Precise large deviations for dependent regularly varying sequences

We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of Nagaev (Theory Probab Appl 14:51–64, 193–208, 1969) and Nagaev (Ann Probab 7:745–789, 1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (Stoch Proc Appl 44:291–327, 1993; 68:1–20, 1997) in the context of central limit theorems with infinite variance stable limits. We illustrate the principle for stochastic volatility models, real valued functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations.     

Asymptotic power utility-based pricing and hedging

Kramkov and Sîrbu (Ann. Appl. Probab., 16:2140–2194, 2006; Stoch. Proc. Appl., 117:1606–1620, 2017) have shown that first-order approximations of power utility-based prices and hedging strategies for a small number of claims can be computed by solving a mean-variance hedging problem under a specific equivalent martingale measure and relative to a suitable numeraire. For power utilities, we propose an alternative representation that avoids the change of numeraire. More specifically, we characterize the relevant quantities using semimartingale characteristics similarly as in ?erný and Kallsen (Ann. Probab., 35:1479–1531, 2007) for mean-variance hedging. These results are illustrated by applying them to exponential Lévy processes and stochastic volatility models of Barndorff-Nielsen and Shephard type (J. R. Stat. Soc. B, 63:167–241, 2001). We find that asymptotic utility-based hedges are virtually independent of the investor’s risk aversion. Moreover, the price adjustments compared to the Black–Scholes model turn out to be almost linear in the investor’s risk aversion, and surprisingly small unless very high levels of risk aversion are considered.     

Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulations. We assume a Black–Scholes market with time-dependent volatilities, and we compute the deltas by means of the Malliavin Calculus as an extension of the procedures employed by Kohatsu-Higa and Montero (Physica A 320:548–570, 2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. To face this challenge we then introduce a new and faster Cholesky algorithm for block matrices that makes the Linear Transformation more convenient. We also propose a new-path generation technique based on a Kronecker Product Approximation. Our procedure shows the same accuracy as the Linear Transformation used for the computation of deltas and prices in the case of correlated asset returns, while requiring a shorter computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004a, Risk 17(5):97–101, b).     

Core equivalences for equilibria supported by non-linear prices

The goal of this paper is to provide some new cooperative characterizations and optimality properties of competitive equilibria supported by non-linear prices. The general framework is that of economies whose commodity space is an ordered topological vector space which need not be a vector lattice. The central notion of equilibrium is the one of personalized equilibrium introduced by Aliprantis et al. (J Econ Theory 100:22–72, 2001). Following Herves-Beloso and Moreno-Garcia (J Math Econ 44:697–706, 2008), the veto power of the grand coalition is exploited in the original economy and in a suitable family of economies associated to the original one. The use of Aubin coalitions allows us to connect results with the arbitrage free condition due to non-linear supporting prices. The use of rational allocations allows us to dispense with Lyapunov convexity theorem. Applications are provided in connection with strategic market games and economies with asymmetric information.     

A Note on the Volume Growth Criterion for Stochastic Completeness of Weighted Graphs

This paper gives complementary results of Folz (Trans Am Math Soc, 2013). We first generalize the weak Omori–Yau maximum principle to the setting of strongly local Dirichlet forms. As an application, we obtain an analytic approach to compare the stochastic completeness of a weighted graph with that of an associated metric graph. This comparison result played an essential role in the volume growth criterion of Folz (Trans Am Math Soc, 2013), who first proved it via a probabilistic approach. We also give an alternative analytic proof based on a criterion in Fukushima et al. (1994).     

Further results on factorization of meromorphic solutions of partial differential equations

This paper is a continuation of Hu-Yang [2]. Here we extend Malmquist type theorem ofalgebraic differential equations of Steinmetz [3] and Tu [4] to higher order partial differential equations. The results also generalize Theorems 4.2 and 4.3 in [2].     

A comparative assessment of different fuzzy regression methods for volatility forecasting

The aim of this paper is to compare different fuzzy regression methods in the assessment of the information content on future realised volatility of option-based volatility forecasts. These methods offer a suitable tool to handle both imprecision of measurements and fuzziness of the relationship among variables. Therefore, they are particularly useful for volatility forecasting, since the variable of interest (realised volatility) is unobservable and a proxy for it is used. Moreover, measurement errors in both realised volatility and volatility forecasts may affect the regression results. We compare both the possibilistic regression method of Tanaka et al. (IEEE Trans Syst Man Cybern 12:903–907, 1982) and the least squares fuzzy regression method of Savic and Pedrycz (Fuzzy Sets Syst 39:51–63, 1991). In our case study, based on intra-daily data of the DAX-index options market, both methods have proved to have advantages and disadvantages. Overall, among the two methods, we prefer the Savic and Pedrycz (Fuzzy Sets Syst 39:51–63, 1991) method, since it contains as special case (the central line) the ordinary least squares regression, is robust to the analysis of the variables in logarithmic terms or in levels, and provides sharper results than the Tanaka et al. (IEEE Trans Syst Man Cybern 12:903–907, 1982) method.     

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations

We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61–97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy–Boussinesq equations, which is a degenerate version of the Navier–Stokes–Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge–Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375–417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141–1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131–151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20–31, 1991, Arch. Ration. Mech. Anal. 185:341–363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450–1461, 1998; Loeper in SIAM J. Math. Anal. 38:795–823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov–Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29–59, 2001) and of the Keller–Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225–234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819–824, 1992; Chalub et al. in Mon. Math. 142:123–141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier–Stokes–Boussinesq equations. Finally, we show how a “stringy” generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology (see Arnold and Khesin in Topological methods in hydrodynamics. Applied mathematical sciences, vol. 125, Springer, Berlin, 1998; Moffatt in J. Fluid Mech. 159:359–378, 1985, Topological aspects of the dynamics of fluids and plasmas. NATO adv. sci. inst. ser. E, appl. sci., vol. 218, Kluwer, Dordrecht, 1992; Schonbek in Theory of the Navier–Stokes equations, Ser. adv. math. appl. sci., vol. 47, pp. 179–184, World Sci., Singapore, 1998; Vladimirov et al. in J. Fluid Mech. 390:127–150, 1999; Nishiyama in Bull. Inst. Math. Acad. Sin. (N.S.) 2:139–154, 2007).     

Burgers type equations, Gelfand’s problem and Schumpeterian dynamics

Burgers?? equations have been introduced to study different models of fluids (Bateman, 1915, Burgers, 1939, Hopf, 1950, Cole, 1951, Lighthill andWhitham, 1955, etc.). The difference-differential analogues of these equations have been proposed for Schumpeterian models of economic development (Iwai, 1984, Polterovich and Henkin, 1988, Belenky, 1990, Henkin and Polterovich, 1999, Tashlitskaya and Shananin, 2000, etc.). This paper gives a short survey of the results and conjectures on Burgers type equations, motivated both by fluid mechanics and by Schumpeterian dynamics. Proofs of some new results are given. This paper is an extension and an improvement of (Henkin, 2007, 2011).     

Quasi-stationary stefan problem with values on the front depending on its geometry

The problem presented below is a singular-limit problem of the extension of the Cahn-Hilliard model obtained via introducing the asymmetry of the surface tension tensor under one of the truncations (approximations) of the inner energy [2, 5–8, 10, 12, 13].     

A New Proof for the Convergence of an Individual Based Model to the Trait Substitution Sequence

We consider a continuous time stochastic individual based model for a population structured only by an inherited vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and the process is viewed in the timescale of mutations. Using averaging techniques due to Kurtz (in Lecture Notes in Control and Inform. Sci., vol. 177, pp. 186–209, 1992), we give a new proof of the convergence of the individual based model to the trait substitution sequence of Metz et al. (in Trends in Ecology and Evolution 7(6), 198–202, 1992), first worked out by Dieckman and Law (in Journal of Mathematical Biology 34(5–6), 579–612, 1996) and rigorously proved by Champagnat (in Theoretical Population Biology 69, 297–321, 2006): rigging the model such that “invasion implies substitution”, we obtain in the limit a process that jumps from one population equilibrium to another when mutations occur and invade the population.     

Option pricing in a conditional Bilateral Gamma model

We propose a conditional Bilateral Gamma model, in which the shape parameters of the Bilateral Gamma distribution have a Garch-like dynamics. After risk neutralization by means of a Bilateral Esscher transform, the model admits a recursive procedure for the computation of the characteristic function of the underlying at maturity, à la Heston and Nandi (Rev Financ Stud 13(3):562–585, 2000). We compare the calibration performance on SPX options with the models of Heston and Nandi (Rev Financ Stud 13(3):562–585, 2000), Christoffersen et al. (J Econom 131(1–2):253–284, 2006) and with a dynamic variance Gamma model introduced in Mercuri and Bellini (J Financ Decis Mak 7(1):37–51, 2011), obtaining promising results.     

Exponential polynomials on commutative hypergroups

Polynomials and exponential polynomials play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative groups. Recently several new results have been published in this field [2–4,6]. Spectral analysis and spectral synthesis has been studied on some types of commutative hypergroups, as well. However, a satisfactory definition of exponential monomials on general commutative hypergroups has not been available so far. In [5,7,8] and [9], the authors use a special concept on polynomial and Sturm–Liouville-hypergroups. Here we give a general definition which covers the known special cases.     

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H ¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L ² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$ Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].     

Defect and Area in Beltrami–Klein Model of Hyperbolic Geometry

Ungar (Beyond the Einstein addition law and its gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrouector Spaces, 2001; Comput Math Appl 49:187–221, 2005; Comput Math Appl 53, 2007) introduced into hyperbolic geometry the concept of defect based on relativity addition of A. Einstein. Another approach is from Karzel (Resultate Math. 47:305–326, 2005) for the relation between the K-loop and the defect of an absolute plane in the sense (Karzel in Einführung in die Geometrie, 1973). Our main concern is to introduce a systematical exact definition for defect and area in the Beltrami–Klein model of hyperbolic geometry. Combining the ideas and methods of Karzel and Ungar give an elegant concept for defect and area in this model. In particular we give a rigorous and elementary proof for the defect formula stated (Ungar in Comput Math Appl 53, 2007). Furthermore, we give a formulary for area of circle in the Beltrami–Klein model of hyperbolic geometry.     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.     

A Robust Spectral Method for Solving Heston’s Model

In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979).     

Classroom-based interventions in mathematics education: relevance, significance, and applicability

This special issue discusses various pedagogical innovations and myriad of significant findings. This commentary is not a synthesis of these contributions, but a summary of my own reflections on selected aspects of the nine papers comprising the special issue. Four themes subsume these reflections: (1) Gestural Communication (Alibali, Nathan, Church, Wolfgram, Kim and Knuth 2013); (2) Development of Ways of Thinking (Jahnke and Wambach 2013; Lehrer, Kobiela and Weinberg 2013; Mariotti 2013; Roberts and A. Stylianides 2013; Shilling-Traina and G. Stylianides 2013; Tabach, Hershkowitz and Dreyfus 2013); (3) Learning Mathematics through Representation (Saxe, Diakow and Gearhart 2013); and (4) Challenges in Dialogic Teaching (Ruthven and Hofmann 2013).