
1
answer
102
views
Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
- 536
6
votes
1
answer
433
views
Triangle inequality for Ito integral?
For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$
Now if ...
- 536
2
votes
1
answer
803
views
On Riemann integration of stochastic processes of order $p$
Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
user85801
2
votes
1
answer
545
views
Multiple Wiener-Ito integral distribution
Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$
Is it possible to find the distribution of multiple Wiener-Ito ...
14
votes
1
answer
2k
views
Why do we mainly integrate with respect to martingales?
Although my resarch focuses on PDEs (optimal transport, these days), I am currently trying to learn stochastic calculus and integration. I am just beginning in this topics, but I was wondering: why do ...
- 5,380
2
votes
1
answer
438
views
A dilemma about the definition of the stochastic integral $\int_a^b\Phi\:{\rm d}W$
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$T>0$
$(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration of $\mathcal A$
$W$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\...
- 167
1
vote
0
answers
251
views
Inflated independent samples for Monte Carlo estimation
In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
- 101
2
votes
2
answers
407
views
How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?
Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$
where $A_i \sim \text{exp}(\lambda), S_i \sim ...
- 139