Large deviations for i.i.d. replications of the total progeny of a Galton--Watson process

The Galton--Watson process is the simplest example of a branching process. The relationship between the offspring distribution, and, when the extinction occurs almost surely, the distribution of the total progeny is well known. In this paper, we illustrate the relationship between these two distributions when we consider the large deviation rate function (provided by Cram\'{e}r's theorem) for empirical means of i.i.d. random variables. We also consider the case with a random initial population. In the final part, we present large deviation results for sequences of estimators of the offspring mean based on i.i.d. replications of total progeny.Comment: Published at http://dx.doi.org/10.15559/16-VMSTA72 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

Large deviations for conditionally Gaussian processes: estimates of level crossing probability

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes -- the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.Comment: Published at https://doi.org/10.15559/18-VMSTA119 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/

Asymptotic results for compound sums in separable Banach spaces

We prove large and moderate deviation results for sequences of compound sums, where the summands are i.i.d. random variables taking values on a separable Banach space. We establish that the results hold by proving that we are dealing with exponentially tight sequences. We present two moderate deviation results: in the first one the summands are centered, in the second one the compound sums are centered.Comment: 18page

Asymptotic results for sums and extremes

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper, we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of i.i.d. random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of i.i.d. exponential random variables.Comment: 1

Short-time asymptotics for non self-similar stochastic volatility models

We provide a short-time large deviation principle (LDP) for stochastic volatility models, where the volatility is expressed as a function of a Volterra process. This LDP holds under suitable conditions, but does not require any self-similarity assumption on the Volterra process. For this reason, we are able to apply such LDP to two notable examples of non self-similar rough volatility models: models where the volatility is given as a function of a log-modulated fractional Brownian motion [Bayer et al., Log-modulated rough stochastic volatility models. SIAM J. Financ. Math, 2021, 12(3), 1257-1284], and models where it is given as a function of a fractional Ornstein-Uhlenbeck (fOU) process [Gatheral et al., Volatility is rough. Quant. Finance, 2018, 18(6), 933-949]. In both cases we derive consequences for short-maturity European option prices and implied volatility surfaces. In the fOU case we also discuss moderate deviations pricing and simulation results.Comment: 25 pages, 3 figure

Large deviations for a class of tempered subordinators and their inverse processes

We consider a class of tempered subordinators, namely a class of subordinators with one-dimensional marginal tempered distributions which belong to a family studied in [3]. The main contribution in this paper is a non-central moderate deviations result. More precisely we mean a class of large deviation principles that fill the gap between the (trivial) weak convergence of some non-Gaussian identically distributed random variables to their common law, and the convergence of some other related random variables to a constant. Some other minor results concern large deviations for the inverse of the tempered subordinators considered in this paper; actually, in some results, these inverse processes appear as random time-changes of other independent processes

Large deviation approaches for the numerical computation of the hitting probability for Gaussian processes

Abstract. We state large deviations for small time of a pinned n-conditional Gaussian process, i.e. the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants, by letting all the past monitoring instants to depend on the small parameter going to 0. Differently from what already developed in Caramellino and Pacchiarott

Optimal importance sampling for continuous Gaussian fields

We consider the problem of selecting a change of mean which minimizes the variance of Monte Carlo estimators for the expectation of a functional of a continuous Gaussian field, in particular continuous Gaussian processes. Functionals of Gaussian fields have taken up an important position in many fields including statistical physics of disordered systems and mathematical finance (see, for example, [A. Comtet, C. Monthus and M. Yor, Exponential functionals of Brownian motion and disordered systems, J. Appl. Probab. 35 1998, 2, 255–271], [D. Dufresne, The integral of geometric Brownian motion, Adv. in Appl. Probab. 33 2001, 1, 223–241], [N. Privault and W. I. Uy, Monte Carlo computation of the Laplace transform of exponential Brownian functionals, Methodol. Comput. Appl. Probab. 15 2013, 3, 511–524] and [V. R. Fatalov, On the Laplace method for Gaussian measures in a Banach space, Theory Probab. Appl. 58 2014, 2, 216–241]. Naturally, the problem of computing the expectation of such functionals, for example the Laplace transform, is an important issue in such fields. Some examples are considered, which, for particular Gaussian processes, can be related to option pricing

Asymptotics for multifactor Volterra type stochastic volatility models

We study multidimensional stochastic volatility models in which the volatility process is a positive continuous function of a continuous multidimensional Volterra process. The main results obtained in this paper are a generalization of the results due, in the one-dimensional case, to Cellupica and Pacchiarotti in \cite{Cel-Pac}. We state some large deviation principles for the scaled log-priceComment: arXiv admin note: text overlap with arXiv:1902.0589