Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
14 views

Poisson point process in 2D with reflecting boundaries

Consider a point process $\{(X_n,T_n)\}$ on a plane $[0,1/\lambda]\times\mathbf R^+$, generated from a Poisson point process $\{T_n\}$ with rate $\lambda$ on $\mathbf R^+$ (i.e. $(T_n-T_{n-1})$ is iid ...
-2
votes
0answers
10 views
1
vote
0answers
20 views

Computing a Diffusion Limit of a Markov Chain

Fix $\alpha >0$, and for $h > 0$, consider the Markov kernel $K_h$ derived by composing the following two `moves': From $x^t$, move to $x^{t+1/2} = x^t + y^t$, where $y^t \sim \text{Gamma}(h, 1)...
0
votes
1answer
35 views

Probability mass function from a generating function

I have the generating function $G_x(\theta) = \frac{\alpha-1}{\alpha-\theta^2}$ and I am trying to determine the probability mass function. I believe I need to determine the Taylor series expansion ...
0
votes
1answer
32 views

Is there a probabilistic Turing machine whose halting probability is non-measurable?

I was wondering if there a probabilistic turing machine whose halting probability is non-measurable? By that, I mean that the probability measure applied to the event "the machine halts" is undefined.
1
vote
0answers
37 views

Convergence (in distribution) of Markov process

For each natural number $N \geq 1$, let $X^N = (X^N_k)_{k = 1}^{\infty}$ be an homogeneous Markov process taking values in $R_+$, with transition kernel \begin{equation} p_N(A|X^{N}_{k-1}) = P[X^{N}_{...
0
votes
0answers
15 views

Modelling Random Variables with Specified PDF and Correlation

I am trying to develop a radar simulation system that is able to generate random processes whose elements are taken from a specified probability density function and have also have a specified ...
3
votes
1answer
25 views

Is the infinitely distributed lag process strong mixing?

Suppose the process $\{X_{t}:t\in Z\}$ is absolutely continuous distributed and strong mixing with the coefficient $\alpha(s)$ defined as \begin{equation} \alpha(s) \equiv \sup \{ |P(A\cap B) - P(A)P(...
-1
votes
1answer
27 views

Do we have that $\mathcal{F}_{\infty} = \sigma(X_{t} \colon t \geq 0)$? [on hold]

If $(\mathcal{F}_{t})_{t \geq 0}$ is the filtration generated by a process $(X_{t})_{t \geq 0}$, one typically sets $$ \mathcal{F}_{\infty} : = \sigma \Big( \bigcup_{t \geq 0} \mathcal{F}_{t} \Big). $$...
0
votes
0answers
17 views

Minimum of Super Martingale and a Constant is Still a Super Martingale

I am having trouble proving the following statement: Let $\alpha>0$ and $M:\Omega\times [0,1]\longrightarrow \mathbb{R}$ be a super martingale. Then $M\wedge \alpha$ is also a super martingale. ...
0
votes
0answers
16 views

Kolmogorov backward equation intuition

The Kolmogorov backward equation equation states that the probability density of a random variable $x$ which follows $dx= \mu dt + \sigma dw$ is given by $-p_t = \mu p_x + 0.5\sigma^2 p_{xx} $ ...
1
vote
0answers
13 views

Forecasting of stationary time series: $WN(0,\sigma^2)$

I am trying to solve the next problem: Let the time series $X_n, n ∈ \mathbb Z$ be a $WN(0, σ^2)$. Find an optimal (in mean square sense) predictor for $X_{n+1}$ if you can observe: 1) $X_n$, 2) $X_{...
1
vote
0answers
33 views

“multiple” of markov chain properties

Let $A_n$ be a markov chain with transition matrix $P$ and $B$ be a chain defined as $B_n = A_{mn}$ for some positive integer $m$. First, I was able to show already that $B_n$ is also a markov chain ...
0
votes
0answers
20 views

Check for stationarity $X(t) = V * \cos(wt+U)$, V and U are indep. rand. variables

Can somebody help me please to figure out how to solve this problem: Are the following process stationary? $X(t) = V \cos(wt+U), t\epsilon R^1$ U and V are independent variables, $U$ ~ $Uniform(-...
1
vote
0answers
28 views

Factorization in the three variable function

Is there any way to estimate the $f(\theta)$ in the following equation? $exp(\lambda_1 \times cos^2\theta) \times exp(\lambda_2 \times cos\theta) = f(\theta) \times exp( \lambda_1+\lambda_2) $ $\...
0
votes
1answer
38 views

Variant on Gambler's ruin problem?

Suppose I started with $W_1=1$ and decided on a ratio $0<\alpha<1$ such that I invest $\alpha W_n$ of my earning in the next round. I either lose my investment, or win it with probability $p=9/...
0
votes
0answers
12 views

Find pdf of rectangular signal with random variable as width

I have a random process which is $\pi_p(t)$ which is the rectangular function in the interval $[0,p]$. $p$ is a uniformly distributed random variable over interval $[0,10]$. I want to find the pdf of ...
1
vote
0answers
22 views

Geometric Brownian Motion Price Processes in high Dimensions

This is my first post so I am open for an suggestions in formating improvement. For some reason I can not find suitable literature for the following problem What I want to do is calculate the option ...
1
vote
0answers
50 views

Asymptotics In Probability Theory Help

I have a problem with a random walk I'm trying to work with. Suppose I have a random walk $$S_n = \xi_1 + ...+\xi_n, n \geq 1, S_0= 0$$ with i.i.d. increments $\{\xi_n\}$ with common distribution ...
1
vote
0answers
15 views

Third Moment of Hitting Time

We recently went through expected hitting times of markov chains in my class and were asked about computing the various moments of hitting times. As such, I'm wondering if my thinking is correct as ...
0
votes
0answers
14 views

how to use moment generating function to calculate co-variance? [on hold]

The time to failure T of an unknown type of a light bulb is a(1+x)-exponential random variable in years, where x is a (μ=1)-Poisson random variable. What is Cov(T,x)?
1
vote
1answer
23 views

Random Walk: Proving that $1 = \sum_{m=0}^{n}P_0(S_{n-m} = 0)P_0(\tau_0 > m)$

I would appreciate a further hint for this question: Let $S_n$ a random walk on $\mathbb Z$, with $S_0=0$. Let $\tau_0 = \inf\{n>0:S_n=0\}$, the hitting time of $0$. Show that $$ 1 = \sum_{m=0}...
1
vote
2answers
18 views

Expected value of the product $E[W(u)W(u+v)W(u+v+w)]$ for Brownian motion

I am trying to understand the Wiener process, and I am not sure if my assumptions make sense. Let $W(t)$ be a Wiener process ($W(0)=0$ and for $t \le s, (W(s)-W(t))$ is a normal distribution with ...
6
votes
0answers
61 views

Random walk on thin ice?

My Question: Is the stochastic process which is described below a (special case of a) well-studied model? What kind of properties are known under which assumptions? Let's think of a random walker on ...
0
votes
0answers
24 views

Time Series Analysis

The balance held in a bank at r% annual interest follows the equation; $X_{n+1} = (1+r)X_n + u_0$ where $r > 0$, $u_0 = -400$ & $\{X_n\}_{n\geq 1}\in T_d$ is a discrete time model which is not ...
1
vote
1answer
16 views

Why is this inequality true in proof of strong law of large numbers for renewal processes

Let $\{N_t\}_{t\ge 0}$ a renewal process with intensity $\lambda>0$. Then $\lim_{t\to\infty} \frac{N_t}{t}=\lambda$ a.s. Here $W_i$ denotes the waiting time of $N_t$ and $T_n:=\sum_{i=1}^n W_i$ ...
0
votes
2answers
62 views

Markov Chain: Optimal stopping to determine the price at which stock is traded

The stock price starts at 100\$. At any given time, there is 50% probability that stock price increases further by 1 and 50% probability that stock price goes back to 100\$. You are paying 1\$ to ...
-1
votes
0answers
14 views

Non homogeneous Markov chain

Let {Xn|n ≥ 0} be a discrete time stochastic process with state space S = {1,2,···}. Let Xn+1 = f(n + 1,Xn,Yn+1),n ≥ 0 where f : {0,1,···}× S × [0,1) → S, Y1,Y2··· are i.i.d. [0,1) valued random ...
1
vote
0answers
33 views

$X$ is martingale

Let be $B$ and $D$ independent Brownian motion. Let be $\alpha \in \mathbb{R}$ constant. Define $$M_t = \alpha (\int_0^t B_s dD_s + \int_0^t D_s dB_s)$$ and $$N_t = \frac{\alpha^2}{2} \int_0^t (B_s^2 +...
1
vote
1answer
60 views

Why do we need absolute continuity of $\langle M \rangle_t(\omega)$ with repect to the Lebesgue measure?

I am trying to understand the proof of proposition 3.2.6 in Stochastic Calculus and Brownian Motion by Karatzas and Shreve. For $X$ bounded they use Lemma 3.2.4 in the same book and eventually claim(...
1
vote
1answer
27 views

Recursive use of mathematical expectation

Given $x_0, y_0, \beta \in \mathbb{R}$, define a random process as: On turn $i$, probability of success is $x_i/y_i$. On success, $x_{i+1} = x_i + \beta$, else, $x_{i+1} = x_i$. And $y_{i+1} = y_i + \...
2
votes
0answers
19 views

Proof verification - application of Girsanov to Brownian running max with nonzero drift

Let $B_t$ be a standard BM w.r.t. $(\Omega, \{\mathcal F_t\}, \Bbb P)$, $\mu$ a constant, $X_t:=\mu t+ B_t$, and $M_t:=\max_{0\le\tau\le t} X_\tau$. Find $\Bbb P(M_t\le a)$ for $a\ge 0$. I have used ...
1
vote
0answers
22 views
+50

Sample instances of random process given all temporal correlation functions?

I asked this question on signal processing stack exchange (question) but I wonder the general answer I am seeking makes the question better suited here. Suppose I have a complex valued random process ...
0
votes
1answer
16 views

Conditional Expectation of Poisson Process Interarrival Events

I'm having some trouble with something my professor said would be on our exam. For a Poisson process $N_t$ with interarrival times $X_i$, how is it that you find $E(X_i | N_t = n)$ (assuming $n\ge i$)?...
0
votes
0answers
15 views

Does this linear system have a unique solution?

$x \in R^n$, $P \in R^{n \times n}$, $P$ is a known stochastic matrix (each row sums to 1), $b \in R^{n}$ is a known non-zero vector. $e = [1, \dots, 1]^T \in R^n$, we have the linear system: $x = P^...
1
vote
1answer
61 views

Heavy-Tailed Distribution

I have a problem with a random walk I'm trying to work with. Suppose I have a random walk $$S_n = \xi_1 + ...+\xi_n, n \geq 1, S_0= 0$$ with i.i.d. increments $\{\xi_n\}$ with common distribution ...
0
votes
1answer
30 views

How to update the weights knowing the loss in Neural Network

Question How update the weight knowing the loss Work explanation I have a really simple network composed by two layers with one neurons in both layers. Considering an input of 1, the final result ...
1
vote
0answers
19 views

How can I find expected value about Rayleigh fading power?

This equation is in https://ieeexplore.ieee.org/document/7413975 In this equation (10) and (11), $h_x$ is Rayleigh fading power. But I don't know why the (10) is transformed to (11). $\mathbb{E}$ is ...
1
vote
1answer
53 views

Poisson Point Process On Real Number Line

Let $\mathbf{\Pi}$ be a homogeneous Poisson point process on the real number line with intensity $\lambda$. Let $r^{+}$ denote the distance from the origin to the closest point of $\mathbf{\Pi}$ on ...
-1
votes
1answer
31 views

If the inter-arrival times of customers are i.i.d. exponential distribution, is it necessary that the number of customers is a Poisson process?

Suppose customers arrive with time interval $U_i$ i.i.d. $Exp(\lambda)$, therefore, $$F(U_i\le t)=1-e^{-\lambda t}$$ The arrival time of customer $i$ is $$T_i=\sum^i_{j=1}{U_j}$$ The number of ...
0
votes
0answers
24 views

Independent of generated $\sigma$-algebra if independent of random variables

Let $(X_t)_{t\geq 0}$ be a stochastic process and let $\mathcal F_t=\sigma(X_s:0\leq s\leq t)$ be the generated filtration. If $Y$ is a random variable independent of $X_s$ for all $0\leq s\leq t$, ...
-1
votes
0answers
24 views

Finite Markov Chain: Distribution and Expected Value

The complete graph on {1,...,N} is the simple graph with these vertices such that any pair of distinct points is adjacent. Let Xn denote simple random walk on this graph and let T be the first time ...
2
votes
0answers
34 views

Mean time to absorption for finite general birth death process with one absorbing state

I have a continuous birth-death chain $\{X(t)\}$ on state space $\{0, 1, \dots, n-1, n\}$, with birth rates $\lambda_i$ and death rates $\mu_i$, where state 0 is absorbing ($\mu_0 = \lambda_0 = 0$), ...
1
vote
1answer
21 views

Show that $𝑋_𝑡=𝑍_1+⋯+𝑍_𝑡$ where $\{𝑍_𝑡\}_{𝑡\geq1}$ is not stationary.

$X_t=𝑍_1+\dots+𝑍_𝑡$ where $\{𝑍_𝑡\}_{𝑡\geq 1}$ is a white noise having the following properties : $𝑍_𝑡 \sim 𝑁\left(0,𝜎^2\right)$ $\forall t\geq 1$ $\gamma_𝑍(h)=0\: ∀ h≠0$ $𝑍_𝑡$ ⫫$𝑍_{𝑡+ℎ}...
0
votes
0answers
16 views

Has the Gaussian Process the Markov Property only when is variance is a diagonal matrix?

I'm studying stochastic process and Markov Chain. I was wondering if a Gaussian Process has the Markov Property (that is the conditional probability distribution (given the present states) of future ...
0
votes
0answers
18 views

Expectation of Ito Process

Suppose $X$ is an Ito process defined by $ dX = adt + b_tdB$ and $X_0 = 0$ where $B$ is a Brownian motion. Ito process consists of two parts: one is deterministic and the other is random. Is the ...
0
votes
1answer
33 views

Stochastic Process Example

Can anyone please give a simple example of what is a stochastic process? I got confused between random variables and stochastic process. I've read somewhere that stochastic process is a collection ...
0
votes
1answer
22 views

Finding variance of a random variable given by two uncorrelated random variables

a) Let $X$ and $Y$ be two uncorrelated random variables. Assume $Var(X) = 1.55$ and $Var(Y) = 0.8$. What is the variance of the random variable $Z = -4X + 5Y - 6$? b) What if $X$ and $Y$ are ...
0
votes
0answers
22 views

How to recognize an ARMA process? [migrated]

By looking at the autocovariance, how could you recognise what discrete model (MA(q), AR(p), or ARMA(p,q)) is more appropriate to describe your data?
0
votes
0answers
10 views

How do I find the expectation of a non-stationary auto-regressive time series, with absorbing states?

Apologies for not knowing Latex! Consider the following recursive function: $$ y(t+1) = y(t) (1+r) + R - e(t) $$ Where $r,R$ are known constants, $r>0$, and $e(t)$ is distributed as a truncated ...