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-1 votes

A simple procedure to simulate multifractional Brownian motion paths

The fractional Brownian motions you generate must not be independent. They are generated from the same Gaussian field $(B_H(t))_{(H,t) \in [a,b] \times \mathbb{R}}$ and there exists a dependence ...
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  • 1
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Is every compound Poisson distribution a mixed Poisson distribution?

(This answer considers mixed/compound Poisson processes instead of mixed/compound Poisson distributions.) I think that the only processes that are mixed Poisson and compound Poisson are ordinary ...
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  • 91
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M/G/1 queue as a Markov renewal process: one-step transition probabilities

Case 1: leaving behind at least one job Suppose first that when job $n - 1$ departs, it leaves behind at least one other job, meaning $I_{n - 1} = i \geq 1$. Then job $n$ is already in the system when ...
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  • 91
1 vote

Law of OU process with time-dependent dynamics

As (assuming $\Sigma\in L^2([0,\infty))$ $$ m_t=\mathbb{E}[X_t] = x + \int_0^t [M_s^1+M_s^2\mathbb{E}[X_s]] ds $$ by the fundamental theorem of calculus, the right hand side is differentiable in $t$, ...
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  • 105
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Short time limits for SDE

No, for example, take $dX_t = 2 dB_t$. By the reflection principle the conditioned process is symmetric around 1, but $Y_1 = 2$.
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  • 656
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Connection between invariant measure and positive recurrence for continuum state space markov chain

Yes, let's assume that sup (which I think should be in the denominator in the last expression) is finite, as otherwise it is always true. Suppose it is bounded by A, the each return time is also ...
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5 votes
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Large noise limit for SDE with general volatility coefficients

The answer is no, as can be seen in the case $\sigma(u) = u$, so that $X_t = \exp(W_t - t/2)$. For the result to be true, Markov's inequality implies that the law of $W$, conditional on $A_M$, would ...
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6 votes
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Why is every Gaussian process a linear process?

$\newcommand\ep\varepsilon\newcommand\si\sigma\newcommand\N{\mathbb N}$Your counterexample is correct. Indeed, if $$X_t=\sum_{j=0}^\infty a_j \ep_{t-j} \tag{1}\label{1}$$ for $t\in\N$, $\sum_{j=0}^\...
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1 vote
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Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)

For $0<\beta \le 1/2$, any limiting distribution of the rescaled process $S_n/n^\beta$ will be fully supported in $[-1,1]$, so it will not be normal. The downward drift will imply (using Hoeffding'...
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2 votes
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Is the solution to this SDE always positive?

To complement Nawaf's answer (1 2), I thought I'd present a short argument how with some additional regularity assumptions, the answer is in fact yes. Suppose that $\sigma\in{C^{2}(\mathbb{R})}$ with $...
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1 vote
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Using gradient descent in probability case

Lets first assume $\zeta_i=0$ and ask the following: under which conditions $w^*$ is a stable fixed point? If it's not a stable fixed point for noise-free case, then you won't end up with a fixed ...
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1 vote

Step in proof of Itô formula

I believe you can condition on $F_0$. Under the regular conditional probability induced by $F_0$, $a_0$ and $b_0$ are deterministic, and of course $w$ is stil a Brownian motion. Then you can apply the ...
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2 votes

Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes

Let's assume that $P$ has finitely many ergodic components so that $P = \sum_{i=1}^m p_i P_i$ with $P_i$ ergodic and $\sum p_i = 1$. Take $X_i$ to be independent stochastic processes with $X_i \sim ...
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1 vote

Construction of SDEs that admit more than one solution

If the diffusion coefficient is not continuous but uniformly positive, there is a general pathwise uniqueness result for homogenous SDE coefficients: Theorem (Pathwise Uniqueness). Suppose the SDE ...
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1 vote

Is every compound Poisson distribution a mixed Poisson distribution?

Q: Is every compound Poisson distribution a mixed Poisson distribution? A: No. Every mixed Poisson distribution has a variance greater than or equal to the mean. The compound Poisson distribution is ...
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2 votes
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Invariance principle: Brownian bridge and random walk conditioned on end point

A more general theorem is proved in [1] for the limits of random walks in the domain of attraction of a stable law. In the case described in the problem, one can also use the strong approximation ...
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2 votes
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A comparison principle for SDE

Don't think so. Take both to satisfy something like $dZ = \sigma |Z| dW$, start one from -1, and one from 1. regardless of the exact parameters, one stays positive and one stays negative.
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asymptotic estimate of random walk involved hitting time and return time

I know for a fact that when $0\leq a\leq b$, we have $\mathbb{P}_a[\tau_0 < \tau_b] = 1-\tfrac{a}{b}$. So my guess is that your probability is equivalent to $\tfrac{1}{m}$ as $m\to+\infty$.
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3 votes
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Solution of SDE with time power law singular diffusion

It seems natural to view the Itô process on a different time scale where it becomes a Itô diffusion, and in turn, invoke existence/uniqueness theory for Itô diffusion. In particular, under the (...
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0 votes

What is the significance of Blumenthal and Getoor's result on the boundedness of paths of a standard Markov process?

It's a simple fact, but not quite as simple as you claim. For example, the function $$ x: [a,b] \to \mathbb R\cup\{\pm\infty\} : t\mapsto \frac 1 {b-t} $$ is càdlàg but not bounded. The rough ...
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0 votes

If $X$ is a Markov process, can we find a mild assumption ensuring that $\frac1t\operatorname E_x\left[\int_0^tc(X_s)\:{\rm d}s\right]\to c(x)$?

I don't have enough reputation to comment, this is just an idea. If you assume that$^1$ $E$ is LCCB, $X$ is càdlàg, $X$ is strong Markov, and $X$ is quasi-left continuous, then for any $t$ the set $$...
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