arXiv subjects

Giorgio Ferrari

Publications and source records attributed to Giorgio Ferrari.

At least 73 records · Page 4Linked to original sources

arXiv2019

Submodular Mean Field Games: Existence and Approximation of Solutions

We study mean field games with scalar It{ô}-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. The mean field game is first defined over ordinary stochastic controls, then extended to relaxed controls. Our approach allows also to treat a class of submodular mean field games with common noise in which the representative player at equilibrium interacts with the (conditional) mean of its state's distribution.

math.OC↗

arXiv2019

An Optimal Dividend Problem with Capital Injections over a Finite Horizon

In this paper we propose and solve an optimal dividend problem with capital injections over a finite time horizon. The surplus dynamics obeys a linearly controlled drifted Brownian motion that is reflected at the origin, dividends give rise to time-dependent instantaneous marginal profits, whereas capital injections are subject to time-dependent instantaneous marginal costs. The aim is to maximize the sum of a liquidation value at terminal time and of the total expected profits from dividends, net of the total expected costs for capital injections. Inspired by the study of El Karoui and Karatzas (1989) on reflected follower problems, we relate the optimal dividend problem with capital injections to an optimal stopping problem for a drifted Brownian motion that is absorbed at the origin. We show that whenever the optimal stopping rule is triggered by a time-dependent boundary, the value function of the optimal stopping problem gives the derivative of the value function of the optimal dividend problem. Moreover, the optimal dividend strategy is also triggered by the moving boundary of the associated stopping problem. The properties of this boundary are then investigated in a case study in which instantaneous marginal profits and costs from dividends and capital injections are constants discounted at a constant rate.

q-fin.MF↗

arXiv2019

On a Class of Infinite-Dimensional Singular Stochastic Control Problems

We study a class of infinite-dimensional singular stochastic control problems with applications in economic theory and finance. The control process linearly affects an abstract evolution equation on a suitable partially-ordered infinite-dimensional space X, it takes values in the positive cone of X, and it has right-continuous and nondecreasing paths. We first provide a rigorous formulation of the problem by properly defining the controlled dynamics and integrals with respect to the control process. We then exploit the concave structure of our problem and derive necessary and sufficient first-order conditions for optimality. The latter are finally exploited in a specification of the model where we find an explicit expression of the optimal control. The techniques used are those of semigroup theory, vector-valued integration, convex analysis, and general theory of stochastic processes.

math.OC↗

arXiv2019

Nonzero-Sum Submodular Monotone-Follower Games: Existence and Approximation of Nash Equilibria

We consider a class of N-player stochastic games of multi-dimensional singular control, in which each player faces a minimization problem of monotone-follower type with submodular costs. We call these games "monotone-follower games". In a not necessarily Markovian setting, we establish the existence of Nash equilibria. Moreover, we introduce a sequence of approximating games by restricting, for each natural number n, the players' admissible strategies to the set of Lipschitz processes with Lipschitz constant bounded by n. We prove that, for each n, there exists a Nash equilibrium of the approximating game and that the sequence of Nash equilibria converges, in the Meyer-Zheng sense, to a weak (distributional) Nash equilibrium of the original game of singular control. As a byproduct, such a convergence also provides approximation results of the equilibrium values across the two classes of games. We finally show how our findings can be employed to prove existence of open-loop Nash equilibria in an N-player stochastic differential game with singular controls, and we propose an algorithm to determine a Nash equilibrium for the monotone-follower game.

math.OC↗

arXiv2019

Optimal Reduction of Public Debt under Partial Observation of the Economic Growth

We consider a government that aims at reducing the debt-to-gross domestic product (GDP) ratio of a country. The government observes the level of the debt-to-GDP ratio and an indicator of the state of the economy, but does not directly observe the development of the underlying macroeconomic conditions. The government's criterion is to minimize the sum of the total expected costs of holding debt and of debt's reduction policies. We model this problem as a singular stochastic control problem under partial observation. The contribution of the paper is twofold. Firstly, we provide a general formulation of the model in which the level of debt-to-GDP ratio and the value of the macroeconomic indicator evolve as a diffusion and a jump-diffusion, respectively, with coefficients depending on the regimes of the economy. These are described through a finite-state continuous-time Markov chain. We reduce via filtering techniques the original problem to an equivalent one with full information (the so-called separated problem), and we provide a general verification result in terms of a related optimal stopping problem under full information. Secondly, we specialize to a case study in which the economy faces only two regimes, and the macroeconomic indicator has a suitable diffusive dynamics. In this setting we provide the optimal debt reduction policy. This is given in terms of the continuous free boundary arising in an auxiliary fully two-dimensional optimal stopping problem.

math.OC↗

arXiv2018

An Optimal Extraction Problem with Price Impact

A price-maker company extracts an exhaustible commodity from a reservoir, and sells it instantaneously in the spot market. In absence of any actions of the company, the commodity's spot price evolves either as a drifted Brownian motion or as an Ornstein-Uhlenbeck process. While extracting, the company affects the market price of the commodity, and its actions have an impact on the dynamics of the commodity's spot price. The company aims at maximizing the total expected profits from selling the commodity, net of the total expected proportional costs of extraction. We model this problem as a two-dimensional degenerate singular stochastic control problem with finite fuel. To determine its solution, we construct an explicit solution to the associated Hamilton-Jacobi-Bellman equation, and then verify its actual optimality through a verification theorem. On the one hand, when the (uncontrolled) price is a drifted Brownian motion, it is optimal to extract whenever the current price level is larger or equal than an endogenously determined constant threshold. On the other hand, when the (uncontrolled) price evolves as an Ornstein-Uhlenbeck process, we show that the optimal extraction rule is triggered by a curve depending on the current level of the reservoir. Such a curve is a strictly decreasing $C^{\infty}$-function for which we are able to provide an explicit expression. Finally, our study is complemented by a theoretical and numerical analysis of the dependency of the optimal extraction strategy and value function on the model's parameters.

math.OC↗

arXiv2018

An All-Electric Single-Molecule Hybridisation Detector for short DNA Fragments

In combining DNA nanotechnology and high-bandwidth single-molecule detection in nanopipettes, we demonstrate an all-electric, label-free hybridisation sensor for short DNA sequences (< 100 nt). Such short fragments are known to occur as circulating cell-free DNA in various bodily fluids, such as blood plasma and saliva, and have been identified as disease markers for cancer and infectious diseases. To this end, we use as a model system a 88-mer target from the RV1910c gene in Mycobacterium tuberculosis that is associated with antibiotic (isoniazid) resistance in TB. Upon binding to short probes attached to long carrier DNA, we show that resistive pulse sensing in nanopipettes is capable of identifying rather subtle structural differences, such as the hybridisation state of the probes, in a statistically robust manner. With significant potential towards multiplexing and high-throughput analysis, our study points towards a new, single-molecule DNA assay technology that is fast, easy to use and compatible with point of care environments.

cond-mat.soft↗

arXiv2018

On a Strategic Model of Pollution Control

This paper proposes a strategic model of pollution control. A firm, representative of the productive sector of a country, aims at maximizing its profits by expanding its production. Assuming that the output of production is proportional to the level of pollutants' emissions, the firm increases the level of pollution. The government of the country aims at minimizing the social costs due to the pollution, and introduces regulatory constraints on the emissions' level, which then effectively cap the output of production. Supposing that the firm and the government face both proportional and fixed costs in order to adopt their policies, we model the previous problem as a stochastic impulse two-person nonzero-sum game. The state variable of the game is the level of the output of production which evolves as a general linearly controlled one-dimensional Ito-diffusion. Following an educated guess, we first construct a pair of candidate equilibrium policies and of corresponding equilibrium values, and we then provide a set of sufficient conditions under which they indeed realize an equilibrium. Our results are complemented by a numerical study when the (uncontrolled) output of production evolves as a geometric Brownian motion, and the firm's operating profit and the government's running cost functions are of power type. An analysis of the dependency of the equilibrium policies and values on the model parameters yields interesting new behaviors that we explain as a consequence of the strategic interaction between the firm and the government.

math.OC↗

arXiv2017

Stochastic nonzero-sum games: a new connection between singular control and optimal stopping

In this paper we establish a new connection between a class of 2-player nonzero-sum games of optimal stopping and certain $2$-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover a differential link between the players' value functions holds across the two games.

math.OC↗

arXiv2017

On an Optimal Extraction Problem with Regime Switching

This paper studies a finite-fuel two-dimensional degenerate singular stochastic control problem under regime switching that is motivated by the optimal irreversible extraction problem of an exhaustible commodity. A company extracts a natural resource from a reserve with finite capacity, and sells it in the market at a spot price that evolves according to a Brownian motion with volatility modulated by a two-state Markov chain. In this setting, the company aims at finding the extraction rule that maximizes its expected discounted cash flow, net of the costs of extraction and maintenance of the reserve. We provide expressions both for the value function and for the optimal control. On the one hand, if the running cost for the maintenance of the reserve is a convex function of the reserve level, the optimal extraction rule prescribes a Skorokhod reflection of the (optimally) controlled state process at a certain state and price dependent threshold. On the other hand, in presence of a concave running cost function it is optimal to instantaneously deplete the reserve at the time at which the commodity's price exceeds an endogenously determined critical level. In both cases, the threshold triggering the optimal control is given in terms of the optimal stopping boundary of an auxiliary family of perpetual optimal selling problems with regime switching.

math.OC↗

arXiv2017

On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an infinite time-horizon. In a general not necessarily Markovian framework, we first show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal stopping problem. We then exploit such link to characterise the optimal control in a two-dimensional Markovian setting in which the state variables are the level of the debt-to-GDP ratio and the current inflation rate of the country. The latter follows uncontrolled Ornstein-Uhlenbeck dynamics and affects the growth rate of the debt ratio. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio under an inflation-dependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully two-dimensional optimal stopping problem.

math.OC↗

arXiv2017

On the Singular Control of Exchange Rates

Consider the problem of a central bank that wants to manage the exchange rate between its domestic currency and a foreign one. The central bank can purchase and sell the foreign currency, and each intervention on the exchange market leads to a proportional cost whose instantaneous marginal value depends on the current level of the exchange rate. The central bank aims at minimizing the total expected costs of interventions on the exchange market, plus a total expected holding cost. We formulate this problem as an infinite time-horizon stochastic control problem with controls that have paths which are locally of bounded variation. The exchange rate evolves as a general linearly controlled one-dimensional diffusion, and the two nondecreasing processes giving the minimal decomposition of a bounded-variation control model the cumulative amount of foreign currency that has been purchased and sold by the central bank. We provide a complete solution to this problem by finding the explicit expression of the value function and a complete characterization of the optimal control. At each instant of time, the optimally controlled exchange rate is kept within a band whose size is endogenously determined as part of the solution to the problem. We also study the expected exit time from the band, and the sensitivity of the width of the band with respect to the model's parameters in the case when the exchange rate evolves (in absence of any intervention) as an Ornstein-Uhlenbeck process, and the marginal costs of controls are constant. The techniques employed in the paper are those of the theory of singular stochastic control and of one-dimensional diffusions.

math.OC↗

arXiv2017

On a Class of Singular Stochastic Control Problems for Reflected Diffusions

Reflected diffusions naturally arise in many problems from applications ranging from economics and mathematical biology to queueing theory. In this paper we consider a class of infinite time-horizon singular stochastic control problems for a general one-dimensional diffusion that is reflected at zero. We assume that exerting control leads to a state-dependent instantaneous reward, whereas reflecting the diffusion at zero gives rise to a proportional cost with constant marginal value. The aim is to maximize the total expected reward, minus the total expected cost of reflection. We show that depending on the properties of the state-dependent instantaneous reward we can have qualitatively different kinds of optimal strategies. The techniques employed are those of stochastic control and of the theory of linear diffusions.

math.OC↗

arXiv2017

A note on a new existence result for reflected BSDEs with interconnected obstacles

In this note we prove existence of a solution to a system of Markovian BSDEs with interconnected obstacles. A key feature of our system, and the main novelty of this paper, is that we allow for the driver $f_i$ of the $i$-th component of the $Y$-process to depend on all components of the $Z$-process. This extends the existing theory on reflected BSDEs, which only addresses problems where $f_i$ depends on $Z^i$.

math.PR↗

arXiv2017

The "go-stop-go" of Italian civil nuclear programs, beset by lack of strategic planning, exploitation for personal gain and unscrupulous political conspiracies: 1946-1987

The Italian civil nuclear projects had a very early origin, with the first ideas originating as far back as 1945. The construction of the first three plants dated back to the period 1956-1964, and at that time Italy ranked third in the world for installed power. However, the very ambitious projects were immediately after thwarted in the early 1960s by the "Ippolito trial". Actually, a whole range of advanced programmes for the development of the country went to a stop, since they clashed with national and international major powers. Italy was relegated to a second rank power. The fourth nuclear plan was designed in 1970, and its commercial operation began in 1981. In the meantime, a strong anti-nuclear movement grew, and the position of the pro-nuclear Italian Communist Party began to change. After the Chernobyl accident, a national referendum was held, which in 1987 put an end to the Italian nuclear programmes.

physics.hist-ph↗

arXiv2017

Nash equilibria of threshold type for two-player nonzero-sum games of stopping

This paper analyses two-player nonzero-sum games of optimal stopping on a class of linear regular diffusions with not non-singular boundary behaviour (in the sense of Itô and McKean (1974), p.\ 108). We provide sufficient conditions under which Nash equilibria are realised by each player stopping the diffusion at one of the two boundary points of an interval. The boundaries of this interval solve a system of algebraic equations. We also provide conditions sufficient for the uniqueness of the equilibrium in this class.

math.PR↗

arXiv2017

A Reconfigurable Cryogenic Platform for the Classical Control of Scalable Quantum Computers

Recent advances in solid-state qubit technology are paving the way to fault-tolerant quantum computing systems. However, qubit technology is limited by qubit coherence time and by the complexity of coupling the quantum system with a classical electronic infrastructure. We propose an infrastructure, enabling to read and control qubits, that is implemented on a field-programmable gate array (FPGA). The FPGA platform supports functionality required by several qubit technologies and can operate physically close to the qubits over a temperature range from 4K to 300K. Extensive characterization of the platform over this temperature range revealed all major components (such as LUTs, MMCM, PLL, BRAM, IDELAY2) operate correctly and the logic speed is very stable. The stability is finally concretized by operating an integrated ADC with relatively stable performance over temperature.

physics.ins-det↗

arXiv2017

Control of the Correlation of Spontaneous Neuron Activity in Biological and Noise-activated CMOS Artificial Neural Microcircuits

There are several indications that brain is organized not on a basis of individual unreliable neurons, but on a micro-circuital scale providing Lego blocks employed to create complex architectures. At such an intermediate scale, the firing activity in the microcircuits is governed by collective effects emerging by the background noise soliciting spontaneous firing, the degree of mutual connections between the neurons, and the topology of the connections. We compare spontaneous firing activity of small populations of neurons adhering to an engineered scaffold with simulations of biologically plausible CMOS artificial neuron populations whose spontaneous activity is ignited by tailored background noise. We provide a full set of flexible and low-power consuming silicon blocks including neurons, excitatory and inhibitory synapses, and both white and pink noise generators for spontaneous firing activation. We achieve a comparable degree of correlation of the firing activity of the biological neurons by controlling the kind and the number of connection among the silicon neurons. The correlation between groups of neurons, organized as a ring of four distinct populations connected by the equivalent of interneurons, is triggered more effectively by adding multiple synapses to the connections than increasing the number of independent point-to-point connections. The comparison between the biological and the artificial systems suggests that a considerable number of synapses is active also in biological populations adhering to engineered scaffolds.

cs.NE↗