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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limiting distribution of $Y_t$ in the mean-reverting Ornstein-Uhlenbeck process

The mean reverting Ornstein-Uhlenbeck process is of the equation: $$dX_t=(a-cX_t) \, dt+\sigma \, dW_t$$ If we are told that both $a$ and $c$ are larger than $0$, what then is the limiting ...
2
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71 views

Is there an information theory for continuous time signals?

Information theory books talk about entropy and mutual information of discrete time processes, such as a sequence of symbols sent with a symbol period $T_s$ and there received sequence. Can we talk ...
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2answers
738 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...
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1answer
99 views

Continuous-time Markov Chain forward/backward equations and MLE

I have two questions: 1) Using Kolmogorov's forward and backward equations, show that $p_{11}(t) + p_{21}(t) + p_{31}(t) = 1$ and $p_{21}(t) = p_{31}(t)$ where $p_{ij}(t) = P(X(t) = j | X(0) = i)$. ...
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339 views

Poisson process counting process

Two individuals, A and B, both require kidney transplants. If she does not receive a new kidney, then A will die after an exponential time with rate $\mu_A$, and B after an exponential time with rate ...
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39 views

Probability of obtaining equal number of each outcome of a fair die at the nth trial

Suppose a fair die is tossed repeatedly. I am concerned in deriving the probability of the occurrence of obtaining equal number of each possible outcome at the nth trial. Clearly this is only possible ...
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130 views

Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$ \tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \} $$ is a stopping time with ...
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145 views

determine type of probability distribution

let us consider following model $$y(t)=A_1 \sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + A_3 \sin(\omega_3 t+\phi_3)+ \ldots +A_p \sin(\omega_p t+\phi_p)+z(t)$$ we have three parameter ...
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1answer
70 views

What is the distribution of the service-starting time lag w.r.t. two concurrent customers from two parallel $M/M/1/1$ queues?

Consider two parallel, independent $M/M/1/1$ queues (denoted $Q_i, Q_j$) with identical arrival rate $\lambda$ and service rate $\mu$, using FCFS (First Come First Served) discipline. Note that the ...
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1answer
36 views

Random variables: functions or equivalence classes

Suppose, we have a continuous parameter stochastic process. Should I consider each of the random variables as functions or equivalence class (in a.e. sense) ?
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156 views

Use Ito's formula to determine the stochastic differential equation satisfied by $V_t$

A stochastic process $V_t$ is defined by $$V_t =\sqrt{t(t+W_t^2)}$$ $W_t$ is the Wiener process and $t$ denotes the time ($t > 0$). Use Ito's formula to determine the stochastic differential ...
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1answer
39 views

Generating two $-1$ correlated Poisson random variables with parameter $5$

Is it possible to generate two random variables $X$ and $Y$ that are both $Poisson(5)$ with $Corr(X,Y)=-1$? Why? I was thinking about generating $3$ independent Poisson random variables $Z_1,Z_2, and ...
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1answer
389 views

Ehrenfest urn model expectation question

Consider the Ehrenfest urn model in which $M$ molecules are distributed between two urns, and at each time point one of the molecules is chosen at random and is then removed from its urn and placed in ...
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51 views

More about the tower property of conditional expectation

If filtrations $\mathcal F_i$ i=1,2 don't have the inclusion relation. i.e. neither $\mathcal F_1$ $\subset$ $\mathcal F_2$ nor $\mathcal F_2$ $\subset$ $\mathcal F_1$. What is E[E[X|$\mathcal ...
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110 views

Time integral of a stochastic process

It seems that the time integral of a stochastic process $X_t$ in the interval $[0,T]$ gives us a random variable. My question is how do we define/calculate such a time integral. For example ...
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95 views

Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$ \tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \} $$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
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1answer
26 views

expectation by reasoning

An unbiased die is successively rolled. Let $X$ and $Y$ denote, respectively, the number of rolls necessary to obtain a six and a five. $E[X]= 6$. find $E[X \mid Y=1]$ Iam stuck on this. Iam ...
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40 views

Covariance of a random function

Suppose $X(s)=\int_0^1 G(s,t)\, dW(t)$, where $W(t)$ is Brownian motion, then what is the variance of $X(s)$ and the covariance of $X(s)$ and $X(r)$. Note that this is not the usual Ito integral ...
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112 views

Poisson process breakdown(waiting times)

Certain electrical disturbances occur according to a Poisson process with rate 3 per hour. These disturbances cause damage to a computer. a) Assume that a single disturbance will cause the computer ...
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413 views

What does limiting fraction of time mean in Poisson processes and renewal processes?

I'm reading on Poisson and Renewal processes and I encountered the term limiting fraction of time. Although it was somewhat defined I couldn't really grasp the meaning of it. Edit Here is an example ...
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66 views

Which Brownian motion property is the most important? [closed]

Which Brownian motion property is the most important? A standard Brownian motion is a stochastic process $(W_t, t\geqslant 0)$ indexed by nonnegative real numbers t with the following properties: ...
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52 views

What is the sample path of a stochastic process

Assume $\Omega $={head, tail}, let T=$\mathbb N$ and $X_t$ $t\in T$ be a collection of i.i.d random variables following Bernoulli distribution. Since a stochastic process is a function of two ...
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139 views

How to solve this SDE ? stuck half way

Problem: $dX_t = \sigma X_tdB_t$, $X_0=x$. $dY_t=X_tdt-Z_tdt$ find $Y_t$, where $Z_t$ is a control and $B_t$ is standard Brownian motion. My attempt: From Ito's lemma, $\partial_BX_t=\sigma X_t$, ...
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65 views

Merton problem: can the stock price keep rising?

I read that the stock price, $S(t)$ of the famous Merton model is given by the following differential equation $dS(t) = µS(t)dt + σS(t)dB(t).$ I gather that this is geometric Brownian motion. A path ...
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1answer
59 views

Random processes

I hope someone could tell me how to explain that "random process is continuous by probability" and "random process is differentiated by probability"? I know that definitions are these: Given a time ...
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1answer
60 views

Lifetime of a spaceship run by three computers

A spaceship is controlled by three independent computers. The ship can function as long as at least two of the three computers are functioning. Suppose the lifetimes of the computers are i.i.d. ...
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1answer
42 views

Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
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1answer
104 views

Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
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1answer
48 views

Fourier transform of n-th power of autocorrelation of a random process

I'm having troubles in understanding how Fourier transform of the n-th power of a time function is obtained. In particular I came across to a particular result with respect to the calculation of the ...
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80 views

Independence of stochastic process $(dB_1t)(dB_2t)$=0?

What does it mean (definition) for two stochastic processes to be independent? like two independent Brownian motion $B_1(t), B_2(t)$. I come across this when I saw a solution of a problem says if ...
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182 views

Sample continuity of Brownian motion

I wanted to know if the Brownian motion and the fractional Brownian motion are almost surely sample continuous or not? Many thanks.
3
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1answer
116 views

What does a customer see when it begins to be served in $M/M/1$ queue?

In queueing theory, the PASTA (Poisson Arrivals See Time Averages) principle [wiki] justifies $a_n = P_n$ where $$a_n = \text{proportion of customers that find } n \text{ customers in the system when ...
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96 views

Making a non-Martingale process a Martingale

Stuck on this question for a very long time: was wondering if any kind soul could help me out: Suppose $B_t$ is a standard Brownian Motion under measure P. Question: Create a martingale process that ...
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91 views

How to compute the limiting distribution of excess life, with uniform density?

I'm working on the reward/renewal process question and get stuck in the following question. could anyone please lend me some help? -- Question: What is the limiting distribution of excess life, ...
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1answer
57 views

Is this a Brownian motion

I am learning SDE, and here is some basic things I have trouble with, Let $B(t)$ be a Brownian motion, and $F \in \mathcal L^2$ is any stochastic process and I know $\int_0^tF(s)dB(t)$ is Ito process ...
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99 views

Expectation of this stochastic process

Let a stochastic process $X(t)= \int_0^t \operatorname{sign}(B(s)) \, dB(s)$, now how to show that $\Bbb E[B(t)X(t)]=0$ ? here $\operatorname{sign}(x)=-1$ for $x<0$, and $1$ otherwise. $B(t)$ is ...
2
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106 views

A right-inverse of Brownian motion local time at zero has stationary independent increments

Let $L_0^t$ be the local time for a standard Brownian motion at $0$ and define $$X_t=\sup\{s\ge0:L_0^s\le t\}, t\ge0. $$ I would like to show that $(X_t)$ has stationary independent increments. That ...
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13 views

number of possible component in sinusoidal model

suppose that we have following model $y[t]=A_1(sin(\omega_1*t+\phi_1)+A_2*sin(\omega_2*t+\phi_2)+....+A_p*sin(\omega_p*t+\phi_p)$+$z(t)$ my question is not related how to determine number of ...
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272 views

Characteristic function of compound Poisson process

It is widely known that the characteristic function of a compound Poisson process is $$ \phi_X(u) = \exp \left(t\lambda \int_{\mathbb{R}} (e^{iux}-1) F(dx) \right). $$ But if I try to derive it via ...
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79 views

Is a core for the generator of a Feller semi-group invariant under the resolvent?

Let $\{T_t:t\geq 0\}$ be a Feller semi-group acting on $C_0(\mathbb{R})$ with generator $(A,\mathcal{D}_A)$. We know a subspace $D\subset \mathcal{D}_A$ is a core for $A$ if $(\lambda-A)D$ is dense in ...
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159 views

Resource for Stochastic Calculus and Ito processes

May someone please recommend a book or website where one can learn Stochastic Calculus and Ito processes from scratch.
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157 views

Poisson integral and discontinuous martingale (Ito-Levy formula)

Consider compounded Poisson process $P$ given by $P_t = \int_0 ^t \int _{\mathbb R}z~ N(dr,dz)$ where $N$ is a Poisson random measure of intensity $dt \otimes \nu$ and $\nu $ is a Levy measure. Why ...
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52 views

Calculate expectation under risk neutral measure: $\mathbb{E_Q}(\max(S-1,0))$

I am busy with a numerical simulation and I want the calculate the following expectation under the risk neutral measure: $\mathbb{E_Q}(\max(S-1,0))$. $S$ is some variable that I calculated using ...
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23 views

Continuity in $x$ of $E^x \int_0^{\tau} f(X_t)dt$

Suppose I have a stochastic diffusion $X$. I am studying an expression of the form $u(x):=E^x\int_0^\tau f(X_t)dt$ where $\tau$ is the exit time of $X$ from my bounded open domain $D$. I am also ...
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1answer
468 views

Inverse of a regular stochastic matrix

Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix? Hope someone could answer ...
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263 views

Conditional probability with Poisson processes

I'm reading a section on conditional logistic models in which a heterogeneous Poisson process is used to make inferences in disease mapping. Basically, the likelihood of a Poisson process is used to ...
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134 views

References for time-inhomogeneous Markov jump processes?

In some central models in life insurance mathematics, the state of the insured is modeled using a continuous-time time-inhomogeneous Markov process with finitely many states. While many results for ...
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1answer
240 views

Expectation and variance of correlated exponential brownian motions

What is the expectation and variance of correlated exponential Brownian motions for the random variable $F$, where $A$ is real constant, $\sigma$ is a real constant and $\rho$ is the correlation. $$F ...
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56 views

Random walk converging to Brownian Motion for $t \in [0, \infty)$

If we define $X_i$ as a Bernoulli random variable with P(X=1) = P(X=-1) = 0.5. E[X_i] = 0, Var(X_i) = 1. As i have understood, by applying the Donsker's theorem, $t \in [0, 1]$, $B_N(t) ...
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1answer
162 views

Birth and Death Process Questions

Consider a birth and death process with the birth rate $\lambda_m = \lambda (m\ge 0)$ and death rate $\mu_m = m \mu (m \ge 1)$. A. How would I derive the stationary distribution? Only information I ...