4,097 research outputs found
An infinite-dimensional approach to path-dependent Kolmogorov equations
No full textIn this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of Lp paths and in the space of continuous paths using the associated stochastic differential equation, thus establishing a relation between path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is shown how to establish a connection between such Kolmogorov equation and the analogue finite-dimensional equation that can be formulated in terms of the path-dependent derivatives recently introduced by Dupire, Cont and Fournié
Spatial dynamics in interacting systems with discontinuous coefficients and their continuum limits
No full textWe consider a discrete model in which particles are characterized by two quantities X
and Y ; both quantities evolve in time according to stochastic dynamics and the equation
that governs the evolution of Y is also influenced by mean-field interaction between
the particles. We allow for discontinuous coefficients and random initial condition and,
under suitable assumptions, we prove that in the limit as the number of particles grows
to infinity the dynamics of the system is described by the solution of a Fokker–Planck
partial differential equation. We provide the existence and uniqueness of a solution to
the latter and show that such solution arises as the limit in probability of the empirical
measures of the system
Optimal portfolio choice with path dependent benchmarked labor income: A mean field model
No full textWe consider the life-cycle optimal portfolio choice problem faced by an agent receiving labor income and allocating her wealth to risky assets and a riskless bond subject to a borrowing constraint. In this paper, to reflect a realistic economic setting, we propose a model where the dynamics of the labor income has two main features. First, labor income adjusts slowly to financial market shocks, a feature already considered in Biffis et al. (2015). Second, the labor income yi of an agent i is benchmarked against the labor incomes of a population y^n≔(y1,y2,...,yn) of n agents with comparable tasks and/or ranks. This last feature has not been considered yet in the literature and is faced taking the limit when n→+∞ so that the problem falls into the family of optimal control of infinite-dimensional McKean–Vlasov Dynamics, which is a completely new and challenging research field. We study the problem in a simplified case where, adding a suitable new variable, we are able to find explicitly the solution of the associated HJB equation and find the optimal feedback controls. The techniques are a careful and nontrivial extension of the ones introduced in the previous papers of Biffis et al. (2015, 0000)
A mean field game model for COVID-19 with human capital accumulation
No full textIn this manuscript, we study a model of human capital accumulation during the spread of disease following an agent-based approach, where agents behave maximising their intertemporal utility. We assume that the agent interaction is of mean field type, yielding a mean field game description of the problem. We discuss how the analysis of a model including both the mechanism of change of species from one epidemiological state to the other and an optimisation problem for each agent leads to an aggregate behaviour that is not easy to describe, and that sometimes exhibits structural issues. Therefore we eventually propose and study numerically a SEIRD model in which the rate of infection depends on the distribution of the population, given exogenously as the solution to the mean field game system arising as the macroscopic description of the discrete multi-agent economic model for the accumulation of human capital. Such a model arises in fact as a simplified but tractable version of the initial one
A mean-field model with discontinuous coefficients for neurons with spatial interaction
No full textStarting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients.
We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles
A simple planning problem for COVID-19 lockdown: a dynamic programming approach
No full textA large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton-Jacobi-Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach
Infinite dimensional calculus under weak spatial regularity of the processes
No full textTwo generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus
Le «buone letture». 2. Giovanni Casati
No full textIl saggio è costituito da due parti, la prima delle quali, dedicata alla fondazione della Federazione italiana delle biblioteche circolanti cattoliche, è stata pubblicata nel precedente numero dei «Nuovi Annali», XXVII (2013), pp. 137-163. In questa seconda parte viene delineata la figura intellettuale di Giovanni Casati, che diresse la «Rivista di letture» dal 1912 al 1944, trasformando il periodico della Federazione in una rivista impegnata nella divulgazione della cultura cattolica. A questo impegno militante Casati fece corrispondere un intenso programma editoriale, che trovò espressione nella pubblicazione di saggi letterari, di manuali e opere repertoriali.The study consists of two parts; the first is dedicated to the history of the Federazione italiana delle biblioteche circolanti cattoliche since its foundation (1904) up to 1912 and was published in the previous volume of the «Nuovi Annali », XXVII (2013), pp. 137-163. In this second part, the author outlines the intellectual figure of Giovanni Casati, who directed the «Rivista di letture» from 1912 to 1944, transforming the magazine of the Federation in a journal engaged in the spreading of Catholic culture. To this militant engagement Casati matched an intense publishing program, which found its expression in the publication of literary essays, manuals and reference works
Comico e tragico del diritto nella novellistica italiana: il novelliere di Giovanni Sercambi
No full textNelle novelle di età tardomedievale ed umanistica, a causa del loro carattere fortemente realistico, il diritto ha un posto importante, poiché rappresenta un elemento centrale e ineliminabile della vita della comunità. Il novelliere di Giovanni Sercambi da Lucca non fa eccezione, anche se l’autore non ha una cultura giuridica né s’interessa in modo specifico al diritto. Il presente saggio vuole analizzare sotto tale profilo queste novelle per verificare quale concezione del diritto emerge da esse e come la narrazione ne risulti arricchita.In the late medieval and humanistic novellas, owing to their very realistic character, the right is an important central and inevitable element in community life. Giovanni Sercambi’s novelliere is no exception even if the author lacks a legal culture, nor is specifically interested in right. This essay intends to analyze these novellas under this profile to verify what kind of right they consider and how the stories are enriched
Semilinear Kolmogorov equations on the space of continuous functions via BSDEs
No full textWe deal with a class of semilinear parabolic PDEs on the space of continuous functions that arise, for example, as Kolmogorov equations associated to the infinite-dimensional lifting of path-dependent SDEs. We investigate existence of smooth solutions through their representation via forward–backward stochastic systems, for which we provide the necessary regularity theory. Because of the lack of smoothing properties of the parabolic operators at hand, solutions in general will only share the same regularity as the coefficients of the equation. To conclude we exhibit an application to Hamilton–Jacobi–Bellman equations associated to suitable optimal control problems
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