# Questions tagged [stochastic-processes]

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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questions

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### De Finetti-style theorem for Point Processes

I am new to point processes. I know there are a number of theorems along the lines that if a point process $\eta$ satisfies:
Complete independence (the random variables $\eta(B_1), \ldots, \eta(B_n)$...

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### Estimating the error of a one-step weak approximation using Taylor

Let $u\in C^{l+1}$ have derivatives of polynomial growth, $X^q,Y^q$ be stochastic processes in discrete time with values in $\mathbb R^d$ for every $q\in Q$ and $L : Q \to \mathbb R$ be a function, ...

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### Mean field games approximate Nash equilibria

I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.
I have a question about a step in theorem 3.8 on page 17. Let ...

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### A probability density function for pink noise

I would like to understand pink (or $1/f$) noise better. However, clearly written resources are difficult to come by, and are usually concerned with its Fourier spectrum, or qualitative descriptions ...

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### Predictable Projection of a Stopped Process (Typo in Jacod & Shiryaev?)

Given a filtered probability space $( \Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$ and an $\mathcal{F} \otimes \mathcal{B}(\mathbb{R}_+)$-measurable bounded process $X: \Omega \times \mathbb{R}...

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### Variance of a random variable obtaining from a linear transformation

Edit: I needed to revise this question as it was suggested.
Suppose there are $N$ realizations of Gaussian process denoted as the vectors $z_{j} \in \mathbb{R}^{n}$ for $j = 1, \ldots, N$. Let $y$ ...

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### 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 ...

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### Random process - autocorrelation

I am currently working on random processes.
Let's consider the random process defined as
$u^s(x,t) = 2\sum_{n=1}^{N} \hat{u}^n \cos(\kappa^n\cdot x + \psi_n + \omega_n t )\sigma^n$
where $\hat{u}^n$,...

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### A theorem by Harald Cramér?

In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:
Suppose $\{X_n\}_{n=2}^\infty$ is a sequence of ...

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### Is this a “contradiction” on stochastic Burgers' equation? How to understand it?

For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...

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### Stochastic processes and continuity of expectation

Let $X$ be a stochastic process with a.s. continuous sample paths on $[0, 1]$ such that $\mathbb E [X_t]$ is finite for all $t \in [0, 1]$. Given any non null subset $Y$ of the probability space, ...

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### “Сross сubic variation” of two Brownian motions and interpretation of the simulation result

Consider two independent 1-dimensional Brownian motions $W_{t},B_{t}$, with an equidistant partition of the interval $[0,T]$, and $n\Delta≡T$.
How to calculate the expression below? Can we rewrite ...

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### What is the Wiener measure of the set of curves with given Hölder constant on a Riemannian manifold?

Let $M$ be a connected Riemannian manifold and $x_0 \in M$. For $0 < \alpha < \frac 1 2$, let
$$H = \{ c : [0,1] \to M \mid c(0) = x_0 \text{ and } \exists C>0 \text { s.t. } d(c(s), c(t)) \...

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73 views

### The probability distribution of “derivative” of a random variable

Disclaimer: Cross-posted in math.SE.
Let me set the stage;
Consider a stochastic PDE, which has to following form
$$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$
where $H$ is a deterministic function, ...

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### Backward Stochastic Differential Equation

Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and
$$
dX_t=f_tdt+B_tdW_t
$$
where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...