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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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      80 views

      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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      30 views

      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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      59 views

      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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      104 views

      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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      1answer
      41 views

      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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      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 ...
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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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      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 ...
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      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 ...
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      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, ...
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      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 ...
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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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      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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      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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