Abstract
Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of L\'evy processes, which includes subordinated stable and symmetric L\'evy processes. We apply this characterisaiton to construct $\varepsilon$-strong simulation ($\varepsilon$SS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our $\varepsilon$SS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.
Highlights
1.1 Setting and motivationThe universality of stable laws, processes and their path transformations makes them ubiquitous in probability theory and many areas of statistics and natural and social sciences
Its relevance in the theory of simulation was highlighted in recent contributions [GCMUB19, GCMUB18a], which developed sampling algorithms for certain path-functionals related to the extrema of Lévy processes
As Lévy meanders arise in numerous path transformations and functionals of Lévy processes [DI77, Cha[97], AC01, UB14, CM16, IO19], it is natural to investigate their simulation problem via their convex minorants, leading to question (Q2): does there exist a tractable characterisation of the law of convex minorants of Lévy meanders given in terms of the marginals of the corresponding Lévy process? This question is not trivial for the following two reasons. (I) A description of the convex minorant of a Lévy meander is known only for a Brownian meander [PR12] and is given in terms of the marginals of the meander, not the marginals of the Brownian motion
Summary
The universality of stable laws, processes and their path transformations makes them ubiquitous in probability theory and many areas of statistics and natural and social sciences (see e.g. [UZ99, CPR13] and the references therein). A complete description of the law of the convex minorant of Lévy processes is given in [PUB12]. As Lévy meanders arise in numerous path transformations and functionals of Lévy processes [DI77, Cha[97], AC01, UB14, CM16, IO19], it is natural to investigate their simulation problem via their convex minorants, leading to question (Q2): does there exist a tractable characterisation of the law of convex minorants of Lévy meanders given in terms of the marginals of the corresponding Lévy process? (I) A description of the convex minorant of a Lévy meander is known only for a Brownian meander [PR12] and is given in terms of the marginals of the meander, not the marginals of the Brownian motion (cf Subsection 1.3.1 below). As Lévy meanders arise in numerous path transformations and functionals of Lévy processes [DI77, Cha[97], AC01, UB14, CM16, IO19], it is natural to investigate their simulation problem via their convex minorants, leading to question (Q2): does there exist a tractable characterisation of the law of convex minorants of Lévy meanders given in terms of the marginals of the corresponding Lévy process? This question is not trivial for the following two reasons. (I) A description of the convex minorant of a Lévy meander is known only for a Brownian meander [PR12] and is given in terms of the marginals of the meander, not the marginals of the Brownian motion (cf. Subsection 1.3.1 below). (II) Tractable descriptions of the convex minorant of a process X typically rely on the exchangeability of the increments of X in a fundamental way [PUB12, AHUB19], a property clearly not satisfied when X is a Lévy meander
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