Stochastic Definition and 166 Threads

If a stoch. process Xt has independent and weak stationary increments. var(Xt) = σ^2 for all t, prove that Cov(xt,xs) = min(t,s)σ^2 I'm not sure how to do this. I tried using the definition of covariance but that doesn't really lead me anywhere. If it's stationary that means the distribution...

I am really stuck on this and can't find any information or examples on it. My textbook has one example in it and it isn't explained at all. It is the following? ∫B{s}ds integrated From 0 to t.It is then changed to a double integral and fubinis theorm is applied to arrive at the solution...

Homework Statement If people entering the engineering building following a PP (9/hora) and you know that between 10:00 am and 11:00 am came exactly 100 persons, what is the probability that between 10:00 am and 10:20 am entered less than 20 people If people entering the engineering...

Hi, I need some help with this hw 1. Suppose that the passengers of a bus line arrive according to a Poisson process Nt with a rate of λ = 1 / 4 per minute. A bus left at time t = 0 while waiting passengers. Let T be the arrival time of the next bus. Then the number of passengers who...

[PLAIN]http://img17.imageshack.us/img17/1061/stochcalcq4.png I am currently taking a class in quantitative finance, part of which includes an introduction to stochastic calculus. This is the first time i have encountered stochastic differential equations, so it is all quite new to me. I am...

Sorry if this is in the wrong section but i have a problem, I have no experience with stochastic equations well analytically anyway. The equation i have is the following; \frac{dv}{dt} = - \alpha v+ \lambda F+\eta Where alpha lambda and F are constants, v is a variable (speed in this...

Hi, I am a math and physics major planning on going into biophysics for grad school, and i want to do computational/mathematical modelling/theoretical work in the field. I have one more math course to take and I am not sure which would be more useful. Here are their very brief course...

Hi, Could some one help me to solve the equations ? dX =sqrt(X) dB where X is a process; B is a Brownian motion with B(0,w) =0;sqrt(X) is squart root of X.

Hello everyone, I'm studying from Oksendal's book, and I'm stuck at an excercise which asks you to find the differential form of: X(t) = (W(t)^{2}-t)^{2} - 4\int (W(s))^{2}ds where W(t) is a Brownian Motion. I tried several possible functions g(t,W(t)) which could have led to a potential...

Hi everyone, I am trying to solve this problem but I am stuck with doubts. Here are my ideas. Homework Statement Busloads of customers arrive at an infinite server queue at a Poisson rate λ Let G denote the service distribution. A bus contains j customers with probability aj = 1...

Hi, I do some calculations on a rf-pulse controlled Spin-1/2 system influenced by noise given by a normal distributed random variable n(t) (which is, I guess, a gaussian stochastic process, as n(t) is a gaussian distributed random variable for all t). Using the Magnus-Expansion...

This is an extract from my lecture notes: "For classical stochastic systems, w(p,x,t)dpdx = prob. particle is in dpdx. w\geq0 \int dp \int dx w(p,x) = 1." 1. Can anyone please explain what a classical stochastic system is? 2. Why is there a question of probability in analysing such a...

hello to everyone, I have a problem solving a stochastic differential equation of the form: dX/dt=aX²+bX+c+sXn(t), where n(t) is white noise with a mean value equal to 0 and variance equal to one. Does anyone know the solution of this stochastic differential equation or how to solve...

Hello, I've studied physics at a university previously and actually earned a degree in theoretical physics, but then switched over to mathematics, where I focused on stochastic analysis/calculus/processes (I'll just call it stochastics). Now, I remember taking a course on stochastics while...

Hello all, I have run into this problem, and being that I know nothing about stochastic DiffyQ I am trying to toy around with it. Basically, the following is a boiled down version of my problem: I have a probability density function that is given: p(t) and let's say we pick 1 value from...

Hi, there. I am not major in physics so maybe I lack some basic knowledge. Imagine one have an ideal sensor, which can convert the emission to some kinds of signal (typically, voltage), then what process can describe the measure data? Is it related to...

1. Assume that earthquakes strike a certain region at random times that are exponentially distributed with mean 1 year. Volcanic eruptions take place at random times that are exponentially distributed with mean 2 years. What is the probability that there will be two earthquakes before the next...

Homework Statement 1. Assume that earthquakes strike a certain region at random times that are exponentially distributed with mean 1 year. Volcanic eruptions take place at random times that are exponentially distributed with mean 2 years. What is the probability that there will be two...

Homework Statement The total population size is N = 5, of which some are diseased and the rest are healthy. During any single period of time, two people are selected at random from the population and assumed to interact. The selection is such that an encounter between any pair of individuals...

I am learning Stochastic Processes right now. Can someone some explain why waiting time is memoryless? Say, if a light bulb has been on for 10 hours, the probability that it will be on for another 5 is the same as the 1st 5 hours. It doesn't make sense to me, because the longer you use it...

Homework Statement Calculate the Kalman-Bucy Filter equations Homework Equations F=(0 1)' K is unknown but y = X_1 + d/dt(v) E=((Fw-Kv)(Fw-Kv)')=FQF' + KRK' Q = E(ww') and R = E(vv') The Attempt at a Solution There is more to this question but I am just having trouble understanding where Q...

Hi, I have several sets of stochastic signals that oscillate about the x-axis over time. I would like to transform these signals into the frequency domain (make a periodogram) so that I can which signal has the most stable frequency. I was thinking about using taking the Fourier transform...

This semester I have a course on mathematical methods in physics. It's in three parts and the first professor is talking about Markov processes (discrete and continuous time) and diffusion. The problem is he doesn't have any notes or a reference textbook. Do you know any textbook on these topics?

Hi to everybody, I'm going to apply for the theme mentioned in the title during my study and further by writing scientific works. Also I'm very excited with it because of its applications. Couldn't anyone suggest some literature for beginners in Stochastic Calculus? P.S. I also have some...

If we have a DE of the following form: \frac{dX}{dt}=b(t,X_t)+\sigma(t,X_t).W_t and look for a stochastic process to represent the (second) noise term. Now my textbook tells me that the only process with 'continuous paths' is Brownian motion. The noise term denotes random, indeterministic...

hello if we have set of stochastic variables representing the random time it takes to do something: X,Y,Z,W and C where C is the sum of X Y Z W, thus the time it takes to do these things in sequence. If: X: N(30,5) Y: N(30,3) Z: N(20,2) W: N(40,7) makes C adding these together right, mean plus...

Homework Statement How to solve this SDE? dX_t = [1/X_t] dt + aX_t dB_t Homework Equations The Attempt at a Solution If I didnt get it wrong, this is not a general linear SDE, and my course in elementary stochastic calcus did not cover SDEs other than the general linear ones...

Homework Statement How to prove that the Ito Integral int_0^t e^s dB_s is normally-distributed, for a given t? Homework Equations The Attempt at a Solution This Ito Integral could be defined as a R-S Integral, and the Riemann Sum should be a linear function of normal r.v.s...

Homework Statement Suppose that passengers arrive at a train terminal according to a poisson process with rate "$". The train dispatches at a time t. Find the expected sum of the waiting times of all those that enter the train. Homework Equations F[X(t+s)-X(s)=n]=((($t)^n)/n!)e^(-$t))...

In the attached equations, for the second last step to the last step why dSdS=sigma2S2dt ?

The attachment is from Shreve's stochastic calculus book In the attachment what does the symbol ^ mean? Thanks

Dear Community, I am faced with a challenge. I can't quite grasp the implications of this. I schould've listened better in statistics lectures! :blushing: I would really appreciate your help. :) I have a not normal stochastic process P on which I can bet. Obviously, I don't know the...

Can we consider a stochastic process being chaotic? consider evolution of only two particular systems with closed initial states (not ensemble or statistical properties of the system)

Can someone explain to me the rigorous meaning of statements like: dt^2 = 0 dW*dt = 0 dW^2 = dt Here W = W(t) is standard Brownian motion. I know that a SDE such as dX = f dW + g dt rigorously means X(t) = X(0) + \int_0^tfdW + \int_0^tgds But what does dt^2 mean? And why...

Would anyone have any resources on the stochastic interpretation of quantum mechanics? It appears to be a relatively new interpretation, proposed this year by Roumen Teskov, based on John Wheeler's "quantum foam." That's really the extent of the information I have, and I'm curious to find...

From my extremely small and inadequate knowledge of stochastic processes (and Wikipedia): A stochastic process is a process in which some later state is determined by predictable actions and by a random element. Now the question: this "random element" is this meant to compensate for...

Urgent Stochastic Calculus help! Hi, I am new to stochastic calculus and finding some difficulty in understanding things. How to approach the solutions for problem under the topics like martingale, linear diffusion SDEs, expectation of martingale, Ito stochastic integral formulas...

Hi, A quick question regarding random functions. Suppose \xi(t) is a stochastic function. In other words, its value at time t is random with some known distribution (Gaussian, say). Is there any way of calculating \frac{d\xi}{dt}? Thanks,

Homework Statement Let P be a stochastic matrix on a finite set I. Show that a distribution π is invariant for P if and only if π(I-P+A) = a, where A = (aij : i,j in I) with aij = 1 for all i and j, and a = (ai : i in I) with ai = 1 for all i. Deduce that if P is irreducible then I-P+A is...

I would like to construct a model using a markov chain that has different stochastic processes for each state in the chain. Is there a term for such a thing, or anything similar to it? Thanks

I had this problem on my last midterm and received no credit for these parts. 1. Express trains arrive at Hiawatha station according to a Poisson process at rate 4 per hour, and independent of this, Downtown local buses arrive according to a Poisson process at rate 8 per hour. a. Given that 10...

I have a few quetions which I'm trying to answer, hope someone can hint me to the right answer. 1.Prove/disprove that there doesn't exist a Markov Chain with transition matrix which all of its entries are positive and which it has infinite transient states and infinite reccurent states. 2...

Hi there, I'm starting revision for Stochastic Analysis and have a few questions relating to the notes I'm reading. I'd much appreciate any clarification as I'm not as up to speed as I'd like. 1) In the definition of classical Wiener space I have H=L_{0}^{2,1}([0,T]; \mathbb{R}^{n}) the...

Suppose that A and B follow geometric brownian motion, where zA, and zB follow wiener process dA/A=a*dt+b*dzA dB/B=c*dt+d*dzB dzA*dzB=e*dt What stochastic process does A/B follow? This is not a homework question(I am sure it's almost trivially easy to those who learned the stuff). I am very...

A component in a manufacturing process breaks down regulary and needs to be replaced by a new component. Assume that the lifetimes of components are i.i.d. random variables. The company adopts this policy: a component is replaced when it breaks down or after it has operated for time "a"...

Homework Statement Question: 18. Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written as P = |1-a b | | a 1-b | where a and b are constants between 0 and 1. (There are two linearly independent steady-state...

Homework Statement http://img411.imageshack.us/img411/4274/50122514bc3.png Homework Equations http://img133.imageshack.us/img133/4624/68596500xm4.png The Attempt at a Solution I don't know how to start I've found this: Let X be the the winnings per bet and let the total profit...

Homework Statement I know that per definition E(N)= \sum P(N=k) \cdot k . But how can I rewrite the above expectation towards the 'usual definition'?

I am taking a project this year. As the title suggests this project is a maths project but I was wondering if anyone can direct me to some science field in which I could use stochastic models. I have very limited knowladge in science so something theoretical would be nice. Any books would be...

Homework Statement This is a question about one single step of a solution of a long equation. http://www.geocities.com/link_herooftime/math.jpg where P, U and V are variables. a, b, c, d are constants and t is the time, which are measured in discrete periods. The question is how...