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

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Keeping time by randomly drifting a $q$-ary string

Imagine I have a string $s$ of length $L$ encoded over an alphabet of size $q$, e.g. $s = 000101$, where $L = 6$ & $q = 2$. For each of $T$ time intervals, $(t_1, ..., t_N) \in T$, I select a bit ...
3
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1answer
256 views

Casino does not win, while clients do lose ? Prob_loss(T) = 1 - .8/sqrt(T)?

Setup. Let casisno generate a color: black or red with equal probability. Let client try to guess the color. If guess is correct - he earns 1 coin from casino, if not - he gives one to casino. If he ...
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1answer
142 views

Colored noise in SDE

I want to numerically study the behavior of a system described by a set of differential equations in the presence of colored noise. It seems that the standard procedure is to use the Langevin ...
5
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662 views

One can earn nothing on the Brownian motion, true ?

Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$). Consider some "trading strategy" ...
2
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1answer
78 views

How to simulate random paths of a non-homogeneous continuous-time Markov process with discrete state space for a given infinitesimal generator matrix?

Let $X=(X_{t},t \in T)$ be a non-homogeneous, continuous time Markov process with a finite state space S={1,...,K}. Let $\alpha_{i,j}(t)$ be the hazard rates of some $\varGamma$-distributed random ...
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63 views

Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
2
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1answer
189 views

Conditional law of an Ito's stochastic integral

Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real ...
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413 views

An “inchworm-like” random walk on an integer interval

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the ...
4
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0answers
87 views

How fast is discrete-time diffusion on a continuous set?

This question is inspired by Joseph O'Rourke's beautiful answer to my previous question. Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...
7
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3answers
767 views

Blue and red balls puzzle

I was sent this puzzle and wondered if it is known or if its origin is known? (I see colored ball puzzles are also in vogue.) Consider a bag with $n$ red balls and $n$ blue balls. At each turn you ...
4
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186 views

Inadmissibility of Simpson's rule

(An earlier version of this at stackexchange got no answers.) Bayesianism says that all uncertainties, or at least all uncertainties about the truth or falsity of propositions, can be expressed by ...
3
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2answers
282 views

Probability distribution for two-state system that depends on residence time

I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...
2
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0answers
59 views

Tail for the integral of a diffusion process

I would like to compute the following tail, $$ \mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right), $$ assuming $$ \mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x), $$ and $X$ is a diffusion ...
3
votes
1answer
145 views

Optimizing a stochastic “flip and prune” procedure for selecting a subset of coins

I place some number of coins, $(c_1, ..., c_N) \in C$ on a table, where each coin is originally tails up. Let's call the "tails" state $0$ and the "heads" state $1$. I then perform the following ...
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1answer
329 views

Mathematical properties of financial prices

Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes. What is known about their mathematical properties ? I know ...
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0answers
104 views

Langevin equation with position-dependant damping: existence of an invariant measure?

The usual Langevin equation for a particle in a 1D harmonic potential $dq(t) = p(t)~dt$ $dp(t) = -q(t)~dt + a ~dW(t) - b~p(t)~dt$ admits as an invariant measure the Gibbs measure ${1\over ...
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370 views

When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb ...
3
votes
1answer
128 views

“Trapping” of discs after random sequential adsorption

Imagine I perform Random Sequential Adsorption (RSA) of discs of some radius $r$ on $[0, 1]^2$, eventually covering the surface to some density $Q \leq 0.543$ with $N$ total discs (where $\approx ...
4
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2answers
97 views

Simulating random sequential adsorption in reverse

Please consider two processes: Process 1 - I simulate random sequential adsorption of discs on the unit square in the continuum limit, randomly selecting real number coordinates and rejecting the ...
16
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2answers
690 views

Age of Stochasticity?

One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here. The question is this: Today ...
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0answers
92 views

A simplified MCMC / MH algorithm. Are there known convergence results?

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
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163 views

Spectral densities and their corresponding covariance functions.

Hey guys, I'm currently doing a course in stochastic processes and have come across something that has been wrecking my mind for a while. So, let's say that I have some even, symmetric function ...
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2answers
155 views

Strictly positive definite autocovariance function of fGn

Hi, let $\gamma(k) = 1/2 (|k+1|^{2H} + |k-1|^{2H}-2|k|^{2H}),k\in\mathbb{Z},$ be autocovariance function of fractional Gaussian noise where $H\in(0,1)$ is parameter. I want to show that $\gamma$ is ...
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1answer
87 views

Empirical distribution of a collection of iid Markov chains

Suppose we have $N$ independent 2-point Markov chains each having a rate matrix $Q = [-1,1;1,-1]$ and stationary distribution $\pi = [0.5,0.5]$. At time $t=0$, we initiate the chains so that the ...
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1answer
69 views

Random Sequential Adsorption of Discs on a Plane - What is the best known lowerbound for the number of circles (of some radius $r$) guaranteed to fit on $[0, 1]^2$?

Imagine I perform a random sequential adsorption (RSA) simulation for circles or discs of some radius $r \leq 1$ in $[0, 1]^2$ (I am open to changing this geometry to the unit circle). As a function ...
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0answers
261 views

How is Kolmogorov forward equation derived from the theory of semigroup of operators?

In Lamperti's Stochastic Processes, given a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$, a ...
5
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1answer
193 views

Memory of Uniformly Random Dyck Paths

Let $D$ be the set of all Dyck paths on square grid of size $n\times n$. For any particular Dyck path, let $S(t)=X_1+X_2+\ldots +X_t$ store the path, where $X_i=\pm 1$. Being a Dyck path, we have ...
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1answer
294 views

Iterated Ito Integral, Gaussian Volterra Process

Let me define $$ J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1} $$ where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice deterministic ...
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97 views

Time integral of a diffusion

Define $\bar\sigma^2_t=\frac{1}{t}\int_0^t\sigma^2(X_s)ds$ where $\sigma(x)\geq0$ is a measurable function and $X_t$ a diffusion process defined by \begin{equation} ...
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84 views

Attractors and solutions to these generalized Ornstein–Uhlenbeck processes?

This is a question about generalized Ornstein–Uhlenbeck processes I asked on MSE, but I haven't received replies about their attractors and solutions yet. So I would appreciate if someone could give ...
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111 views

Adding a damping term to a dynamical system or Markov process: what happens to invariant measures?

Consider the continuous-time Markov process on ${\mathbb R}^n$ described by the SDE $\dot{x}(t) = F(x(t)) + \xi(t)$ where $F:{\mathbb R}^n \to {\mathbb R}^n$ is a smooth mapping, and $\xi(t)$ is a ...
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1answer
490 views

Hitting time probability in a Random Walk with possibility to die.

A Random Walker can move of one unit to the right with probability $p$, to the left with probability $q$ and it can jump again to the starting point with probability $r$ and die. Naturally $p+q+r=1$. ...
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0answers
151 views

Have you seen this one parameter family of distributions before?

This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the ...
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0answers
160 views

What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?

There is a theorem as follows: Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto ...
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1answer
257 views

Is this probabilistic principle for stochastic processes known?

In the course of a proof, I used the following principle, which seems so intuitive that it should have a name: Suppose one has a stochastic process $X_t$, for $t \in \omega$, on a (possibly infinite) ...
3
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2answers
793 views

stopping time expectation for gambler's ruin

2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin. I let $X_n = x + \sum_{i=1}^{n}\xi_i$ where $\xi_i$ are i.i.d. ...
5
votes
2answers
215 views

Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to ...
2
votes
1answer
81 views

Maximal probability of “infinitely often” over MDP

Let us consider a Markov Decision Process (MDP) with a Borel state space $X$. Often, the optimization problems over MDP involve optimization of some objectives dependent on the reward function $$ ...
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0answers
51 views

Sufficiency of stationary policy for negative stochastic dynamic programming

Consider a Markov Decision Process with Borel state space $X$ and Borel action space $U$, like the one defined in the book "Stochastic Optimal Control: Discrete-time case" by Bertsekas and Shreve. All ...
0
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1answer
122 views

Learning stochastic calculus, want to know what the notation of this function means

$f(x) = \sum_if_i 1_{[ai;bi)}(x)$ This is a function that is piecewise constant equal to f_i on finite set of intervals [ai; bi) in a set F: I am a little confused about what this 1 in the summation ...
2
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3answers
286 views

Ito formulae for stochastic processes with finite cubic, quartic … n-tic variation

Many stochastic processes that you encounter are kind of well-behaved, i.e. have infinite variation, yet finite quadratic variation. My question revolves around stochastic processes that have ...
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583 views

asymmetric random walk, hitting time probability

Let's consider an asymmetric Random Walk on $Z$, with transition probabilities $p_{i, i+1}=p$, $~~p_{i, i+1}=q$, $\forall i \in \mathcal{Z}$, $p+q=1$ and $p>q$. I am interested in the probability ...
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1answer
99 views

Nonstandard definition for the generator of a standard Ito diffusion

For a standard Brownian motion, the generator of the diffusion is $$ L = \frac12 \frac{d^2}{dx^2}. $$ Is there a nonstandard definition of this generator?
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139 views

Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence, a noise is black if it is not isomorphic to standard Gaussian white noise. Tsirelson showed the existence of black noise through the scaling limit ...
0
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1answer
195 views

Reflected Brownian Motion

Let $Y(t)$ be a reflected Brownian motion, and $G(t)$ is the process which keeps count of number of times that $Y(t)$ has hit the X axis. How do I approach to find distribution of $G(t)$, or almost ...
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94 views

Trying to get an idea of the maths I could use for this optimization problem

Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this ...
4
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2answers
156 views

If $\mathcal{F}_t$ is separable why is $\mathcal{F}_\infty$ generated by a random variable?

I am reading this introduction to enlargement of filtration and at the beginning of section 2.4 there is a claim that I cannot justify but seems like it should be well known. The author claims that ...
4
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1answer
95 views

A terminal coalgebra of a certain functor on Mes

Let $\mathfrak C = \mathsf{Mes}$ be the category of meausurable spaces and measurable maps. For any object $X\in \mathfrak C_0$ we assign a measurable space $\mathcal P(X)$ whose elements $\mu$ are ...
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1answer
133 views

Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...
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1answer
239 views

On martingale representation theorem

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\mathcal{F_{t}})_{0\le t\le T}$ a filtration generated by standard Brownian motion $W_t$. Let $f(x)$ be $C^1$ function such that $|f'(x)| ...