Stochastic Definition and 166 Threads

What textbooks would you recommend for self studying Nonlinear Dynamics? I am a undergraduate junior who will be doing research on nonlinearity of spiking neurons. I have taken courses on ODE, vector calculus, probability, statistics, and linear algebra.

I was wondering how useful a course in basic stochastic processes is if you want to pursue a career in physics? And especially for a theoretical physicist or astronomer. Im going to have to choose two courses next semester and I think I'm going to choose Special relativity and Mathematical...

The problem is:Let $W(t)$, $t ≥ 0$, be a standard Wiener process. Define a new stochastic process $Z(t)$ as $Z(t)=e^{W(t)-(1/2)\cdot t}$, $t≥ 0$. Show that $\mathbb{E}[Z(t)] = 1$ and use this result to compute the covariance function of $Z(t)$. I wonder how to compute and start with the...

Are there any tutorials that apply stochastic differential equations to the settings of elementary physics problems ? - for example, an object sliding down a not-frictionless ramp. The ramps of everyday life don't have a constant coefficient of friction. A better model for them would be...

Hi guys, I´ve just started with SDE and Itô´s lemma but don't really know where and how to begin. I´ve realized that both a Reimann integral and Itô integral is needed both cannot figure out how to solve these questions. Would be much appreciated if someone would help me.

I had a brief question regarding SDEs. Typically, I've seen models like the Ornstein-Uhlenbeck process that generally revert back to the mean over time. However, I've been trying to find a stochastic differential equation/process that avoids the mean, such as a sharp increase followed by a sharp...

I am new to SDE, and especially Ito's Lemma. I have a question that I simply cannot answer. It attached. Can someone please help?

I am studying an article which involves stochastic variables http://www.rmki.kfki.hu/~diosi/prints/1985pla112p288.pdf. The author defines a probability distribution of a stochastic potential V by a generator functional G[h] = \left<exp\left(i\int...

I've been reading Oksendal, and it's quite tedious. It want to see if my understanding of the motivation and process is correct. 1) Differential equations that have random variables need special techniques to be solved 2) Ito and Stratonovich extended calculus to apply to random...

Hello Is it true that if a counting process has stationary increments, then it is time homogeneous? stationary increments → time homogeneous? And is it also true that independent increments gives Markovian property? that is : independent increments → markovian property ? The...

The first image contains questions And the second image shows my answer which i manage to do so far, but some of them i could not do it.

Homework Statement Suppose I wish to fit a plane z = w_1 + w_2x +w_3y to a data set (x_1,y_1,z_1), ... ,(x_n,y_n,z_n) Using gradient descent Homework Equations http://en.wikipedia.org/wiki/Stochastic_gradient_descent The Attempt at a Solution I'm basically trying to figure out the...

An introductory text is preferable. Topics relevant (not a deal-breaker if not covered): Poisson process, Markov chains, renewal theory, models for queuing, and reliability. Also, in the future I'd like to dabble in stochastic calculus, but my background in measure theory is non-existent. I've...

Good day. For $k\geq0$ a continuous function on $\mathbb{R}_+$ and $\{W_t\}$ a standard Brownian motion, could you help me find the Taylor's expansion of the following exponential: $e^{-\int_0^t k(s) dW_s}.$ For the case where $e^{-k W_t}$ where $k>0$ is a constant, I was able to recompute...

Can anyone provide references for stochastic processes where future steps do depend on the past state of the system? Most of the material I'm finding deals purely with memoryless processes. Thanks!

I really need your help for a solution to these exercises. I will be so grateful.1/ passengers arrive at a train station according to a poisson process of rate lambda per minute and trains depart station according to a renewal process with inter-departure times uniformly distributed between a...

Two independent random variables X and Y has the same uniform distributions in the range [-1..1]. Find the distribution function of Z=X-Y, its mean and variance. =Using change of variables technique seems to be easiest. fX(x) = 1/2 fY(y) =1/2 f = 1/4 ( -1<X<1 , -1<Y<1) Using u =x -y...

Two independent random variables X and Y has the same uniform distributions in the range [-1..1]. Find the distribution function of Z=X-Y, its mean and variance. =Using change of variables technique seems to be easiest. fX(x) = 1/2 fY(y) =1/2 f = 1/4 ( -1<X<1 , -1<Y<1) Using u...

Random variable X is distributed according to Gaussian distribution P(x)= (1/sqrt(2πσ^2) * exp(- (x-μ)^2 / (2σ^2) ) 1. Calculate its first three moments 2. What are the distributions of Y= X+b; Z=σX; W=X^2 P(x)= (1/sqrt(2πσ^2) * exp(- (x-μ)^2 / (2σ^2) ) so 1.1. the moment of first...

Find characteristic functions of 1. The random variable X uniformly distributed on[-1..1] 2. The random variable Y distributed exponentially (with exponent λ) 3. The random variable Z=X+Y

Hello there. The stochastic calc book I'm going through ( and others I've seen ) uses the phrase "\mathscr{F}-measurable" random variable Y in the section on measure theory. What does this mean? I'm aware that \mathscr{F} is a \sigma-field over all possible values for the possible values of...

Hello, when we have a deterministic signal f:ℝ→ℝ that is square integrable we can typically write f \in L^2(\mathbb{R}). However, what if \{ f(t): \; t\in \mathbb{R} \} are random variables, i.e. f is a continuous-time stochastic process? What is the notation to denote the space of "square...

Homework Statement Given X(t) = cos(2\pi50t + ω), where the stochastic variable ω is uniformly distributed between 0 and 2\pi. Suppose the sampling frequency fs is 30 Hz. What frequency interval is covered after the sampling? Homework Equations Normalized frequency when sampling can be...

In his book "The Demon-Haunted World" Carl Sagan talks about this, I wonder what kind of physics phenom is that. t.i.a.

Imagine there is a complex system and we are interested in its basic statistical properties, like the stationary probability distribution. For example for a single electron wandering in defected lattice of semiconductor. Physics offers two basic ways of answering such question: - from one side...

My university is offering a course called "Stochastic Process". The only prerequisites to this course according to my university is a course in Probability which uses the book by Rosen. I've read elsewhere that the course actually requires more of analysis (functional analysis and measure...

As the title says, while using Stochastic Calculus, can someone explain some of the properties of differentials? Why does dB_t dB_t=dt Also, why does dt dt=0 and dB_t dt=0 I don't really get why these work?

Hi, I'm trying to find a probability distribution (D) with the following property: Given two independent stochastic variables X1 and X2 from the distribution D, I want the difference Y=X1-X2 to have a uniform distribution (one the interval [0,1], say). I don't seem to be able to solve it...

Dear forum members, I am trying to solve the following system of equations. ψ(x,y,z)=∫∫ψ(x',y',z)K(x',y',z)dx'dy' z=f(ψ) What I do is to solve the integral equation with a Monte Carlo method, evaluate "z" and do a loop until convergence. My question to you is whether it is...

Would anyone be able to help me with a problem involving martingales and another problem dealing with Poisson processes perhaps? I'm completely stuck on my last homework assignment in the class this term. Thanks for anyone who can help.

Dear All: Given two random variables X and Y, if I have established the relationship E[X]>=E[Y], does this necessarily imply that X must have a first-order-stochastic dominance over Y? I know that first order stochastic dominance implies that the mean value of the dominating random...

Hello. I plan on doing independent study on the Stochastic Process and time series models. I have already learned two semesters worth of statistics (Mathematical Statistics and Applications by Wackerly, Mendenhall and Scheaffer). And I have taken a semester of multiple regression models. I...

I hope someone can put me on the right track here. I need to derive the infinitesimal generator for a bridged gamma process and have come a bit stuck (its for a curve following stochastic control problem - don't ask). Any tips, papers, books that could guide me out of my hole would be greatly...

Consider stochastic process ##X(t)## with properties $$ \langle X(t) \rangle = 0, $$ $$ \langle X(t) X(t-\tau) \rangle = C_0e^{-|\tau|/\tau_c}. $$ For example, the position of a Brownian particle in harmonic potential can be described by ##X##. In that case, the probability...

One of my math teachers discussed stochastic ("random") variables today. In an example, he discussed the probability of picking a random number n, such that n\inℝ, in the interval [0,10]. He proceeded to say that the probability of picking the integer 4 (n = 4) is 0, supporting his claim with...

Why does: \int_0^t d(e^{-us} X(s)) = \sigma \int_0^t e^{-us} dB(s) for stochastic process X(t) and Wiener process B(t)? Also, why is the following true: \int_0^t d(e^{-us} X(s)) = e^{-ut}X(t) - X(0)

What does the right arrow mean in this context: " ...Then the process t \mapsto \int_{0}^t \phi_s dM_s are well-defined continuous local martingales, whose quadratic variations are given by ... " Is this supposed to mean "the process X that is the mapping X: t \mapsto \int_{0}^t \phi_s dM_s"

Touch of background, about half way through a mechatronics engineering degree, I found that I love fluid mechanics and from previous studies I know I enjoy EM. To throw a curve ball in I also am fascinated by the concept of randomness and the associated mathematics. Now for the question, could...

Hi I am reading about the Langevin stochastic differential equation \frac{d}{dt}p = -\alpha p + f(t) where p is the momentum and f(t) the Langevin force. By definition <F(t)>=0 and <f(t)f(t')> = 2Dg(t-t'), where g is the second order correlation function. My question is, why is...

I am trying to reproduce results of a paper. The model is: dS = (v-y-\lambda_1)Sdt + \sigma_1Sdz_1 \\ dy = (-\kappa y - \lambda_2)dt + \sigma_2 dz_2 \\ dv = a((\bar{v}-v)-\lambda_3)dt + \sigma_3 dz_3 \\ dz_1dz_2 = \rho_{12}dt \\ dz_1dz_3 = \rho_{13}dt \\ dz_2dz_3 = \rho_{23}dt \\...

Homework Statement I'm trying to figure out how to use Ito's Lemma, but all I have are notes and proofs. It would help if someone could go through one actual example with me: Use Ito's Lemma to solve the stochastic differential equation: X_t=2+\int_{0}^{t}(15-9X_s)ds+7\int_{0}^{t}dB_s and...

This is not actually a problem but I need clarification for stochastic ordering From Wikipedia, there is stated: (http://en.wikipedia.org/wiki/Stochastic_ordering) Usual stochastic order: a real random variable A is less than a random variable B in the "usual stochastic order" if Pr(A>x)...

Homework Statement A company incurs manufacturing costs of $q per item. The product is sold at a retail price of $p per item with p > q. The customer demand K (e.g. number of items that will be sold when the number of items offered is large enough) is a discrete random variable in N. The...

Homework Statement Suppose a book of 600 pages contains a total of 240 typographical errors. Develop a poisson approximation for the probability that three partiular successive pages are error-free. The Attempt at a Solution I say that the number of errors is poissondistributed...

Hi everyone, I am comparing the following optimization problems: Prob Space =(Omega, F_T, (F_t)_{t=1, ..T}, P). Let X be an adapted process. I denote E[|F_t] as the conditional expectation given F_t. 1.Z_t(omega)= max{ E[X_s | F_t](omega) : s=t, ..,T} 2. Y_t(omega)=max{E[X_tau |...

I can find the stationary distribution vector $\boldsymbol\pi$ for a stochastic matrix $P$ using: $\boldsymbol\pi P=\boldsymbol\pi$, where $\pi_1+\pi_2+\ldots+\pi_k=1$ However, I can't find a textbook that explains how to do this for a doubly stochastic (bistochastic) matrix. Could somebody...

Homework Statement I can find the stationary distribution vector \boldsymbol\pi for a stochastic matrix P using: \boldsymbol\pi P=\boldsymbol\pi, where \pi_1+\pi_2+\ldots+\pi_k=1 However, I can't find a textbook that explains how to do this for a doubly stochastic (bistochastic) matrix...

The technical definition of an adapted stochastic process can be found here https://en.wikipedia.org/wiki/Adapted_process. I understand the following chain of consequences from this definition: {X_i} is adapted \Rightarrow Each random variable X_i is measurable with respect to the...

Hey! Just as the title suggests I am looking for a good book on stochastic processes which isn't just praised because it is used everywhere, but because the students actually find it thorough, crystal-clear and attentive to detail. Hopefully with solved exercises and problems too! Anyone...