Abstract. We consider the numerical solution of Hamilton-Jacobi-Bellman equations arising in stochastic control theory. We introduce a class of monotone approximation schemes relying on monotone interpolation. These schemes converge under very weak assumptions, including the case of arbitrary degenerate diffusions. Besides providing a unifying framework that includes several known first order accurate schemes, stability and convergence results are given, along with two different robust error estimates. Finally, the method is applied to a super-replication problem from finance.

Background and ObjectivesCovid-19 vaccines approved by the EU, UK and USA have been found to be safe and effective. The cost effectiveness of these vaccines depends upon a number of factors. The aim of this paper is to explore the cost effectiveness of a COVID-19 vaccine and to analyse how the price of the vaccine and the cost of administrating it influence its cost effectiveness. Methods We considered an epidemiological model developed by an expert group within 'Statens Serum Institut', which is a unit under the auspices of the Danish Ministry of Health. The model allowed us to differentiate between two population groups, those aged ≥60 years and those aged <60 years. We used the model to consider four scenarios: (i) vaccination of 25% of the total population (corresponding to approximately 1.5 million persons) but targeting vaccines towards the population aged ≥60 years, (ii) vaccination of 25% of the total population, targeting vaccines only towards the population aged <60 years, (iii) vaccination of 40% of the total population where 15% are aged <60 years and 25% are aged ≥60 years (corresponding approximately to the full Danish population aged >60 years), and (iv) 40% of the total population is vaccinated but vaccines are targeted solely towards those aged <60 years. The time horizon of the analysis was six months, and the perspective was that of the Danish healthcare sector. ResultsThe results show that inclusion of the elderly population aged ≥60 years was more cost effective than a vaccination strategy that targeted a population aged <60 years old only, when productivity losses were not included. Furthermore, the results show that an extension of the target group from the elderly population only, to also include the younger population comes with an increasing cost per life-year gained. The incremental cost-effectiveness ratio depends on the price of the vaccine, hereunder also the administration costs, and the discount rate used for the estimation of life-years or quality-adjusted life-years gained from a vaccine. Furthermore, inclusion of productivity losses in the analyses influenced the cost effectiveness of vaccination of the population aged <60 years of age. ConclusionThe cost effectiveness of a COVID-19 vaccine is sensitive to whether or not productivity losses are included in the analyses. Without productivity losses, the elderly population should always be part of the target group for a COVID-19 vaccination programme. Taking productivity losses into account, at least in the case of low vaccine prices, vaccinating the younger population first can actually be cost effective.

In recent years, implicit stochastic Runge-Kutta (SRK) methods have been developed both for strong and weak approximations. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes such as simple iteration, modified Newton iteration or full Newton iteration. We employ a unifying approach for the construction of stochastic B-series which is valid both for Itô-and Stratonovich-stochastic differential equations (SDEs) and applicable both for weak and strong convergence to analyze the order of the iterated Runge-Kutta method. Moreover, the analytical techniques applied in this paper can be of use in many other similar contexts.

We present and analyse a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with separation between the (fast) timescale of individual trajectories and the (slow) time-scale of the macroscopic function of interest. The algorithm combines short bursts of path simulations with extrapolation of a number of macroscopic state variables forward in time. The new microscopic state, consistent with the extrapolated variables, is obtained by a matching operator that minimises the perturbation caused by the extrapolation. We provide a proof of the convergence of this method, in the absence of statistical error, and we analyse various strategies for matching, as an operator on probability measures. Finally, we present numerical experiments that illustrate the effects of the different approximations on the resulting error in macroscopic predictions.

The class of stochastic Runge-Kutta methods for stochastic differential equations due to Rößler is considered. Coefficient families of diagonally drift-implicit stochastic Runge-Kutta (DDISRK) methods of weak order one and two are calculated. Their asymptotic stability as well as mean-square stability (MS-stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.