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# Questions tagged [stochastic-processes]

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

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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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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 $X: \Omega \times \mathbb{R}... 0answers 32 views ### Variance of a random variable obtaining from a linear transformation Edit: I needed to revise this question as it was suggested. Suppose there are$N$realizations of Gaussian process denoted as the vectors$z_{j} \in \mathbb{R}^{n}$for$j = 1, \ldots, N$. Let$y$... 1answer 89 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 ... 0answers 99 views ### 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$\hat{u}^n$,... 1answer 376 views ### 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 ... 1answer 232 views ### 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 ... 1answer 102 views ### 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, ... 2answers 73 views ### “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 ... 0answers 33 views ### 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 $$H = \{ c : [0,1] \to M \mid c(0) = x_0 \text{ and } \exists C>0 \text { s.t. } d(c(s), c(t)) \... 0answers 73 views ### The probability distribution of “derivative” of a random variable Disclaimer: Cross-posted in math.SE. Let me set the stage; Consider a stochastic PDE, which has to following form$$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$where H is a deterministic function, ... 2answers 82 views ### Backward Stochastic Differential Equation Let W_t be a standard Brownian motion. Let T be the terminal date, X_T=x, and$$ 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 ...

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