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We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper we investigate the r\^ole of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators.

While solar-like oscillations in red giants have been observed at massive scale by the Kepler mission, few features of these oscillation mode frequencies, other than their global properties, have been exploited for stellar characterization. The signatures of acoustic glitches in mode frequencies have been used for studying main-sequence stars, but the validity of applying such techniques to evolved red giants, particularly pertaining to the inclusion of nonradial modes, has been less well-examined. Making use of new theoretical developments, we characterize glitches using the $\pi$ modes associated with red giant stellar models, and use our procedure to examine for the first time how properties of the He II acoustic glitch -- specifically its amplitude and associated acoustic depth -- vary over the course of evolution up the red giant branch, and with respect to other fundamental stellar properties. We find that the acoustic depths of these glitches, in conjunction with other spectroscopic information, discriminates between red giants in the first-ascent and core-helium-burning phases. We critically reexamine previous attempts to constrain acoustic glitches from nonradial (in particular dipole) modes in red giants. Finally, we apply our fitting procedure to Kepler data, to evaluate its robustness to noise and other observational systematics., Comment: 15 pages, 12 figures; accepted to ApJ

We investigate the problem of finding upper and lower bounds for a Choquet risk measure of a nonlinear function of two risk factors, when the marginal distributions of the risk factors are ambiguous and represented by nonadditive measures on the marginal spaces and the joint nonadditive distribution on the product space is unknown. We treat this problem as a generalization of the optimal transport problem to the setting of nonadditive measures. We provide explicit characterizations of the optimal solutions for finite marginal spaces, and we investigate some of their properties. We further discuss the connections with linear programming, showing that the optimal transport problems for capacities are linear programs, and we also characterize their duals explicitly. Finally, we investigate a series of numerical examples, including a comparison with the classical optimal transport problem, and applications to counterparty credit risk., Comment: 30 pages, 3 figures, 4 tables

Consider the inverse first-passage problem: Given a diffusion process $\{\frak{X}_{t}\}_{t\geqslant 0}$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a survival probability function $p$ on $[0,\infty)$, find a boundary, $x=b(t)$, such that $p$ is the survival probability that $\frak{X}$ does not fall below $b$, i.e., for each $t\geqslant 0$, $p(t)= \mathbb{P}(\{\omega\in\Omega\;|\; {\frak{X}}_s(\omega) \geqslant b(s),\ \forall\, s\in(0,t)\})$. In earlier work, we analyzed viscosity solutions of a related variational inequality, and showed that they provided the only upper semi-continuous (usc) solutions of the inverse problem. We furthermore proved weak regularity (continuity) of the boundary $b$ under additional assumptions on $p$. The purpose of this paper is to study higher-order regularity properties of the solution of the inverse first-passage problem. In particular, we show that when $p$ is smooth and has negative slope, the viscosity solution, and therefore also the unique usc solution of the inverse problem, is smooth. Consequently, the viscosity solution furnishes a unique classical solution to the free boundary problem associated with the inverse first-passage problem., Comment: 24 pages

This paper aims to extend downside protection to a hedge fund investment portfolio based on shared loss fee structures that have become increasing popular in the market. In particular, we consider a second tranche and suggest the purchase of an upfront reinsurance contract for any losses on the fund beyond the threshold covered by the first tranche, i.e. gaining full portfolio protection. We identify a fund's underlying liquidity as a key parameter and study the pricing of this additional reinsurance using two approaches: First, an analytic closed-form solution based on the Black-Scholes framework and second, a numerical simulation using a Markov-switching model. In addition, a simplified backtesting method is implemented to evaluate the practical application of the concept.

We consider the problem of determining an upper bound for the value of a spectral risk measure of a loss that is a general nonlinear function of two factors whose marginal distributions are known, but whose joint distribution is unknown. The factors may take values in complete separable metric spaces. We introduce the notion of Maximum Spectral Measure (MSP), as a worst-case spectral risk measure of the loss with respect to the dependence between the factors. The MSP admits a formulation as a solution to an optimization problem that has the same constraint set as the optimal transport problem, but with a more general objective function. We present results analogous to the Kantorovich duality, and we investigate the continuity properties of the optimal value function and optimal solution set with respect to perturbation of the marginal distributions. Additionally, we provide an asymptotic result characterizing the limiting distribution of the optimal value function when the factor distributions are simulated from finite sample spaces. The special case of Expected Shortfall and the resulting Maximum Expected Shortfall is also examined.

Consider the set of probability measures with given marginal distributions on the product of two complete, separable metric spaces, seen as a correspondence when the marginal distributions vary. In problems of optimal transport, continuity of this correspondence from marginal to joint distributions is often desired, in light of Berge's Maximum Theorem, to establish continuity of the value function in the marginal distributions, as well as stability of the set of optimal transport plans. Bergin (1999) established the continuity of this correspondence, and in this note, we present a novel and considerably shorter proof of this important result. We then examine an application to an assignment game (transferable utility matching problem) with unknown type distributions.

If a Lagrangian defining a variational problem has order $k$ then its Euler-Lagrange equations generically have order $2k$. This paper considers the case where the Euler-Lagrange equations have order strictly less than $2k$, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivative variables, with a specific upper bound on the degree of the polynomial. The paper also provides an explicit formulation, derived from a geometrical construction, of a family of such $k$-th order Lagrangians, and it is conjectured that all such Lagrangians arise in this way.

In this paper we consider three types of embedded options in pension benefit design. The first is the Florida second election (FSE) option, offered to public employees in the state of Florida in 2002. Employees were given the option to convert from a defined contribution (DC) plan to a defined benefit (DB) plan at a time of their choosing. The cost of the switch was assessed in terms of the ABO (Accrued Benefit Obligation), which is the expected present value of the accrued DB pension at the time of the switch. If the ABO was greater than the DC account, the employee was required to fund the difference. The second is the DB Underpin option, also known as a floor offset, under which the employee participates in a DC plan, but with a guaranteed minimum benefit based on a traditional DB formula. The third option can be considered a variation on each of the first two. We remove the requirement from the FSE option for employees to fund any shortfall at the switching date. The resulting option is very similar to the DB underpin, but with the possibility of early exercise. Since we assume that exercise is only permitted at discrete, annual intervals, this option is a Bermudan variation on the DB Underpin. We adopt an arbitrage-free pricing methodology to value the option. We analyse and value the optimal switching strategy for the employee by constructing an exercise frontier, and illustrate numerically the difference between the FSE, DB Underpin and Bermudan DB Underpin options.