# Questions tagged [stochastic-analysis]

For questions about stochastic analysis or stochastic calculus, for example the Itô integral.

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### How to analyze a particular service queuing model?

If a job service system schedules queued jobs in the system at fixed time intervals (such as 1 hour) to execute (possibly multiple jobs at once), each job runs for 1 hour, and the number of jobs ...
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### numerical integration of a function satisfying a ode

I need to numerically approximate an integral of the form $$\int_0^\tau f(X_t)\:{\rm d}t,\tag1$$ where $(X_t)_{t\ge0}$ is the solution of a SDE $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag2.$$...
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### Numerical solution to non-linear ODE (not stochastic DE) with one normally distributed initial condition

Are there any books, research on solving a non-linear ordinary differential equation (not a stochastic differential equation) with one normally distributed initial condition? One simple way would be ...
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### Help me understand this proof of "the covariance of a Gaussian measure is trace-class"

So I am reading an introductory script on stochastic analysis in Hilbert spaces and there is a step in the proof of "Gaussian measures have trace-class covariance" that I don't understand: ...
1 vote
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### Local Lipschitz continuity and explosion time in SDE.

I am self-studying the following material, whose source is https://sayanmuk.github.io/StochasticAnalysisManifolds.pdf However, I am stuck with one step of the proof of the only theorem that follows ...
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### Extending Schilling's proof of Ito process approximation by simple processes for one-dimensional case to multivariate case

Below is the proof of Lemma 18.5 from Rene Schilling's Brownian motion which states that an Ito process can be approximated uniformly in probability by a simple Ito process. Now it is stated in the ...
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### Expectation Value of the Product of a Time integral and a Ito Integral

Consider a stochastic process $X_t$ \begin{equation} dX_t = a(X_t)dt + \sigma dW_t \end{equation} where $W_t$ is a Wiener Process. I know the expectation value of the product of two stochastic ...
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### Is there a measure theoretic interpretation of rough path integrals?

In case of an almost surely continuous function $f$ we know that the Lebesgue Measure coincides with the Riemann Integral. With this in mind, supposing $\mathbf{X}$ is a rough path lift of $X$, is ...
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### Can we show that a (random) time discretized Markov process is still Markov?

Let $(E,\mathcal E)$ be a measurable space; $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$; $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a ...
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### Is there a continuous-time Markov process whose generator domain is not contained $C^2$?

Let $(X_t)_{t\ge0}$ be a Hilbert space $H$ (take $H=\mathbb R^d$, for simplicity) valued time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. The latter can be considered as ...
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### White noise with non-constant variance

Is there a name for white noise that has non-constant variance? I have some examples from experimental data where the variance of the white noise increases with time. However, I am not sure how to ...
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Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u\in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$ I was wondering if there ...
Find parameters $a,b,c$ for process $aW_t^2+cW_{bt^2+a}$ to make it gaussian. The process is gaussian if  E \left [ \exp \left ( i \sum_{l=1}^{k} s_l Y_{t_l} \right ) \right ] = \exp \left ( -\...