Mild solutions to the dynamic programming equation for stochastic optimal control problems

We show via the nonlinear semigroup theory in $L^1(\mathbb{R})$ that the $1$-D dynamic programming equation associated with a stochastic optimal control problem with multiplicative noise has a unique mild solution $\varphi\in C([0,T];W^{1,\infty}(\mathbb{R}))$ with $\varphi_{xx}\in C([0,T];L^1(\mathbb{R}))$. The $n$-dimensional case is also investigated

Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model

We study a family of mean field games with a state variable evolving as a multivariate jump diffusion process. The jump component is driven by a Poisson process with a time-dependent intensity function. All coefficients, i.e. drift, volatility and jump size, are controlled. Under fairly general conditions, we establish existence of a solution in a relaxed version of the mean field game and give conditions under which the optimal strategies are in fact Markovian, hence extending to a jump-diffusion setting previous results established in [30]. The proofs rely upon the notions of relaxed controls and martingale problems. Finally, to complement the abstract existence results, we study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity, and provide some numerical results.Comment: 37 pages, 6 figure

ε-Nash equilibrium in stochastic differential games with mean-field interaction and controlled jumps

We consider a symmetric n-player nonzero-sum stochastic differential game with jump–diffusion dynamics and mean-field type interaction among the players. Under the assumption of existence of a regular Markovian solution for the corresponding limiting mean-field game, we construct an approximate Nash equilibrium for the n-player game for n large enough, and provide the rate of convergence. This extends to a class of games with jumps classical results in mean-field game literature. This paper complements our previous work Benazzol et al. (2017) on the existence of solutions of mean-field games for jump–diffusions

Minimal flavor-changing Z′Z' models and muon g−2g-2 after the RK∗R_{K^*} measurement

There has been a steady interest in flavor anomalies and their global fits as ideal probes of new physics. If the anomalies are real, one promising explanation is a new $Z'$ gauge boson with flavor-changing coupling to bottom and strange quarks and a flavor-conserving coupling to muons and, possibly, electrons. We point out that direct production of such a $Z'$, emerging from the collision of $b$ and $s$ quarks, may offer a complementary window into these phenomena because collider searches already provide competitive constraints. On top of that, we analyse the same $Z'$ scenario in relation to another long-standing discrepancy between theory and experiment that concerns the anomalous magnetic moment of the muon. By scanning the allowed $Z'$ coupling strengths in the low-mass region, we assess the compatibility of the signals from LHCb with the $Z'$ searches in the high energy LHC data and the measurements of the anomalous magnetic moment of the involved leptons. We also argue that observations of the latter can break the degeneracy pattern in the Wilson coefficients $C_9$ and $C_{10}$ presented by LHCb data. The $Z'$ model we consider is compatible with the new measurement of $R_{K^*}$, therefore it can potentially account for the long-standing deviations observed in $B$-physics.Comment: 13 pages, 5 figures. Missing factors corrected in eqs. 2.7-2.9 with slight updates in fig.

An example of a three-type interference pattern in the Tuscan Nappe, South-western sector of Apuan Alps (Northern Apennines, Italy)

The study area is located in the Northern Apennines, which is composed of oceanic and continental- derived tectonic units, stacked toward NE during Apennine subduction (Oligocene to present). The continental-derived units, representative of the Adria paleomargin involved into the collisional phases of the Alpine orogeny, are, from the lower to the upper structural levels: the Apuan Alps Unit, the Massa Unit and the Tuscan Nappe. The Apuan Alps Unit consists of pre-Mesozoic metamorphic basement and metasedimentary cover rocks, ranging from Triassic to Oligocene, deformed and metamorphosed under greenschists facies conditions. The Massa Unit, is also characterized by pre-Mesozoic metamorphic basement unconformably covered by Middle to Upper Triassic metasedimentary rocks. It recorded higher metamorphic conditions respect to underlying Unit. The Tuscan Nappe, instead, is detached from its basement and it is composed by Late Triassic to Early Miocene nonmetamorphic sedimentary rocks. This Unit was deformed at shallow structural levels. In the southeastern sector of the Apuan Alps, sedimentary rocks of the Tuscan nappe crop out. Particularly, in the study area, this unit includes only LateTriassic - Early Cretaceous rocks (Carosi et al., 2005). Field observations and structural data allowed us to elaborate a 1:5.000 scale structural map. Structural analyses highlighted a comlplex deformation history, composed by five deformation events (from D1 to D5), documented a different scale. The D1 event is testified by S1 foliation classifiable as slaty cleavage. In thin section, S1 is emphasised by preferential orientation of phyllosilicates and lenticular domains composed of quartz, calcite, detrital micas, albite and oxides. Open to isoclinal similar folds (F2) with NW-SE trending axes are associated to the D2 event. F2 folds are associated to a well developed S2 axial plane classifiable as crenulation cleavage without significant re-crystallisation. The D3 event is not well developed in this area and it is locally recorded in the fine-grained rocks where it produced isoclinal folds with scattered axes and axial planes oriented N-S. The previous architecture is reworked by folds with chévron geometry and sub-horizontal axial plane associated to the D4 event. Finally, D5 event is characterized by open folds with sub-vertical axial plane and three systems of normal to transtensive faults. These brittle structures, could be interpreted as conjugate faults of the Monte Croce-Pescaglia faults system. In conclusion, field observations and structural analysis conducted both at the micro and at the mesoscopic scale, indicate that Tuscan Nappe exposed in the study area was affected by a kmscale three-type interference pattern (Ramsay, 1967). Furthermore, our studies suggest that the faults played a role of great importance in the present architecture of this sector of the belt

Default Contagion in Financial Networks

The preset work aims at giving insights about howthe theory behind the study of complex networks can be profitablyused to analyse the increasing complexity characterizinga wide number of current financial frameworks. In particularwe exploit some well known approaches developed within thesetting of the graph theory, such as, e.g., the Erd˝os and Rénymodel, and the Barab´asi-Albert model, as well as producingan analysis based on the evolving network theory. Numericalsimulations are performed to study the spread of financial peakevents, as in the case of the default of a single bank belonging toa net of interconnected monetary institutions, showing how theknowledge about the underlying graph theory can be effectivelyused to withstand a financial default contagion