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Esquível, M. L., P. Mota, and J. P. Pina. "On a Stochastic Model for a Cooperative Banking Scheme for Microcredit." Theory of Probability & its Applications. 66.2 (2021): 326-335. AbstractWebsite

We propose and study a simple model for microcredit using two sums, with a random number of terms, of identically distributed random variables, the number of terms being Poisson distributed; the first sum accounts for the payments { the payables{ made to the collective vault by the participants and the second sum, subtracted to the first, accounts for the loans received by the participants, the receivables.
Under a global independence hypothesis we de fine, by mean of moment generating functions, an easily
implementable condition for the sustainability of the collective vault. That is, if, for all the participants and at any time, on average, the sum of the loans is strictly less than the sum of the payments to the collective vault then the probability of the collective vault failing can be made arbitrarily small, provided the loans only start to be accepted after a sufficiently large delay. We present numerical illustrations of collective vaults for exponential and chi-squared distributed random terms. For the practical management of such a collective vault it may be advisable to have loan granting rules that destroy independence of the random terms. We present a first simulation study that shows the effect of such a breaking dependence loan granting rule on maintaining the stability of the collective vault.

Esquível, M. L., NP Krasii, and P. Mota. "Auto and Externally Induced Regime Switching Diffusions." Communications On Stochastic Analysis. 14.1-2 (2020): 27-47. AbstractWebsite

In the current literature we can find mainly two approaches to the
SDE regime switching modeling. The traditional one, the externally induced
regime switching diffusions is described by the switching being derived from
a separate continuous time Markov process, with a finite, or denumerable,
state space { indexing the regimes { the random times of the regime switches
being exactly the jump times of the finite valued Markov process. There is a
first alternative approach in which the regime switching occurs whenever the
trajectory enters in some prescribed region on the state space; the regions we
consider will be mainly open intervals defined by unknown thresholds for the
trajectories; thresholds that, in principle, should also be estimated. In this
approach the partitioning of the the state space is already defined in the drift
and volatility of the SDE. In a second alternative approach the switching occurs
in a random way but at some random times defined when the trajectories hit
some prescribed thresholds, that again, must be estimated. We may designate
these two alternative approaches as auto-induced regime switching diffusions
as there is no external noise source to force the switching occurrence. We prove
a generalization of an existence result of the existence of auto-induced regime
switching SDE solutions for irregular coefficients and a result that encompasses
some of the cases of both externally and auto-induced regime switching SDE
solutions.

Esquível, Manuel L., and Pedro Mota. "On Some Auto-Induced Regime Switching Double-Threshold Glued Diffusions." Journal of Statistical Theory and Practice. 8 (2014): 760-771. AbstractWebsite

Regime switching processes are usually defined with an external random source driving the regime changes. In this article, we define and study a regime switching diffusion considering two thresholds, and regime switching occurring, by a change in the diffusion drift and volatility, whenever the trajectory touches the upper threshold after having crossed, or touched, the lower threshold or touches the lower threshold after having crossed, or touched, the upper threshold. We develop an estimation procedure for the thresholds and for the regime parameters of the diffusions. We show that a generalized Black–Scholes model with the regime switching diffusion as the law of the risky asset is arbitrage free and complete under an additional hypothesis on the diffusion coefficients of the two regime diffusions.

Esquível, M. L., N. Machado, NP Krasii, and P. Mota. "On the Information Content of Some Stochastic Algorithms." Recent Developments in Stochastic Methods and Applications. Eds. A. N. Shiryaev, K. E. Samouylov, and D. V. Kozyrev. Cham: Springer, 2021. 57-75. Abstract

We formulate an optimization stochastic algorithm convergence theorem, of Solis and Wets type, and we show several instances of its application to concrete algorithms. In this convergence theorem the algorithm is a sequence of random variables and, in order to describe the increasing flow of information associated to this sequence we define a filtration – or flow of σ -algebras – on the probability space, depending on the sequence of random variables and on the function being optimized. We compare the flow of information of two convergent algorithms by comparing the associated filtrations by means of the Cotter distance of σ-algebras. The main result is that two convergent optimization algorithms have the same information content if both their limit minimization functions generate the full σ-algebra of the probability space.

Esquível, Manuel L., Pedro P. Mota, and João Tiago Mexia. "On some statistical models with a random number of observations." Journal of Statistical Theory and Practice. 10 (2016): 805-823. AbstractWebsite

We extend some classical statistical inference to the case of a random number of observations with a stabilized distribution: namely, in the normal model, inference for the mean with known and unknown variance and inference for the variance. We describe some useful models for the number of observations obtained by truncation or translation of usual models given by integer-valued random variables: Poisson, binomial, geometric, and negative binomial. We present an efficient random search algorithm for the computation of the quantiles of the relevant statistics, we describe an interval estimation procedure for the extended model, and we propose a parametric bootstrap simulation study to validate the proposed procedure.

Esquível, M. L., NP Krasii, P. Mota, and N. Machado. "On a parallelised diffusion induced stochastic algorithm with pure random search steps for global optimisation." Mathematics. 9.23 (2021). AbstractWebsite

We propose a stochastic algorithm for global optimisation of a regular function, possibly unbounded, defined on a bounded set with regular boundary; a function that attains its extremum in the boundary of its domain of definition. The algorithm is determined by a diffusion process that is associated with the function by means of a strictly elliptic operator that ensures an adequate maximum principle. In order to preclude the algorithm to be trapped in a local extremum, we add a pure random search step to the algorithm. We show that an adequate procedure of parallelisation of the algorithm can increase the rate of convergence, thus superseding the main drawback of the addition of the pure random search step.