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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Activity of $\Gamma$-type OU process.

Consider an OU type process $$ dV_t = -\lambda dt + dZ_{\lambda t} $$ as studied in chapter 17 in Sato (deterministically subordinated). Assume $V$ has denstity $u$ with respect to its Levy measure ...
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51 views

the infinite sum of symmetric random variables is also symmetric

Definition. Let $(\Omega, {\mathcal F}, \mathbb{P})$ be a probability space and $X$ a random variable in $\Omega$. $X$ is said to be ${\mathbf symmetric}$ (about $0$) if $X$ and $-X$ are equal in ...
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98 views

Existence of a Continuous Modification of Fractional Brownian Motion

For a course on stochastic processes, I've been working on an exercise on fractional Brownian Motion. Showing that this process has a continuous modification is one of the final steps of the exercise, ...
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35 views

Càdlàg of a stochastic process when absolutely convergent

I have a stochastic process given as $X_t(\omega)=\sum_{m=1}^\infty a_m 1_{\{0<b_m\leq t\}}(\omega) $, where $(a_m,b_m)\in(\mathbb{R}^d,\mathbb{R}_{+})$. I know that that the sum converge ...
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22 views

Lebesgue measure for a voronoi cell

Assuming that in a two dimensional space, I have a random set of points distributed over that space and that I created Voronoi cells based on those points. Now I have added to the two dimensional ...
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37 views

Radon-Nikodym Derivatives between Ito Processes

I am curious about the following problem: Let $B_t$ be a standard Brownian motion on $(\Omega, \mathcal F, \mathcal F_t, \mathbb P_a)$, where the filtration is generated by $B_t$. On a finite ...
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33 views

Proving that the Markov chain is recurrent - Confusion/Help

Giving the following transition matrix [ 0.9 0.1 ] [ 0.8 .2 ] Classify the states From drawing the graph I know that both stats are recurrent. However I'm really failing to prove mathematically ...
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38 views

How to prove this Brownian motion convergence?

Let $W_t$ be a Brownian motion. How do I show the following? $$ \alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.} $$ Showing convergence of this ...
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16 views

Birth-coagulation process

I'm reading a paper where the birth-coagulation process is described as: $$ A \rightarrow A+A ;\, \text{rate:} \gamma $$ $$ A + A \rightarrow A ;\, \text{rate:} \chi $$ The total density of ...
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20 views

Explain what is an empirical processes.

I need some help understanding what meant by empirical processes and empirical measures. I'm familiar with the basics concepts of measure theory and stochastic processes.
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20 views

Characteristic function of basic affine process by rotation count algorithm

Hi everyone, I've had this frustating, silly problem for a while now. I've looked at the problem for a loooong time now, which may be one of the reasons I can't see the solution. I am trying to ...
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18 views

left limit of filtration

Let $N(t) = I_{(X\le u, \delta = 1)}, X = min(T,C), \delta = I_{(T\le c)} $ $F_s = \sigma \{ N(u), I_{(X\le u, \delta = 0)} , 0\le u \le s\rbrace$ and $F_{s^{-}}$ = $ \sigma \{ \cup_{(u<s)} ...
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87 views

Formula for the variance of a renewal process

Let $N(t)$ be a renewal process, with a sequence of IID inter-arrival times $X_{1}, X_{2}, \dots$ having finite second moment: $EX_{i}^{2} < \infty$. How would I show that $$\mathrm{Var}N(t)= 2 ...
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32 views

Show that a function of a symmetric random walk is a martingale

Suppose $S_n = (X_n,Y_n)$ is a symmetric random walk on $\mathbb{Z}^2$. Show that $G_n = X_n^2 + Y_n^2 - n$ is a martingale. What is true about $E_{(x_0,y_0)}[|S_n|]$? Find an upper bound for ...
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30 views

Is the variance of an Ito process strictly increasing?

Consider the Ito equation: $dX_t = f(t, X_t) dt + G(t, X_t) dW_t$ where $f:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$, $G:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^{n\times m}$, $X_t \in ...
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52 views

Finding the Kolmogorov's Backward equation

SO this question is probably really easy, I am just struggling in understanding how to do it It goes like this: we have a system with 3 components, at time $t=0$, component 1 is active and the other ...
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48 views

expectation of the last arrival time of a poisson process

Consider a Poisson process with rate $λ$ and let $L$ be the time of the last arrival in the interval $[0, t]$, with $L = 0$ if there was no arrival. (a) Compute $E(t − L)$. (b) What happens when we ...
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30 views

What is the distribution of $\sup\limits_{t\geq 0}( B_t-xt)$

I would like to find the distribution of $\sup\limits_{t\geq 0}( B_t-xt)$, where $(B_t)_{t \geq 0}$ is a Wiener process and $x > 0$. I don't know how to begin. Any help is appreciated.
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31 views

Why does Brownian motion have finite $L^2$ norm?

The title might be a bit misleading. Sorry for that but here is the question. For predictable processes $X$, the $L^2$ norm over the set $[0,T]\times\Omega$ under the Doleans measure $\mu_M$, $M$ ...
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23 views

Epidemic model (Pure birth process)

Consider a population of $N$ individuals, some of them infected. Contacts between any two members of the population are according to Poisson process with rate $\lambda$ per day. When contact occurs, ...
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32 views

Explanation of a stochastic process

It's rather crazy of me to try to read a book about stochastic processes without ever having read deeply into the necessary preliminaries, but after some thinking, in order for me to continue, I only ...
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32 views

Interpretation Of 4th-Order Cumulants For Complex Random Variables

I asked about this over at the DSP site several days ago but have not gotten any responses. I'll replace the word "signal" with "random variable" and hope that someone from a pure math background can ...
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31 views

Regularity of a parabolic equation

Consider the following parabolic equation on $\mathbb{R}^d$: \begin{equation} \partial_t\mu=\mathrm{div}(b\mu) + \mathrm{div}(D\nabla\mu), \end{equation} where the drift ...
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39 views

Deriving the autocorrelation function for the ARMA model

Definitions The ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. Question To derive the ...
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17 views

A question involving Poisson processes

If $(N_t)_{t \in \mathbb{R}_+}$ is a point process (or a Poisson process), what does it mean $$ N_t - N_{t-} \in \{ 0, 1 \}? $$ Notation: $N_{t-} = \underset{u \rightarrow t^-}{lim} N_u$. Thank you!
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46 views

Strong Markov property of Bessel processes

I am thinking about the following: If $(B_t)_{t \geq 0}$ is a Brownian motion in $\mathbb{R}^3$, how can we show that the Bessel process (of order $3$) $(|B_t|)_{t \geq 0}$ has the strong Markov ...
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39 views

Yule walker equation limited matrix size

Definitions For an ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. It is straightforward to show that ...
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28 views

Is this simply assuming an Ito semimartingale.

I am reading a paper where they start by assuming some process follows $$ \frac{dX_t} {X_{t-}} = \alpha_t dt + \sqrt{V_t} dW_t + \int_{x > -1} x \tilde{\mu}(dt, dx) $$ with $\alpha_t$ and $V_t$ ...
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42 views

Proving that $X_t = W_t~ I (0<t\le T) + (2W_T - W_t) ~I(t > T)$ is a brownian motoin

The steps to showing that a process is a BM are as follows: (1)$X_0 = 0$ (2) $ \forall t ~~~X_t$ is continuous (3)$X_t \sim N(0,t)$ (4)$X_{t+s}-X_{s} \sim N(0,t)$ (5)$X_{t+s}-X_{s} \bot \mathscr ...
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25 views

Seeking help understanding steps in a proof, in “Convergence of probability measures by P.Billingsley”.

To not waste anyones time: this question is directed at people who either have the mentioned book in their possesion, or have at some point read it. I am reading through Patrick Billingsley's book: ...
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28 views

From which domain to which co-domain does the sample function maps it's function values?

I am learning from Knill's probability theory and stochastic processes with applications book. In chapter 4 about "continuous time stochastic processes" I encountered the following. For any fixed ...
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36 views

Help with this Markov Chain Proof please

Problem: Consider a finite Markov Chain with N states $(1,2,...,N)$. Let $P(n) = [P_{i,j} (n)]$, be an n-step transition matrix. Suppose that $lim_{n\to\infty} P_{i,j} (n) = \pi_{j} $ for any $1 ...
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22 views

Calculation of quadratic covariation of stopped processes

I am stuck in computing the quadratic covariation of the following two processes: Let $0< y <r$ and let $(B_t)$ be a Brownian motion started at $y$. Let $T_0 = \inf \{ t \geq 0 : B_t = 0 \}$ ...
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32 views

Some basic questions related to independence of random variables

I attend a lecture about Stochastic Processes even though I have not studied mathematics and some of the basics in probability theory are missing. So I hope you can help me with the following ...
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38 views

Question about noise term in SDEs

Do any properties/assumptions of SDEs prevent the noise term from being extremely large? Using a simple population growth model as an example, $\frac{dNt}{dt} = (r_{t} + W_{t})Nt , N0$ given, ...
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54 views

Right-continuity of the augmented filtration

In most books about stochastic processes, the authors write about "the usual augmentation of a filtration". I am having trouble with proving that their construction is correct, i.e. that the augmented ...
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67 views

Conditional expectation of Wiener process

I want to calculate $E(W_t | W_1)$, $E(W_t^2 | W_1)$ and $E(W_t^2 | W_1, W_2)$, where $(W_t)_{t\geq0}$ is a Wiener process. For the first one I used the conditional distribution formula for the ...
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20 views

Exponential Distribution and Expected Times

Suppose two people A,B are assigned to do an individual task and then a group task. Person A completes his individual task on average around 30 minutes. Person B completes his individual task on ...
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129 views

Divergence of random series

So, suppose that $X_n$ is a sequence of independent identically distributed random variables with Bernoulli distribution with parameter $p$. Now, consider the random series $$\sum_{n=1}^{\infty} ...
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40 views

Calculating a stochastic differential

Let $f$ be a real-valued function with bounded continuous second derivative $f''$, and $w(t)$ be a Wiener process. Let $$ V(t,w(t)) = f(w(t)) - \frac{1}{2} \int_a^t f''(w(s))ds. $$ I want to apply I ...
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34 views

An application of Ito-Doob differential formula

I want to apply the following formula $$dV(t,w(t)) = \left(\frac{\partial}{\partial t}V(t,w(t)) + \frac{1}{2} \frac{\partial^2}{\partial x^2}V(t,w(t))\right)dt + \frac{\partial}{\partial ...
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Expected range of simple random walk in $\mathbb{Z^2}$

Let $(Y_k)_{k\geq0}$ be a simple random walk process. The range of an $n$-step random walk, $R_n$, is a random variable that characterizes the number of distinct points visited at time $n$: ...
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30 views

Does Markov Chain converge in Variance Norm?

Assume the chain $\{X_n\}_{n\in\mathbb{N}}$ on the statespace $(S,\mathcal{F})$ is aperiodic, irreducible and positive recurrent. We denote with $\pi$ its (unique) stationary distribution. Is it true ...
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24 views

Expected Times for Exponential Distribution Problem

Two craftsman A,B are making birdhouses at rates 2,3 per hour respectively. What is the expected time until each of the craftsman finish making one bird house? So I have tried conditioning on each: ...
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18 views

How would I solve for long run average profit?

I was looking at a problem, and I was wondering how I would set this up. Any help would be welcome. Thank you! A store stocks a particular item. The demand for the product each day is 1 item with ...
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89 views

How do I compute the variance of expected number of fair coin flips for HTH sequence using linear system of equations?

Assuming fair coin flips, I know how to compute the expected number of coin flips to see HTH sequence by writing out the linear system of equations from the state transition diagram below. ...
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2answers
77 views

Is this stochastic process a martingale?

I have the following process: $X_t=tB_t-\int^{t}_{0}B_s \ ds$ where $B_t $ is a Brownian motion. Is this a Gauß-process and/or a martingale? Can someone help me with this? And how can I calculate ...
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51 views

Malliavin derivative of a Lebesgue integral.

Let $X_t$ a random process such that its Malliavin derivative is well defined for all $t$. Then I have read that : $D_s(\int_0^t \! X_u \, \mathrm{d}u)=\int_s^t \! D_s(X_u) \, \mathrm{d}u.$ What I ...
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27 views

Question about poisson process waiting times

This question is a conceptual question about understanding the answer to another problem. (original problem here: Average waiting time in a Poisson process) The original problem asked for an average ...
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20 views

Expectation of Compound Poisson Process

$\mathbb{E}[e^{(\sigma-\lambda)X_t } \mathbb{1}\{X_t \geq X^*\}] $ I am not too sure how to compute the expectation of a compound Poisson process multiplied with a indicator function. The Question ...