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.

learn more… | top users | synonyms

0
votes
0answers
8 views

Random process with stationary independent increments determined by first order distribution?

It says in my random processes book that if a random process $X_t$ with stationary independent increments has value $0$ at the start ($X_0 = 0$) then it is completely determined by it's first order ...
0
votes
0answers
18 views

Problem about Random walk and Stopping time.

Here is an example in "Probability with Martingales" My questions are: (1)Does equation (a) hold for $T=\infty$? (2)The equation:$$\mathbb{E}M_T^\theta=1=\mathbb{E}[(sech \theta)^Te ...
1
vote
1answer
70 views

A Markov Chain Problem.(Change the color of ball)

There are $n$ different color balls in a box. Take two balls in turns, and change color of the second ball to the first. (This is one operation). Let $k$ be the (random) number of operations needed to ...
0
votes
1answer
7 views

In an irreducible, aperiodic, null-recurrent Markov chain holds $\sum_n p_{ij}^{(n)} = \infty$

My lecture notes state the following theorem: Theorem 2. Let $(X_n)$ be an irreducible, aperiodic, null-recurrent Markov chain. Then $$\forall i,j \in S : p_{ij}^{(n)}\to 0 \text{ and } \sum_n ...
1
vote
0answers
3 views

Is a discrete random process issued from a sampled continuous ergodic WSS process also ergodic?

I have a continuous time process $\{X_t,t\in\mathbb{R}\}$ that is WSS and ergodic for the 1st and 2nd moments. I create a random discrete process $\{Y_n,n\in\mathbb{N}\}=\{X_t,t=nT\}$ by discretizing ...
0
votes
0answers
19 views
+50

Changing a queueing processes

I am wondering if there are any general results related to how queue behavior changes if one is allowed to repeatedly make one-off behavior changes. Situation Consider a general queueing system ...
0
votes
1answer
30 views

Integrating probabilities

My following problem is of general nature, here is an example to illustrate it. For example let $\left(\xi_i\right)_{i \geq 1}$ be independent and identically Exp(1) distributed random variables. We ...
9
votes
2answers
306 views

What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
0
votes
0answers
23 views
+50

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
-2
votes
0answers
10 views

Advice on Stochastic Drift Sequence

I recently wrote a paper for university reviewing stochastic differential equations and it pretty well. I now have an idea for a paper which I would like to write in my spare time (and maybe if I can ...
0
votes
0answers
18 views

Jeffrey's Prior for Bivariate Lognormal

Exactly what the question says, I'm working on code for an MCMC simulation and need to set some uninformative or weakly informative priors. I haven't been able to find the prior for the sigma ...
0
votes
1answer
21 views

Poisson Process Suitable Scenarios

I have a couple of doubts about if these scenarios are suitable to be modeled as a Poisson process. I will like to have your views and arguments why. Packets are lost due to packet overflow in the ...
1
vote
1answer
30 views

Poisson process has independent and stationary increments

Being $N_t$ a Poisson process, defined as $$N_t:=\sum_{n\geq 1} n \mathbb{1}_{[T_n,T_{n+1}[}(t)$$ where $T_n$ are sums of independent exponential random variables, how can I prove it has stationary ...
-1
votes
0answers
13 views

Prove this is markov chain [on hold]

Let $Y_i = i$ with probability $p_i$. Let $X_i = max[Y_i , Y_i-1, ...]. $ We have $i={0,1,2,3,4,...}. $ Prove that $X_i$ is a Markov chain and write down its matrix.
0
votes
0answers
23 views

Will the branching process go extinct with probability 1?

I am trying check whether the branching process goes extinct with probability one. Single Type Branching Process with Pk = (1/2n)(n/k), for k = 0,.....,n with n > 2. Assuming, i can be able to ...
0
votes
1answer
38 views

How to apply the strong Markov property in this case?

I'm trying to understand the following proof: Theorem: Let $(X_n)$ be an irreducible $(\alpha, \mathbf p)$-Markov chain with a finite state space $S$. Then $(X_n)$ is positive recurrent. ...
1
vote
2answers
326 views

birth and death processes

Suppose we have a system of N balls, each of which can be in one of two boxes. A ball in box I stays there for a random amount of time with exponential(lambda) distribution and then moves ...
2
votes
1answer
411 views

Stopping theorem for continuous martingale

I've a question about a proof in my lecture notes. We want to prove the following theorem. $M=(M_t)_{t\ge 0}$ be a $(P,F)$-martingale, where $P$ is a probability measure and $F=(\mathcal{F}_t)$ a ...
0
votes
1answer
36 views

For a Brownian motion prove that (a) $N (t) -λt $ and (b) $e^{(\log(1-u) N (t) + uλt)}$, are martingales

For a Brownian motion ${z (t)}$ and for any $β ∈ R$, be $V (t) = \exp\{ βz (t) - (t β ^ 2) / 2\}, t≥0 $ Show that ${V (t)}$ is a martingale with respect to a Brownian filtration. Also ${N (t)}$ be a ...
2
votes
0answers
28 views

Explicit solution SDE?

I have the following SDE: $$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$ where ...
3
votes
1answer
71 views

Long-memory process and convergence of finite dimensional distributions

We assume that $(X_t)_{t\in\mathbb{N}}$ is a stationary sequence of standard normal random variables such that for the autocovariance function holds $\gamma _X (k)\sim Ck^{2d-1}$ with $d \in (0,1/2)$ ...
0
votes
0answers
48 views

Mean and variance of a stochastic process

Let \begin{equation} \begin{array}{l} y_1(t)=e^{-\kappa_1 t}y_1(0)+\displaystyle\int_0^t\kappa_1 e^{\kappa_1(s-t)}\theta_1ds +\sigma_1\displaystyle\int_0^te^{\kappa_1(s-t)}\sqrt{y_1(s)}dZ_1(s),\\ ...
2
votes
3answers
34 views

Is $W_{2t}-W_t$ a brownian motion?

Is $W_{2t}-W_t$ a brownian motion? $(W_t)_{t\geq 0}$ is a brownian motion, I have to show that $X_t:=W_{2t}-W_t$ is a brownian motion as well. $$W_{2t}= 1/\sqrt{2} W_t$$ (by scaling property) then ...
5
votes
1answer
77 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
2
votes
2answers
166 views

Expectation of a stopping time of a Wiener process

How can we calculate $\mathbb{E}(\tau)$ when $\tau=\inf\{t\geq0:B^2_t=1-t\}$? If we can prove that $\tau$ is bounded a.s. (i.e. $\mathbb{E}[\tau]<\infty$), then we can use the fact that ...
1
vote
0answers
38 views

Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
2
votes
1answer
31 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
1
vote
1answer
19 views

Centered independent increments process is a martingale

Let $(X_n)$ be an centered integrable process with independent increments (which as far as I understand means that $(X_{n+1}-X_n)_{n\in \mathbb N}$ is independent). While showing that $(X_n)$ is a ...
1
vote
2answers
42 views

$E[M_t|H_t]$ is a martingale with respect to $H=(H_t)_{t\geq 0}$, $H_t \subset \mathcal{F}_t \forall t$

Being $(M_t)_{t \geq 0}$ an $\mathcal{F}$-martingale, I have to show that $E[M_t|H_t]$ is a martingale with respect to $H$ ($H=(H_t)_{t\geq 0}$, $H_t \subset \mathcal{F}_t \forall t$). I proceded ...
1
vote
1answer
42 views

Find one-dimensional distribution function $F(y\mid t)$ of random process $Y(t)$

$ Y(t)=tZ^2;\quad Z\sim U(-2;2); \quad t\ge0. \quad$ I need to 1) find one-dimensional distribution function $F(y|t)$ of random process $Y(t)$. 2) calculate probability that trajectory of the ...
1
vote
1answer
30 views

Do we need $\tau \leq \nu$ to show $E(X_\tau)=E(X_\nu)$?

My lecture notes claim that if $(X_n)$ is a martingale and $\tau$ is a stopping time bounded by $N$ then $$E(X_\tau)=E(X_{\tau \wedge N})=E(X_{\tau \wedge 0})=E(X_0)$$ and then remarks that if $\tau$ ...
1
vote
2answers
19 views

Random number distribution from a different distribution

Suppose I have a random number generator that generates random numbers $x$ with a normal distribution $p(x) \propto e^{-x^2}$ (modulo normalization, but lets keep it simple). Now, out of these ...
1
vote
1answer
53 views

An exponential martingale [closed]

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
2
votes
1answer
33 views

Proof that a random variable has exponential distribution.

Supose that $X_1$ is a continuous and positive (real) random variable with exponential distribution, namely $$P(X_1>t)=e^{-\lambda t}\quad t>0$$ Now suppose that $X_2$ is another continuous and ...
0
votes
2answers
375 views

Finding the distribution of a poisson distribution with random variable lambda

So suppose $X$ is a rv with a Poisson distribution with $\lambda$ being a random variable as well. $\lambda$ has an exponential distribution with mean $1/c$ and $f_\lambda(t) = c\times\exp(-ct)1_{[0, ...
1
vote
2answers
258 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
0
votes
0answers
20 views

characteristic function of a stochastic process with stationary and independent increments

Let $(X_t)_{t\geq 0}$ be a stochastic process with independent and stationary increments. I have to show that $E[e^{itX_1}]=\phi^n(t)$ Since increments are independent, I can write ...
1
vote
3answers
280 views

Markov property with respect to a filtration

Suppose $\{ X_t: t \in \mathbb{R} \}$ is a stochastic process on a probability space $(\Omega, \mathcal{F}, P)$, and it is adapted to a filtration $\{\mathcal{F}_t \}$ on the probability space. $\{ ...
4
votes
0answers
58 views

Filtration and measure change

I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand the concept of "filtration". Yes, the definition of filtration is straight forward, it's ...
0
votes
1answer
32 views

Stochastic processes with full memory

Markov processes are stochastic processes with no memory. How are called the stochastic processes with full memory? Can't found anything on google.
1
vote
1answer
57 views

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
0
votes
1answer
38 views

Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$ P(t) = \exp(tQ), $$ see for ...
0
votes
0answers
3 views

How to calculate radius density of observation using locality sensitive hashing?

How do I calculate radius density of observation using locality sensitive hashing? I am new to the locality sensitive hashing(LSH). LSH based learning and Querying was difficult to understand.
6
votes
0answers
282 views

An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price ...
1
vote
1answer
18 views

Max of independent and identical random variables is Markov

I'm supposed to show that given a sequence $\{Y_n\}$ of i.i.d the stochastic process $$X_n=\max(Y_0, Y_1...,Y_n)$$ is a Markov of chain. I think I could do it by induction but I would rather see how ...
1
vote
1answer
291 views

Renewal Processes for Uniform and exponential Distributions

Suppose the lifetime of a component Ti in hours is uniformly distributed on [100, 200]. Components are replaced as soon as one fails and assume that this process has been going on long enough to reach ...
0
votes
0answers
14 views

Approximating the probability of an event by finite-dimensional distributions

Let $(X(t))_{t\ge 0}$ be a stochastic process on $\mathbb{R}^d$, say an Ito diffusion (with continuous sample paths). Let $A\subset \mathbb{R}^d$ be a measurable set and $t>0$. Does the following ...
0
votes
1answer
38 views

Stopping times problem: $ \tau_+ = \inf \{t \ge 0 \mid W_t>0\}$

Stopping times problem, $\tau_+ = \inf \{t \ge 0 \mid W_t>0\}$ I can not prove the following : P/S: When I look at the stopping time, I feel that $\{W_0 > 0\} = \{\tau_+ = 0\}$ , is that ...
1
vote
0answers
27 views

If a random integral has moments of all orders, is the same true for its kernel?

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...
0
votes
1answer
42 views

Expected value and variance of random process

Let $U,V$ be random variables with distributions $\mathcal{U}(-1,1)$ ,$\mathcal{E}(2)$ (uniform and exponential). If $U$ and $V$ are independent what is the variance and expectation of the random ...