Lévy_processes on The Dan MacKinlay family of variably-well-considered enterprises
https://danmackinlay.name/tags/l%C3%A9vy_processes.html
Recent content in Lévy_processes on The Dan MacKinlay family of variably-well-considered enterprisesHugo -- gohugo.ioen-usSat, 25 Jul 2020 10:27:21 +1000Lévy processes
https://danmackinlay.name/notebook/levy_processes.html
Sat, 25 Jul 2020 10:27:21 +1000https://danmackinlay.name/notebook/levy_processes.htmlGeneral form Intensity measure Subordinators Spectrally negative Martingales Sparsity properties Bridge processes Recommended readings \(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\)
Stochastic processes with i.i.d. increments over disjoint intervals of the same length, i.e. which arise from divisible distributions. Specific examples of interest include Gamma processes, Brownian motions, certain branching processes, non-negative processes…
Let’s start with George Lowther:
Continuous-time stochastic processes with stationary independent increments are known as Lévy processes.Polynomial chaos expansion
https://danmackinlay.name/notebook/polynomial_chaos_expansion.html
Fri, 22 May 2020 10:13:53 +1000https://danmackinlay.name/notebook/polynomial_chaos_expansion.htmlVanilla Generalized Placeholder.
Vanilla Wikipedia says:
Polynomial chaos (PC), also called Wiener chaos expansion,is a non-sampling-based method to determine evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters. PC was first introduced by Norbert Wiener where Hermite polynomials were used to model stochastic processes with Gaussian random variables. It can be thought of as an extension of Volterra’s theory of nonlinear functionals for stochastic systems.Stochastic differential equations
https://danmackinlay.name/notebook/stochastic_differential_equations.html
Mon, 18 May 2020 12:23:18 +1000https://danmackinlay.name/notebook/stochastic_differential_equations.htmlPlaceholder. A time-indexed, causal, measure-valued stochastic process. As seen in state filters, optimal control, financial mathematics etc.
Cosma’s explanation of SDEs looks good for cannibalising for parts when I write my own.
Useful tools: infinitesimal generators, martingales.
One difficulty in this field is that many references take SDEs to be synonymous with Itō processes, whose driving noise is Brownian. In full generality, e.g. (Kallenberg 2002) they are a lot more general than that.Gamma processes
https://danmackinlay.name/notebook/gamma_processes.html
Fri, 28 Feb 2020 09:18:43 +1100https://danmackinlay.name/notebook/gamma_processes.htmlGamma distribution Moments Multivariate gamma distribution with dependence Gamma superpositions The Gamma process Gamma bridge Time-warped gamma process Matrix gamma processes Centred gamma process As a Lévy process \[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]
Gamma processes provide the classic subordinator models, i.e. non-decreasing Lévy processes. By “gamma process” in fact I mean specifically a Lévy process with gamma increments. Other processes that happen to have gamma marginals, e.Potential theory in probability
https://danmackinlay.name/notebook/potential_theory_probability.html
Wed, 12 Feb 2020 09:57:03 +1100https://danmackinlay.name/notebook/potential_theory_probability.htmlPlaceholder. I am unfamiliar with potential theory a thing in itself. I keep running in to it, in Markov stochastic proccesses and in graphical models and would like to know that I understand the tools I am using properly. Some at least of the results seems to be terminological updates of words I know already, other perhaps not.
Doyle, Peter G, and J Laurie Snell. 1984. Random Walks and Electric Networks.Branching processes
https://danmackinlay.name/notebook/branching_processes.html
Fri, 07 Feb 2020 17:33:31 +1100https://danmackinlay.name/notebook/branching_processes.htmlTo learn We do not care about time Discrete index, discrete state, Markov: The Galton-Watson process Continuous index, discrete state: the Hawkes Process Continuous index, continuous state Parameter estimation Discrete index, continuous state Special issues for multivariate branching processes Classic data sets Implementations A diverse class of stochastic models that I am mildly obsessed with, where over some index set (usually time, space or both) there are distributed births of some kind, and we count the total population.Infinitesimal generators
https://danmackinlay.name/notebook/infinitesimal_generators.html
Wed, 05 Feb 2020 09:33:10 +1100https://danmackinlay.name/notebook/infinitesimal_generators.htmlAt first I found it hard to visualise infinitesimal generators but perhaps this simple diagram will help
\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]
This note exists because no one explained to me satisfactorily to me why I should care about infinitesimal generators. These mysterious creatures pop up in the study of certain continuous time Markov processes, such as stochastic differential equations driven by Lévy noise.Polynomial chaos expansion
https://danmackinlay.name/notebook/polynomial_chaos.html
Wed, 05 Feb 2020 09:33:10 +1100https://danmackinlay.name/notebook/polynomial_chaos.htmlPoisson processes
https://danmackinlay.name/notebook/poisson_processes.html
Wed, 29 Jan 2020 10:56:30 +1100https://danmackinlay.name/notebook/poisson_processes.htmlBasics Poisson distribution Moments Poisson bridge Hitting time Taylor expansion \[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]
Poisson processes are possibly the simplest subordinators, i.e. non-decreasing Lévy processes. They pop up everywhere, especially as a representation of point processes, and as the second continuous-time stochastic process anyone learns after the Brownian motion.
Basics A Poisson process \(\{\mathsf{n}\}\sim \operatorname{PoisP}(\lambda)\) is a stochastic process whose inter-occurrence times are identically and independently distributed such that \(\mathsf{t}_i-\mathsf{t}_{i-1}\sim\operatorname{Exp}(\lambda)\) (rate parameterization).Divisibility, decomposability, stability
https://danmackinlay.name/notebook/divisible_distributions.html
Tue, 28 Jan 2020 12:48:19 +1100https://danmackinlay.name/notebook/divisible_distributions.htmlInfinitely divisible Decomposable Self-decomposable Stable Induced processes 🏗 all of these are about sums; but presumably we can construct this over other algebraic structures of distributions, e.g. max-stable processes.
For now, some handy definition disambiguation.
Infinitely divisible The Lévy process quality.
A probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of any arbitrary natural number of independent and identically distributed random variables.Markov bridge processes
https://danmackinlay.name/notebook/bridge_processes.html
Mon, 20 Jan 2020 16:33:25 +1100https://danmackinlay.name/notebook/bridge_processes.html\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\]
A bridge process for some time-indexed Markov process \(\{\Lambda(t)\}_{t\in[S,T]}\) is obtained from that process by conditioning it to attain a fixed value \(\Lambda(T)=Y\) starting from \(\Lambda(S)=X\) on some interval \([S,T]\). We write that as \(\{\Lambda(t)\mid \Lambda(S)=X,\Lambda(T)=Y\}_{t\in[S,T]}.\) Put another way, given the starting and finishing values of a stochastic markov process, I would like to rewind time to find out the values of its path at a midpoint which is “compatible” with the endpoints.Gaussian processes
https://danmackinlay.name/notebook/gaussian_processes.html
Tue, 03 Dec 2019 10:11:26 +1100https://danmackinlay.name/notebook/gaussian_processes.htmlRelationship between addition of covariance kernels and of processes “Gaussian Processes” are stochastic processes/fields with jointly Gaussian distributions of observations. The most familiar of these to many of us is the Gauss-Markov process, a.k.a. the Wiener process. These processes are convenient due to certain useful properties of the multivariate Gaussian distribution e.g. being uniquely specified by first and second moments, nice behaviour under various linear operations, kernel tricks….