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Questions tagged [stochastic-processes]

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Geometric Brownian Motion With Mean Reversion

Is it possible to generate this mean-reverting Ornstein-Uhlenbeck stochastic differential equation $$ dx_t = \theta(\mu - x_t)\text{d}t + \sigma \text{d}W_t $$ in Excel? Are there any mean-reverting ...
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Expectation of random variable times stopping time

Let $(X_t)_{t\geq0}$ by a jump-diffusion process and define $\tau = \inf\{t\geq 0 : X_t \geq B \}$ the first time that $X_t$ exceeds $B$. Let $C>0$. I am having trouble understanding intuitively ...
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Intuitive and/or visual understanding Slyvnyak’s Theorem?

I am having difficulty in understanding Slyvnyak’s theorem. I am following a highly cited and seminal paper on Stochastic geometry for cellular networks titled, 'Stochastic Geometry and Random Graphs ...
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Terminology: Dynamic Bayesian network with hidden process

I came across a problem which can be modelled using a special type of dynamic Bayesian network. I'm looking for a name for this kind of network, but could not find anything so far. It resembles a "...
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18 views

Construction of Independent Brownian Motions

Let $(\Omega, \mathcal{F}, \mathbb{F} = (\mathcal{F}_t)_{t \in [0,T] },P) $ be a filtered probability space. Can we construct for all $n\in \mathbb{N}$ a family $ \{B^1, \dots, B^n \} $ of ...
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Constructing a stochastic process indexed by $\mathbb{Z}$

Consider $\{S_n\}_{n\ge 0}$ to be the simple symmetric random walk with $S_0=0$ and $S_n=X_1+\cdots+X_n$ where $X_i$ are iid and $P(X_1=1)=P(X_1=-1)=1/2$. Is it possible to construct a process $\{Y_n\}...
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covariance of the integral of brownian motion over disjoint intervals

Let $W_i = \int_{t_{i-1}}^{t_i} B(t) \,dt$ be the integral of Brownian motion over the time $t \in [t_{i-1},t_i]$. I'm reading a paper which says that $W_i$ and $W_j$ are independent Gaussian random ...
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Right Continuous Adapted Processes without right-continuous Filtration.

Let $\mathscr{F}$ be a filtration and $\mathscr{F}^+$ be the right continuous version. ($\mathscr{F} = \{\mathscr{F}_t\}_{t\in [0,T]})$ Suppose $X$ is right-continuous $\mathscr{F}$-adapted process. ...
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1answer
12 views

Relation between stationary distribution and derivatives of autocorrelation for Gaussian process

Let $x(t)$ be a stationary Gaussian process with derivative $v(t)$. Let the stationary distribution be $p(x,v) = e^{-A x^2 - B v^2}$. Let the normalized autocorrelation function of $x(t)$ be $\rho(t)$....
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essential supremum notation on stochastic processes

I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They use the notation $\mathbb{H}_{T}^{\infty}(\mathbb{R}^{d})$ for the space of $\mathbb{R}...
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How do I optimally sample to fit a function to data?

Setup I have categorical (true/false) data $H$ of whether a neuron fires an action potential (spike) under various stimulation conditions $z = [S, D]$ (strength, duration). The neuron is described ...
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1answer
21 views

Markov Chain Monte Carlo thermolization time estimation (not by eye)

For a given MCMC algorithm, there are two important time(=step) scale. $\tau_{thermolization}$ also known as burn-in time, intialization time. $\tau_{indenpendent}$ the time scale to make $X_i$ and $...
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Expected price change decomposition from finance: bogus or can it be made rigorous?

A relatively well-known finance book called Fixed Income Securities by Bruce Tuckman and Angel Serrat, features the following argument. Let the price of a security at time $t$, under a (random) factor ...
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1answer
31 views

Claim $\lim_{\delta \to 0} E\int_{|y| \le 1}\int_{\mathbb{R}^d}| u(x)-u(x+\delta y)|^2\rho(y) \,dx\,dy = 0$

Let $(\Omega,\mathcal{F},P)$ be a probability space, with $u \in L^2( \Omega \times\mathbb{R}^d)$. Let $\rho$ be the standard mollifier approximation to Dirac's delta dunction defined on $\mathbb{R}^d$...
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1answer
36 views

Two entrance times associated with Brownian motion are equal almost surely

Let $(W_t)_{t \in \mathbb{R+}}$ be a Wiener process and define two random times for fixed $a>0$. $S_a = \inf\{t > 0: W_t> a\}$ $T_a = \inf\{t > 0: W_t \geq a\}$ It is well known that $...
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68 views

Brownian motion on Ito's process [on hold]

Let $W={W_t:t≥0}$ be a Brownian motion on $(Ω,F,F=(F_t)t≥0,P)$. Fix $α,β∈R$ and consider the following SDE: $$dX_t=β−X_tT−tdt+dWt,0<t<T $$ and $$X_0=α,XT=β.$$ A solution to this SDE, with ...
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Covariance for stochastic variables

if $X$ and $Y$ are stochastic variables with $\operatorname{Var}(X)=1.34$ and $\operatorname{Cov}(X,Y) = 0.64$, find $\operatorname{Cov}(2X, 3X+2Y)$. No ideas on this one, as I don't see any way of ...
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1answer
27 views

Convergence in $L^2( [0,T] \times \Omega )$ implies uniform convergence

Let $X^n$ be a family of continuous stochastic processes such that $E[ \sup_{t \in [0,T} \mid X^n_t \mid ^2 ] < \infty $ for all $n.$ We assume that $$ \lim_{n \to \infty} E\left[ \int_0^T \mid X^...
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1answer
18 views

Branching process expectancy given an initial condition

Given a branching process {$X_n$} with offspring probabilities $p_0 = 1/6, p_1 = 1/3,$ and $p_2 = 1/2$, find ${E[X_2|X_0 = 10]}$. I know that $E[X_n|X_{n-1}] = X_{n-1}\mu$. I tried stating that $E[X_2|...
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23 views

Is my stochastic process wide sense-stationary?

If I have a process Xn that is made of two sequences of independent random variables. For n being even , Xn is either +1 or -1 with probability 1/2 , but for n odd , Xn is either 1/5 or -5 with ...
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1answer
45 views

Compute expectation of stopped Brownian motion

Let $B_t$ be a standard Brownian motion starting from $0$. Let $\tau_a$ be the hitting time of Brownian motion hitting $a$ and $a>0$. I want to calculate $E[X_T] = E[B_{T \wedge \tau_a}]$ with $X_t$...
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1answer
23 views

Example of a stochastic non Markov process?

A Markov process is defined as: $$P(X_t| X_{1:t-1}) = P(X_t|X_{t-1})$$ Is there a non Markov process that can be generated by a computer and cannot be converted to a Markov process by changing the ...
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May I ask why the following operator between Banach spaces is continuous please?

I am reading the paper "On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces" (Here is the link for electronic version). But when reading it, I meet a problem and would ...
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1answer
49 views

Let $W=\{W_t:t\geq0\}$ be a Brownian motion. Find $\operatorname{Var}(W_1^3)$

sorry for my newbie question. I'm learning stochastic process on my own and I got confused about a lot of concepts and notations. I need your help with this small exercise: Let $W=\{W_t:t\geq0\}$ be ...
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Levy construction of Brownian motion by Haar function and Schauder function

For every $t \in [0,1]$, we set $h_0(t) = 1$, and then, for every integer $n \geq 0$ and every $k \in \{0,1,2,...,2^n-1\},$ $$h^n_k(t) = 2^{n/2}\mathbb{1}_{[(2k)2^{-n-1},(2k+1)2^{-n-1})}(t) - 2^{n/2}\...
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1answer
37 views

Two Brownian motions and stopping time

X and Y are two independent Brownian motions both equal to $1$ at time $0$. Consider the first time, T, that Y process hits $0$. What is the probability that $X_T>0$? My intuition: I think that at ...
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1answer
20 views

Probability bound for the tail of a random serie

Let $X_n$ a random sequence such that $E(X_n) = 0$ and $E(X_n^2)= 1$ If $a_n$ is a positive decreasing sequence such that $\sum_{n=1}^\infty a_n < \infty$. Is the following statement abount the ...
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1answer
33 views

Gambler´s ruin problem

A gambler starts with an initial fortune of 9. He wins 1 with p=1/3 and losses 1 with q=2/3. The game ends when the gambler looses all of his money or when we reaches 15. What is the probability ...
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1answer
29 views

Basic stochastic inequality

Let $X$ and $Y$ be independent random variables. Prove that $$P(X \leq x\mid X >Y) \leq P(X \leq x).$$ I try to use the definition of conditional probability, but I can't get success. $$P(X \leq x\...
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Integration of the Jump part of SDE

Let $L_t^{\alpha}$ be the alpha stable Levy motion, I am not sure how to compute the integral $\int_{t_{i-1}}^{t_i} dL_t^{\alpha}$.
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Finding stationary distribution of a stochastic process

Consider the following SDE for $z_t = (x_t,v_t)\in \mathbb{R}\times [-1,1]$: $$dx_t=-\mu x_t dt+ v_0 v_t dt + \sigma dW_t,$$ $$dv_t=-\frac{a^2}{2} v_t dt - a\sqrt{1-v_t^2} dB_t,$$ where $\mu, v_0, a, ...
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1answer
27 views

Is the finite difference of an iid discrete-time stochastic process also iid, or a Markov process?

Let $\{X_n\}$ be a discrete-time stochastic processes, i.e. a sequence of random variables $X_n,\ n \in \mathbb{N}$. Assume that the $X_n$ are independent and identically distributed (iid). Let $\{Y_n ...
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1answer
27 views

On the convergence of a sequence of random variables indexed by random variables

Let $X_k$ be a decreasing and uniformly bounded sequence of nonnegative random variables which is completely determined by $\mathcal F_k$ (not necessarily $\sigma(X_1,...,X_k)$). Mathematically $ X_{...
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62 views

Is stochastic process a martingale?

Is the stochastic process $$ X(t):=W\left(\int_0^t f(W(s))\mathrm{d}s\right) $$ a martingale? Here, $f(x):\mathbb{R}\rightarrow\mathbb{R}^+$ is a positive function and $W(s)$ is a standard Brownian ...
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Distribution of $T_k$ where $T_k - T_{k-1}$ is a Geometric with parameter p

Consider the stochastic process $\{X_n, n=0,1,...\}$ a Bernoulli process with parameter $p$. Let $S_n = \sum_{i=1}^{n}X_i $, with $S_0 = 0$. Define $\{T_k, k=0,1,...\}$ by $$T_k = min_{n}\{S_n >= k\...
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38 views

Dealing with different definitions of the Ornstein–Uhlenbeck process

I've run up against a wall in reconciling two different definitions of the Ornstein–Uhlenbeck process, and would appreciate some help. On the one hand, as discussed here, we can define an Ornstein–...
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1answer
69 views

Distribution of random variable given by SDE at some point in time

Suppose we have an stochastic differential equation given by, $$\mathrm{d}X = N(X,t)\,\mathrm{d}t + M(X,t)\,\mathrm{d}B,$$ where $B$ is a brownian motion. As far as I understand we can think of this ...
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What's the transition semigroup of the Markov chain generated by the Metropolis-Hastings algorithm?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,\mathcal E)$ be a measurable space $Q$ be a Markov kernel with source and target $(E,\mathcal E)$ $x_0\in E$ $\alpha:E\times E\to[...
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Markov Property fos Ising type Models

We are interested in proving the Markov property for the long range Ising type model in $\mathbb{Z}^d$. Setting: Define $\Omega = \{-1, +1\}^{\mathbb{Z}^d}$ the space of all possible configurations ...
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1answer
61 views

A curious variant of the classical 2D random walk: allowing duplication and vanishing

Background. Recall the standard random walk on a 2D grid (i.e. $\mathbb{Z}^2$). A person starts at the origin. At every iteration, the person moves in one of the four directions (up, down, left, or ...
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Reversible distribution of a CTMC on a graph, where the edges are cut according to another process.

Hi : I am trying to formulate a theorem i know to be true, but i cant find the proof anywhere, could anyone finish my proof or point me in the direction of the theorem stated properly. THEOREM : ...
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1answer
25 views

(Poisson process) probability of least one arrival of $N_2$ and $N_3$, between 2 successive arrivals of $N_1$.

Let $N_1(t),~N_2(t),~N_3(t)$ be independent Poisson processes of rates $\lambda_1,~\lambda_2,~\lambda_3>0$, respectively. Evaluate the probability between two successive arrivals of $N_1$ we get ...
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Solution of the differential equation $\ddot{x}(t)+\sin(\omega t)x(t)=cos[\eta(t)]$

The differential equation: $$\ddot{x}(t)+\sin(\omega t)x(t)=cos(\eta t)$$ has an analytical solution involving Mathieu functions. This is valid if both $\omega$ and $\eta$ are constant. Suppose the ...
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1answer
51 views

Fractional Brownian Motion and Fractional Laplacian

It is well known that the Laplacian is the infinitesimal generator of a Brownian Motion, that is, $$ \lim_{t \to 0} \frac{E[f(x+B_t)-f(x)]}{t}= \Delta f(x). $$ Is it true that for the Fractional ...
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1answer
16 views

Isonormal Gaussian process

I am reading Nualart's book The Malliavin Calculus and Related Topics and there is some issues that I am stuck wtih. Let $H$ be a real separable Hilbert Space with. A stochastic process $W=\{W(h)\;;h\...
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Time to first jump for doubly stochastic poisson process (cox process)

Let $N_t$ denote a doubly stochastic Poisson (Cox) process with intensity process $\lambda_t$. Lando (1998) defines the time to first jump $\tau$ as: $$\tau = \inf\{ t: \int_{0}^{t} \lambda_u du \geq ...
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131 views

Analysis of super-structures emerging in a spiral representation of prime numbers

The fact that each prime number (greater than $9$) ends with one of the four digits $1,3,7,9$, allows us to classify the tens in which the primes are found according to which of these four digits, ...
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2answers
32 views

When does the conditional expectation of the sum of random variables match with the sum their respective conditional expectations?

I am studying stochastic processes. While studying random walk I acquainted with a notation $N_i$ where $$N_i = \mathrm {Total\ number\ of\ times\ of\ visit\ to\ i}.$$ Let $(X_n)_{n \geq 0}$ be a ...
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1answer
18 views

Finding the probability that $P \{W(2)> |W(1)|\}$, where $W(t)$ is the standard Wiener process. Brownian motion.

I know that the increments of the Brownian motion are independent, and it looks like that might be of use in calculating this probability. I thought that this might be the case: $$P\{W(2)> |W(1)|...
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1answer
36 views

If $X_n$ is a martingale with respect to $\{Y_n\}$ then does this hold with respect to $\{Y_n^2\}$

To me this is a bit of a curve-ball as we usually only deal with the form $E[\cdot \mid Y_n]$. The usual way to go about would be to prove $E[X_{n+1} \mid Y_n^2] = X_n$. I do not quite know where to ...