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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A question about a Markov Chain

I encountered a question about Markov Chains which looks interesting. Given a homogeneous, irreducible, non cyclic Markov Chain with $K$ possible states and a transition matrix $Q$. We define $T_i$ ...
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17 views

Comparing hitting time of two random walks

There are two random walks, $S^t_i=S^{t-1}_i+ X_i^t$ for $i=1,2$, $X^t_i$ i.i.d they have boundaries $h_1$ and $h_2$ respectively. I'm wondering if it's possible to calculate the probability that one ...
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36 views

Mean value theorem inside the Expectation

Consider a stochastic process $X_t$ with continuous paths. I'd like to apply the mean value theorem inside the expectation, i.e. write something like $$ \operatorname{E} \left[ \int_0^t X_s \, ...
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1answer
74 views

Derivation of Black-Scholes equation by riskless portfolio

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
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19 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 ...
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1answer
28 views

Markov property for a stochastic process with discrete state space.

Consider a stochastic process $\{X_s\}_{s\in\mathcal S\subseteq\mathbb R}$ with value in $(\mathbb R,\mathcal B(\mathbb R))$ adapted to a filtration $\{\mathcal F_s\}$ (we can suppose that ...
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230 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 ...
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1answer
15 views

Find the probability generating function of a GW process

Consider a Galton-Watson process with offspring distribution $\mathrm{Poisson}(1)$. That is, $\textbf{p}(k) = \frac{e^{-1}}{k!}$. Given this information, and that $P(z) = ...
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1answer
37 views

Distribution of number of Poisson arrivals in interval

$X_1$ and $X_2$ are both Poisson processes. $N$ is the number of arrivals of $X_1$ in between two subsequent arrivals of $X_2$. Derive the probability density $f_N(n)$ of $N$. I wanted to start from ...
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26 views

Distribution of maximum/minimum proportion in a sampling process

I am facing something that can be explained as a balls & urns problem. Suppose you have $B$ black and $W$ white balls inside an urn. They are randomly chosen, one by one, without replacement, and ...
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19 views

Proof of finite expectation of renewal process (2) [duplicate]

I don't know if it is allowed here, to repost again his own question. I hope it is ok... I already asked this question here: Finite expectation of renewal process But I don't understand the last steps ...
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27 views

Supply the transition matrix for these (possible) Markov chains

Reading Grimmet, Stirzaker: Probability and Random Processes, which unfortunately doesn't have solutions. Trying to make sure I understand Markov chains. A die is rolled repeatedly. Which of these ...
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1answer
32 views

Covariance of a function of random variables

I want to find the covariance $K_X(t,t')$ of the following signal $X(t)$: $X(t)=\sum\limits_{n=-\infty}^{+\infty} A_np(t-nT)$ where $ p(t) = \begin{cases} \ 1 & \text{if } 0<t\leq T/2 ...
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1answer
38 views

The meaning of the connection between power spectral density and auto correlation

I know that if we have a signal $x(t)$, then its Fourier transform would be the signal in the frequency space, which I understand to be how much of each frequency exists in the x(t) signal. $ ...
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1answer
85 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": $(dt)^2=0$ ...
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1answer
18 views

Multi-dimensional Feynman Kac Theorem

I am trying to understand how to prove the multi-dimensional version of the Feynman-Kac formula. The single-dimensional version is proved on this page: en.wikipedia.org/wiki/Feynman–Kac_formula ...
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1answer
52 views

Expected value and covariance of compound Poisson process

$Y_1,Y_2,...$ are independent random variables with a distribution identical to that of $Y$. $N(t)$ is a poisson process with parameter $\lambda$. $$X(t)=\sum\limits_{n=1}^{N(t)}Y_n$$ Find the ...
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12 views

Determining Causal Relationships Between Two Processes (Pierce-Haugh)

Pierce and Haugh did some research on causality in temporal systems. For simplicity, consider two time series $\{X_{t}\}$ and $\{Y_{t}\}$. Suppose that both follow a causal and invertible $ARMA$ ...
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14 views

Quadratic Variation of Diffusion Process and Geometric Brownian Motion

I'm looking to find out the stochastic differential equation satisfied by the quadratic variation of Geometric Brownian Motion, Diffusion Process. For example, for a diffusion process that ...
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16 views

Asymptotic Distribution of Cross-Correlation Function Between Two AR(1) Processes

Let $$X_{t} = \phi_{x}X_{t-1} + e_{t}$$ and $$Y_{t} = \phi_{y}Y_{t-1} + \epsilon_{t}$$ be two $AR(1)$ processes. My textbook writes that $\sqrt{n} \cdot r_{XY}(k) \sim N \left(0, \frac{1 + ...
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16 views

Probabilities in a Markov Model

I am reading a paper on Markov Models and I am trying to figure out how to compute the probabilities for the $\alpha$-pass. I am given an $N\times N$ matrix $A$, that has the probabilities of ...
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19 views

Change of variable in stochastic integral

Let $B$ be a standard Bronwian motion. Can we do a change of variable in the sense $s=\theta+h$ $$\int_{0}^{t+h}X_sdB_s=\int_{-h}^{t}X_{\theta+h}dY_\theta.$$ In this case what is the process ...
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54 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
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1answer
48 views

Conditional expectation for Poisson process

Let $X(t)$ be a Poisson process with rate $\lambda = 6$ describing arrivals per hour of customers at a bank. Let the probability of a customer being male be $2/3$. Suppose 10 males has arrived during ...
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10 views

A Markov Decision Process problem

Consider an Stochastic shortest path problem where all stationary policies are proper. A stationary policy is said to be proper if, when using this policy, there is positive probability that the ...
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11 views

Determining Stationarity of (Multivariate) Processes

A necessary and sufficient condition for a vector autoregressive process to be stationary is for its eigenvalues to all have modulus less than one. I learned this result in the case of a model of this ...
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1answer
23 views

Filtered Poisson Process

I have a Poisson Process with rate $\lambda$ and also a filter which is applied on this process. After first event is issued, during time window $T$, all the following events are filtered. After the ...
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1answer
39 views

Invariant mesure of a reflected random walk

Let $(X_n), n \geq 0$ be a Reflected Random Walk defined by: $X_0 = 0$ and: $ X_{n+1}=\max( 0 , X_n + \xi )$ $\xi $ is a random variable such that $P(\xi=a)=\theta$ and $P(\xi=-b)=1-\theta$ for a ...
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55 views

Ito formula proof for bounded functions using stopping time

I'm self studying with the Oksendal book "Stochastic differential equations" and trying to do some exercises by myself. P.57 the exercise asks for the following (a screenshot will save us typing ...
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13 views

Fitting of the Lévy triple

Given a Lévy process and its triplet $(\mu,\Sigma,\nu)$ i.e. the triplet such that for each $t\ge 0$ $ X(t) = bt + W_A(t) + \int_{|x|<1} x \tilde N (t, dx) + \int_{|x|\ge 1} x N(t,dx)$ where ...
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1answer
61 views

The Lévy-Khintchine formula and integrability conditions of a random measure

I am trying to see the connection between the Lévy-Khintchine and the integrability conditions of a Lévy measure. The literature seems to always connect both, but I cannot make sense of this relation ...
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34 views

Reflected random walk

Suppose that $X_n$ is a reflected (in 0) random walk with parameter $\theta$. So $X_{n+1}-X_n = 1$ with probability $\theta$ , and -1 with probability $1-\theta$ when $X_n \geq 1$, if $X_n=0$ then ...
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45 views

Independence of two processes

Suppose $X_t$ is the solution of the SDE $$dX=a(X)dt+b_1(X)dW_1+b_2(X)dW_2$$ $Y_t$ is the solution of the following SDE $$dY=p(Y)dt+q_1(Y)dW_1+q_2(Y)dW_2$$ Here, $W_1$ and $W_2$ are independent ...
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1answer
48 views

Stochastic integral wrt the compensated Poisson random measure

I am solving the exercises in a book I have about Lévy processes ("Lévy Processes and Stochastic Calculus", Applebaum, 2003), and I cannot get my head around an exercise that seems rather simple. I ...
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1answer
22 views

Integration with respect to two different Brownian motions

Let $B$ be the standard Brownian motion. The process $W_s=B_{s+a}-B_a$ is also a Brownian motion. I just want an example of a process $X_s$ such that $$E\int_0^tX_sdB_s\neq E\int_0^tX_sdW_s.$$
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10 views

Time homogeneous asset dynamics model

I'm studying asset process. As i know, Black scholes model and CEV model is time homogeneous diffusion model. Are there time homogeneous model ???
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36 views

The projective limit of probability spaces and the Kolmogorov-Daniell theorem

Does the "projective limit" concept exist for probability spaces? The only result that I know of seems to be the Kolmogorov-Daniell theorem, but this is just a particular case where the spaces ...
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1answer
51 views

An example of stochastic process

I use the following definition for a stochastic process. Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection ...
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1answer
50 views

Given “N” number of events, calculate the number of condition to check whether all the events are statistically independent.

please help me out here, i dont even know where to start with this question :(. Any guidelines anything at all that may give me an idea to answering the question will be greatly appreciated. Please ...
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13 views

For a general absorbing markov chain, if we have that $I-Q$ can be inverted, is it possible to prove the chain covers all stationary distributions?

If I have a general absorbing markov chain, there are nice properties when $I-Q$ is invertible. In my book, it claims it can be shown that a vector: $(0,0,0,...,0,v_1,...,v_{N-r+1} \in ...
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1answer
20 views

Autocorrelation of a Wiener Process proof

Given a Wiener process X, how do I prove this? $R_x(s,t) = E[X(s)X(t)] = min(s,t)$ There seems to be a trick with dividing to two cases of $s<t$ and $s>t$, but I can't figure out how this ...
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29 views

Discontinuous Lévy-Processes with normal increments

Does there exist a Lévy-Process with normal increments but with paths that aren't even continuos when modified on null sets? I'm asking because when defining Brownian motion as Lévy-Process, ...
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96 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 ...
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0answers
24 views

Differential of stochastic process

How do I find the dynamics of $X_t=\int_0 ^t \sigma (s,t) dW_s$? It seems that the simple solution of $dX_t = \sigma(t,t)dW_t$ is not correct since I get $X_t = \int _0 ^t \sigma(s,s) dW_s$ if I ...
3
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1answer
85 views

Every Lipschitz function is the primitive of a measurable function

I was doing exercise 5 of this exercise sheet and I don't know how to conclude. I need to prove that if $f \colon [0,1]\to \mathbb{R}$ is Lipshitz, $X$ is a uniform$(0,1)$ random variable and ...
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38 views

Autocovariance Function of this $AR(1)$ Process

Consider the $AR(1)$ process given by $(1-0.6B)(X_{t} - 3) = a_{t}$ where $a_{t} \sim WN(0,1)$ and $B$ is the backshift operator ($X_{t}B = X_{t-1}$). We can rewrite the process in the more ...
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1answer
24 views

How to find the dynamics of stochastic process?

We have $Y_t=e^{\int_0^t W_sds}$. How do I obtain the dynamics of $Y_t$ (i.e. $dY_t$)? It seems that we can't use Ito Lemma because $\int_0^t W_sds$ is not in the form $X_t = \int_0 ^t \sigma_s dW_s ...
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31 views

Dynamics of short rate in HJM

According to a simplified HJM framework, we have: Forward Rate: $f(t,T)=\sigma W_t +f(0,T) +\int_0^t{\alpha(s,T)}ds$, where $W_t$ is brownian motion. Dynamics of forward rate: ...
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exercise 1.21 of chapter 1 of Revuz and Yor's

This is the exercise 1.21 of chapter 1 of Revuz and Yor's: Let $X=B^+$ or $|B|$ where $B$ is the standard linear BM, $p$ be a real number $>1$ and $q$ its conjugate number ($q^{-1}+p^{-1}=1$). ...
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36 views

Mean time spent in a state in a CT Markov Chain

I consider a continuous-time homogenous Markov chain: with discrete state $X$ taking values in $\mathcal{F}=\{1,\cdots,N\}$ with the transition rates satisfying: \begin{equation} \begin{cases} ...