[Paper Review] On the location of the maximum of a process: Lévy, Gaussian and multidimensional cases

Sergio I. López, Leandro P. R. Pimentel|arXiv (Cornell University)|Nov 7, 2016

Random Matrices and Applications6 references3 citations

TL;DR

This paper extends functional analytic techniques for proving uniqueness of the maximizer in stochastic processes by relaxing continuity assumptions and generalizing to multidimensional and non-diffusive processes. It establishes that the existence of a derivative in a perturbed expectation functional implies almost-sure uniqueness of the maximizer location, with explicit formulas for the expected maximizer location in Gaussian and Lévy processes.

ABSTRACT

In this short article we show how the techniques presented in arXiv:1207.4469 can be extended to a variety of non continuous and multivariate processes. As examples, we prove uniqueness of the location of the maximum for spectrally positive Lévy processes, Ornstein-Uhlenbeck process, fractional Brownian Motion and the Brownian sheet among others gaussian processes.

Motivation & Objective

• To extend the functional derivative criterion for maximizer uniqueness beyond continuous processes to càdlàg and multidimensional processes.
• To establish conditions under which the location of the maximum of a stochastic process is almost surely unique, even when sample paths are discontinuous.
• To derive explicit formulas for the expected location of the maximizer in Gaussian and Lévy processes using perturbation of the process with strictly increasing functions.
• To generalize previous results on Gaussian processes to non-Markovian and non-diffusive processes such as fractional Brownian motion and the Brownian sheet.

Proposed method

• Introduces a generalized notion of quasi-maximizers for càdlàg processes, where a point is a quasi-maximizer if it is locally maximal in any neighborhood.
• Proposes Theorem 1: for a càdlàg process on a compact set, the existence of the derivative of the perturbed expectation map $ a \mapsto \mathbb{E}[S^a] $ at $ a=0 $ implies almost-sure uniqueness of the quasi-maximizer.
• Derives Theorem 2 for multidimensional continuous processes: uniqueness of the maximizer location is equivalent to the existence of the gradient of the perturbed expectation map at zero.
• Applies the method to spectrally positive Lévy processes, showing that uniqueness holds when $ \sigma > 0 $ or $ c \neq 0 $, and gives a probabilistic characterization when $ \sigma = c = 0 $.
• Uses conditional expectation and Gaussian process representation to compute the gradient of the perturbed expectation, leveraging the joint Gaussianity of the process at finitely many points.
• Applies the method to fractional Brownian motion and the Brownian sheet by verifying the conditions on the covariance structure and the existence of the derivative.

Experimental results

Research questions

• RQ1 Under what conditions is the location of the maximum of a càdlàg stochastic process almost surely unique?
• RQ2 Can the functional derivative criterion for maximizer uniqueness be extended to non-diffusive and non-Markovian Gaussian processes such as fractional Brownian motion?
• RQ3 How does the expected location of the maximizer relate to the process's covariance structure in multidimensional settings?
• RQ4 What is the probability that a spectrally positive Lévy process attains its maximum at a unique point when it has no diffusion component?
• RQ5 Can the perturbation method with strictly increasing functions be used to derive explicit formulas for the expected maximizer location in Gaussian processes?

Key findings

• For a spectrally positive Lévy process with $ \sigma > 0 $, the maximum is almost surely attained at a unique point.
• When $ \sigma = 0 $ but $ c \neq 0 $, the maximum is still almost surely unique, as the drift ensures monotonicity in the limit.
• When $ \sigma = c = 0 $, the probability of a unique maximizer equals $ \mathbb{P}(\tau = 0) $, where $ \tau $ is the first exit time from zero.
• For fractional Brownian motion with Hurst parameter $ H > 1/2 $, the method confirms almost-sure uniqueness of the maximizer location.
• For the Brownian sheet, the method establishes that the maximizer location is almost surely unique due to the existence of the gradient of the perturbed expectation map.
• The expected value of the maximizer location $ \mathbb{E}[Z] $ in a Gaussian process is given by $ \left( \frac{\mathrm{Cov}(S, X(t^i))}{\sigma_{i,i}} \right)_{i=1}^d $, linking the maximizer to the process's covariance structure.