A variational inequality arising from optimal surrender of variable annuity with lookback benefit

We introduce a variable annuity (VA) contract with a surrender option and lookback benefit, that is, the benefit of the VA contract is linked to the maximum process of the policyholder’s account value. In contrast to the constant guarantee model provided in Bernard et al. (Insur. Math. Econ. 55:116–128, 2014), it is optimal for the policyholder of the VA contract with lookback benefit to surrender the VA contract when the policyholder’s account value is below or equal to the optimal surrender boundary. Thus, from the perspective of the insurer to construct a portfolio of VA contracts, utilizing the VA contracts with lookback benefit along with VA contracts with constant guarantee provides the diversification of early surrenders. The valuation of this contract can be described as a two-dimensional parabolic variational inequality. By converting this into the one-dimensional problem, we obtain the integral equations for the value function and the free boundary. The recursive integration method is applied to obtain the numerical solutions. We also provide comparative statics of the optimal surrender boundaries with respect to various parameters.

Survey of Calculation Methods for Polytropic Efficiencies

Calculating polytropic efficiencies is a basic task used for quantifying performance of power cycles involving compression and/or expansion. The incremental definition of a “polytropic curve” of gases by Gustav Zeuner in 1905 may be the oldest mention of the word “polytropic” in a thermodynamic context [1]. In Turbomachinery blading, the typical changes of state are nearly adiabatic and polytropic. L. S. Dzung was probably the first defining an incremental polytropic efficiency in 1944 [3]. Recursive integration of this has become the best thermodynamic quality measure of a blading. Both Zeuner and Dzung started their consideration with an incremental definition. However, they integrated analytically assuming ideal gas data. This resulted in the well-known formula (1) p v n = constant Most thermodynamic textbooks declare this the definition of a polytropic change of state. However, the incremental definition survived too. Stodola [2], Dzung and later scientists established it as another definition of a polytropic change of state. Thus, we face now two definitions of a polytropic change of state, which are theoretically identical for ideal gases but different for real gases and vapors. In educational context, this is disturbing and forcing to a logical detour. We trace the historic roots and show that the initial incremental definition is the physically healthier one. Recursive integration allows direct application to turbomachinery with any finite pressure ratio and to any real fluid.

Transient Dynamics in the Random Growth and Reset Model

A mean-field type model with random growth and reset terms is considered. The stationary distributions resulting from the corresponding master equation are relatively easy to obtain; however, for practical applications one also needs to know the convergence to stationarity. The present work contributes to this direction, studying the transient dynamics in the discrete version of the model by two different approaches. The first method is based on mathematical induction by the recursive integration of the coupled differential equations for the discrete states. The second method transforms the coupled ordinary differential equation system into a partial differential equation for the generating function. We derive analytical results for some important, practically interesting cases and discuss the obtained results for the transient dynamics.

Information-Theoretic Estimation of Econometric Functions

This chapter introduces an information-theoretic approach to specify econometric functions as an alternative to avoid parametric assumptions. We investigate the performances of the information-theoretic method in estimating the regression (conditional mean) and response (derivative) functions. We have demonstrated that they are easy to implement and are advantageous over parametric models and nonparametric kernel techniques. For the implementation of our estimation method, a recursive integration procedure is introduced. An empirical illustration is also presented to study the Canadian earning function. The asymptotic properties of the regression and derivative functions are established. In addition, a test for normality is also proposed.