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Esercizi di matematica per l'economia. Serie, integrali, algebra lineare, programmazione lineare
No full textOptimal control for stochastic heat equation with memory.
Full text linkIn this paper, we investigate the existence and uniqueness of solutions for a class of evolutionary integral equations perturbed by a noise arising in the theory of heat conduction. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise
Time-consistency of risk measures: how strong is such a property?
No full textQuite recently, a great interest has been devoted to time-consistency of risk measures in its different formulations (see Delbaen in Memoriam Paul-André Meyer, Lecture notes in mathematics, vol 1874, pp 215–258, 2006; Föllmer and Penner in Stat Decis 14(1):1–15, 2006; Bion-Nadal in Stoch Process Appl 119:633–654, 2009; Delbaen et al. in Finance Stoch 14(3):449–472, 2010; Laeven and Stadje in Math Oper Res 39:1109–1141, 2014, among many others). However, almost all the papers address to coherent or convex risk measures satisfying cash-additivity. In the present work, we study time-consistency for more general dynamic risk measures where either only cash-invariance or both cash-invariance and convexity are dropped. This analysis is motivated by the recent papers of El Karoui and Ravanelli (Math Finance 19:561–590, 2009) and Cerreia-Vioglio et al. (Math Finance 21(4):743–774, 2011) who discussed and weakened the axioms above by introducing cash-subadditivity and quasi-convexity. In particular, we investigate and discuss whether the notion of time-consistency is too restrictive, when considered in the general framework of quasi-convex and cash-subadditive risk measures. Finally, we provide some conditions guaranteeing time-consistency in this more general framework
Invariant measures for stochastic differential equations on networks
No full textAbstract. We study existence and uniqueness of an invariant measure for infinite dimensional
stochastic differential equations with dissipative polynomially bounded nonlinear terms. We also
exhibit the existence of a density with respect to a Gaussian measure. Moreover, we decompose
the solution process into a stationary component and a component which vanishes asymptotically
in the L 2 -sense. Applications are given to neurobiological networks where the signals propagation
is modelled by a system of coupled stochastic FitzHugh-Nagumo equations
Small noise asymptotic expansions for stochastic PDE's.The case of a dissipative polynomiallly bounded non linearity
No full textWe study a reaction-diffusion evolution equation perturbed by a Gaussian
noise. Here the leading operator is the infinitesimal generator of a C0-semigroup of strictly
negative type, the nonlinear term has at most polynomial growth and is such that the whole
system is dissipative.
The corresponding Itô stochastic equation describes a process on a Hilbert space with
dissipative nonlinear, non globally Lipschitz drift and a Gaussian noise.
Under smoothness assumptions on the nonlinearity, asymptotics to all orders in a small
parameter in front of the noise are given, with uniform estimates on the remainders. Applications
to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded
domain are included. As a particular example we consider the small noise asymptotic expansions
for the stochastic FitzHugh-Nagumo equations of neurobiology around deterministic
solutions
Small noise asymptotic expansions for stochastic PDE's driven by dissipative nonlinearity and L\'evy noise
Full text linkFeedback optimal control for stochastic Volterra equations with completely monotone kernels.
Full text linkIn this paper we are concerned with a class of stochastic Volterra integro-dierential problems with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provide other interesting result and require a precise descriprion of the properties of the generated semigroup. The rst main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The main technical point consists in the dierentiability of the BSDE associated with the reformulated equation with respect to its initial datum x
An analytic approach to stochastic Volterra equationswith completely monotone kernels
Full text linkWe apply the semigroup setting of Desch and Miller to a class of stochastic integral equations
of Volterra type with completely monotone kernels with a multiplicative noise term; the corresponding
equation is an infinite dimensional stochastic equation with unbounded diffusion operator that we solve
with the semigroup approach of Da Prato and Zabczyk. As a motivation of our results, we study an optimal
control problem when the control enters the system together with the noise
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