Lévy processes
Stochastic processes with independent increments, jump diffusion
May 29, 2017 — November 17, 2021
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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. […] it was seen that processes with independent increments are described by three terms — the covariance structure of the Brownian motion component, a drift term, and a measure describing the rate at which jumps occur. Being a special case of independent increments processes, the situation with Lévy processes is similar. […]
A d-dimensional Lévy process \(\Lambda(\cdot)\) is a stochastic process indexed by \(\bb{R}\) taking values in \({\mathbb R}^d\) such that it possesses
independent increments: \(\Lambda(t)-\Lambda(s)\) is independent of \(\{\Lambda(u)\colon u\le s\}\) for any \(s<t.\)
stationary increments: \(\Lambda({s+t})-\Lambda(s)\) has the same distribution as \(\Lambda(t)-\Lambda(0)\) for any \(s,t>0.\)
continuity in probability: \(\Lambda(s)\rightarrow \Lambda(t)\) in probability as \(s\rightarrow t.\)
1 General form
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2 Intensity measure
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3 Subordinators
See subordinators.
4 Spectrally negative
Lévy processes with no positive jumps are called spectrally negative and have some nice properties as regards hitting times from below (Doney 2007).
5 As martingales
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6 Sparsity properties
In the context of stochastic differential equations, Lévy processes give a model of a driving noise that is sparse compared to the usual Wiener process driving noise model. The subcategory of such Lévy SDEs that are additionally linear are called sparse stochastic processes (M. A. Unser and Tafti 2014; M. Unser, Tafti, and Sun 2014; M. Unser et al. 2014) and I would enjoy having a minute to sit down and understand how they work.
7 Bridge processes
There are various interesting uses for Lévy bridges. 🏗 For now, see bridge processes.
8 Recommended readings
Albin’s lectures based on Sato (1999) seem pretty good, as does Sato’s book itself. Also Bertoin (2000) is good. Kyprianou (2014) is a good introductory treatment. People recommend Applebaum (2009) a lot, and it is a fine and readable book but it does not emphasise areas I personally need (SDEs, non-negative processes), so I do not actually ever use it after chapter 2 or so. If I wanted to be excessively general I would probably go for Kallenberg (2002). M. A. Unser and Tafti (2014) has a theory of linear Lévy SDEs from the perspective of signal processing which is an unusual and helpful angle. If you want to get straight to business, perhaps favour the minimalist introduction in the stochastic simulation textbooks (Asmussen and Glynn 2007; Rubinstein and Kroese 2016) or, my favourite, (Aurzada and Dereich 2009).
9 Gamma process
See Gamma processes.
10 Brownian motions
TBD
11 Subordinators
See Subordinators!…
12 Student-Lévy process
See t-processes.