A stochastic process describes evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples - Gaussian process, Markov Process, etc.

learn more… | top users | synonyms

2
votes
0answers
37 views

Unsure whether my continuous time Markov Chain distribution is correct

I' am reading Introduction to Stochastic Processes by Lawler and have hit problem 3.3(c) that I' am not sure I have correct. $\textbf{3.3}$ Suppose $X_t$ and $Y_t$ are independent Poisson processes ...
0
votes
0answers
8 views

Non Homogeneous Poisson Process Insurance example [on hold]

An insurance company finds that for a certain group of insured driver , the number of accidents over each 24 hours period rises from midnight to noon and then declines until the following ...
0
votes
0answers
9 views

Deriving the expectation of an OU model

I have an Ornstein-Uhlenbeck model (listed below) and I know what the expectation and variance should be, but I am having difficulty deriving them for myself. Can anybody point me towards a good ...
1
vote
0answers
18 views

Two-alternative forced choice [on hold]

Suppose that $p[r|+]$ and $p[r|-]$ are both Gaussian functions with means $\langle r \rangle_+$ and $\langle r \rangle_-$ and common variance $\sigma_r^2$. How can I show that $$P[correct] = ...
1
vote
0answers
44 views

Which machine learning technique is appropriate for my problem?

I'm new in machine learning topics and I've problem in modeling my environment which has multi parameters with different value ranges and a few actions to perform when value of each parameter is not ...
0
votes
0answers
12 views

Choice of Boltzmann Exponent for Simulated Annealing

I was wondering if there was a good source on the optimiality of different choices for the "Boltzmann Exponent" for Simulated Annealing. Mathematica defines it here, though it would generally apply to ...
0
votes
2answers
19 views

Is Stochastic gradient descent and Online gradient descent compatible with Map-Reduce?

My initial thoughts to this problem was no, since online and stochastic both use single values at a time. But what if say you have different online servers that act independently for a limited period ...
1
vote
0answers
23 views

Using an RNN/LSTM to generate sequences with a unique output

I'm trying to train a LSTM recurrent neural network where my data consists of a sequence of animal migration data ...
3
votes
1answer
34 views

What do the realizations of X(t)=Usin(t)+Vcos(t) where U and V are random variables with mean 0 and and variance 1 look like from -2pi to 2pi?

I'm not sure what the realizations of a time series really mean, and how to implement any kind of drawing with random variables. Any hints or descriptions would be very helpful.
0
votes
0answers
9 views

Setting up transition matrix for balls in an urn

Here is the question I'm trying to answer. I don't need a solution for this, but I had a question about how I would approach the problem. An urn contains two red and two green balls. The balls are ...
7
votes
3answers
219 views

What is the distribution for the time before K successes happen in N trials?

What is the distribution for the time before K successes happen in N trials? Suppose there is a telephone center, and N people, each of whom will either call the telephone center in time T with ...
2
votes
1answer
27 views

Is it a good idea to use $\eta = \arg \max( L(y, f(x) )$ to choose the step size for stochastic gradient descent (SGD)?

I wanted to have a good (as optimal as possible) automatic way of choosing the step size for minimizing the generalization error $\mathbb{E}_{ (x,y) \sim p_{x,y} }[L(y, f(x))]$, where $L$ is the loss ...
1
vote
0answers
23 views

Probability of conditional Wiener process

lets assume that we have $W_t$ - Wiener Process. What will be probability $$P[W_4-W_2>1|W_3=0.5]$$ I know that if instead of $W_3$ we would have $W_2$ then it would be independent, and ...
3
votes
3answers
52 views

Is it possible to generate data for stochastic process with specific distribution and autocorrelation?

It seems though that there is a disconnect between constructing paths of a stochastic process with both a specific distribution and autocorrelation. It seems like you can have either one property or ...
0
votes
1answer
32 views

Why can a process with independent increments never be a stationary process?

Why can a process with independent increments never be a stationary process? I don't understand the reasoning behind this. Thanks !
1
vote
1answer
45 views

writing down likelihood for dynamic state space model?

I have a discrete-time state space model where observations depend on a latent rate $X$. The prior on the rate is $X \sim \mathrm{Normal}(\mu_x, \sigma_x)$. Each observation $Y_t$ is generated using ...
0
votes
1answer
63 views

Write a computer program so that # heads/n diverges? [closed]

Suppose a computer program tosses a coin, and counts the times when it tosses 'head', denoted by $\#heads$. Let $$X_n=\frac{\#~heads}{n}.$$ Is there a way to write the computer program so $X_n$ ...
0
votes
0answers
20 views

How to find Kolmogorov Forward Equations, given generator matrix Q? [duplicate]

I am having difficulty in forming Kolmogorov Forward Equations. I understand how the KFE is derived and that $$\frac{d}{ds} p_{ij} (s) = \sum_{k \neq j} p_{ik} (s) \lambda_{k} r_{kj} - p_{ij} ...
1
vote
1answer
42 views

Question on notation in Brownian motion

I' am reading about the zero set of Brownian motion from Introduction to Stochastic Processes by Lawler. Let $X_t$ be a Brownian motion and define the following random set: $Z=\{t:X_t=0\}$ ...
2
votes
1answer
66 views

Confused about an example of Brownian motion

I' am reading Introduction to Stochastic Processes by Lawler and have hit a particular example given in the book about Brownian motion that confuses me. I'll give most of example here: Let ...
1
vote
0answers
60 views

Stochastic Volatility and SDEs

Consider the following SDEs $$\begin{align} &\dfrac{\text{d}S}{S}(t) = \alpha(t)\text{ d}t + \sigma(t)\text{ d}Z^{(1)}(t) \\ &\dfrac{\text{d}\sigma}{\sigma}(t) = \beta(t)\text{ d}t + ...
1
vote
1answer
99 views

How to compute expectation of square of Riemann integral of a random variable?

How does one compute $E[(\int_0^T W_s ds)^2]$ where $(W_t)_{t \in [0,T]}$ is standard Brownian motion in $(\Omega, \mathscr F, \mathbb P)$? Apparently proving $$\int_0^T W_s ds = \int_0^T (T-s) dW_s ...
2
votes
0answers
55 views

Random walk with continuously distributed steps on $[-1,1]$

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability $$P(S_n \textrm{ reaches } a \textrm{ before} -b) ...
0
votes
1answer
26 views

What is the variance for $t \rightarrow 0^+$ of a Cox-Ingersoll-Ross process?

Let us define a Cox-Ingersoll-Ross process as: $$ d X(t) = \alpha (\mu - X(t))dt + \sigma \sqrt{X(t)} dW(t)$$ with $X(t)$ distributed as a non-central Chi-square and $W(t)$ as a Wiener process. The ...
0
votes
0answers
12 views

Inference arrival rate by samples of forward recurrence time for renewal process

There is a regenerative renewal process with iid. inter arrival time $X$, which follow the distribution $F(x)$. $N(t)$ is the counting process and $S_{N(t)} = X_0+X_1+\dots+X_{N(t)-1}$ is the time ...
8
votes
0answers
121 views

Special probability distribution

If $p(x)$ is a probability distribution with non-zero values on $[0,+\infty)$, for what type(s) of $p(x)$ there exist a constant $c>0$ such that $\int_0^{\infty}p(x)\log{\frac{ ...
3
votes
3answers
48 views

How do you see a Markov chain is irreducible?

I have some trouble understanding the Markov chain property irreducible. Irreducible is said to mean that the stochastic process can "go from any state to any state". But what defines whether it can ...
0
votes
0answers
41 views

Is there such thing called a Uniform point process? (not Poisson point process)

We know a Poisson spatial point process is characterized by the following properties: The Total number of points N follows a Poisson distribution Given N, the point process is a uniform distribution ...
0
votes
0answers
24 views

Hidden Markov Models and Viterbi Algorithm: Fair and Biased Die

So following is the problem that I am trying to solve using Viterbi algorithm and HMM: Before attempting to write a program, I want to do this problem by hand for the first 3 observations($651$). ...
0
votes
1answer
21 views

approaches to predict probability of “game result”

sorry for undetermined terms, I just never got familiar with this type of tasks. Suppose I have a dataset with M results of games between N teams (M >= N). Like t1 vs t2 = t1 won t1 vs t3 = t3 won ...
1
vote
2answers
62 views

Why is Geometric Brownian motion not a Lévy process?

I'm trying to price an option. The old-style (e.g. Black Scholes 1973) pricing models use the GBM to model the underlying asset, which suffers of some deficiencies wrt volatility smiles and term ...
4
votes
1answer
129 views

Meaning of $\lim\limits_{n\rightarrow \infty}\mathbb{E}(X_n^2)=0$?

This is related to convergence in probability: What's the meaning of $\lim\limits_{n\rightarrow \infty}\mathbb{E}(X_n^2)=0$ for random variables $X_1,X_2,...$.
3
votes
1answer
168 views

Using Keras LSTM RNN for variable length sequence prediction

I have a set of sequences. Each sequence is the form $\{(s_1,l_1),(s_2,l_2) \ldots\}$ where $s_i$'s are real valued numbers and $l_i$s are labels from a fixed alphabet. It is important to note that ...
1
vote
2answers
53 views

Help me understand this: $ Pr(T < t \mid y) = \int_{0}^{\infty }\int_{\Omega }k(x)p(t, x\mid y)\,dx\,dt \ $?

How do I read this equation (especially the left side) in terms of a Continuous Markov Process model? $$ Pr(T < t \mid y) = \int_{0}^{\infty }\int_{\Omega }k(x)p(t, x\mid y)\,dx\,dt \\ $$ Where $ ...
0
votes
0answers
29 views

Will n-th order Markov Chain have a better prediction than first order chain?

I am a student learning Stochastic Process right now. The Markov process was defined on a history independence hypothesis, however with some situations, the history-dependent data with Markovian could ...
1
vote
0answers
28 views

Stochastic process: Number of states visited in $n < \infty$

I am from Spain and I don't know the exact English term, so I will try to explain the definition. In a discrete stochastic process, let: P be the transition matrix with all is states being ...
2
votes
1answer
74 views

What's the forecast of a Random Walk with Noise model?

I have a RW with noise model defined as: $$ y_{t} = z_{t} + v_{t}$$ where $ z_{t} = z_{t-1} + e_{t}$. $v_{t}$ and $e_{t}$ are mutually independent with expectation $0$ and variance $\sigma_{v}^{2}$ ...
0
votes
1answer
24 views

Spatial correlation and tolerance limits

I have previously studied the problem of defining tolerance limits for normal distributions for a given small sample of observations. Now I would like to take into consideration the fact that the ...
1
vote
0answers
23 views

Estimating the parameters of a stochastic asset price models using Matlab

I am simulating asset prices using different existing stochastic models as well as my own proposed stochastic models. I would like to estimate the parameters of each model by using the historical spot ...
0
votes
0answers
11 views

Actual volatility and option pricing under stochastic volatility model

I have read stochastic volatility (SV) model but finding difficult to visualize the model. I have few questions: What is proxy for actual volatility in stochastic volatility model as it assumes ...
1
vote
1answer
188 views

Buy Till You Die(BTYD) - Model Validation in R

I have used the BTYD Package in R to estimate the number of transactions that a customer is expected to make in the future. I have 2 questions: 1) Is the number of transactions always a non-integer ...
4
votes
2answers
101 views

Derivative of a Gaussian Process

I believe that the derivative of a Gaussian process (GP) is a another GP, and so I would like to know if there are closed form equations for the prediction equations of the derivative of a GP? In ...
1
vote
1answer
93 views

Prove/Disprove $\sigma(Z_0, Z_1, …, Z_n) = \sigma(V_0, V_1, …, V_n)$

Given a stochastic process $Z = (Z_n)_{n \geq 0}$ on a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$ where $\mathscr{F_n} = \mathscr{F_n}^{Z} \doteq \sigma(Z_0, ...
2
votes
0answers
40 views

What do people mean when they say a function is “linear” or “non-linear”

It seems like every function/process can be "linear" given the right basis. e.g. the function x**2 is linear in the state space which includes polynomials of degree 1 to n. Similarly, a process like ...
2
votes
1answer
69 views

How to use “kernel trick” in Stochastic gradient descent?

How to use "kernel trick" in Stochastic gradient descent? I can find kernel perceptron on Wikipedia but I can't find "kernel sgd" anywhere that gives me a clear algorithm to do that. Can someone teach ...
1
vote
0answers
21 views

Large sample distribution of $\beta$

Suppose we have a linear model : $y = X \beta + \epsilon$. Could anyone provide me with some reference or an answer about the unconditional, large sample distribution of the vector of regression ...
1
vote
1answer
35 views

Busy sharing cars

Given a pool of C cars available to use by P people and that the average person uses a car t percent of the time, how often will there be zero cars available for use? For example, if there are 500 ...
0
votes
0answers
6 views

Simulation of diffusion bridge when transition density is known

Consider an Ito process $dX_t = a(t, X_t) dt + b (t, X_t) dW_t$ with known transition density $P(X_t|X_0)$. We can assume the functions $a$ and $b$ are differentiable. Suppose we have data for 2 ...
1
vote
1answer
48 views

Expected Value in Poisson Point Process with Prior Knowledge

I have a setup with a homogeneous Poisson Point Process (PPP) of intensity $\lambda$ in $W \subseteq \mathbb{R}^d$ and a set $A \subseteq W$. I'm looking for the expected value of points in set $A$, ...
0
votes
1answer
24 views

What is the 'Accuracy' of Markov's inequality upper bound estimate

I'm confused with part e) in this question since nowhere at all in my notes was 'accuracy' of the upper bound brought up. What I calculated for c) was: P(N3>0)=0.422 for p=0 P(N3>0)=0.2929 for ...