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AbstractWe investigate the effect of marking‐to‐market on an optimal futures hedge under stochastic interest rates. An intertemporal optimal hedge ratio that accounts for basis risk and marking‐to‐market is derived. This ratio includes all previous hedge ratios, with constant interest rates as special cases. In a preliminary empirical study using S&P 500 index futures contracts, we demonstrate that the futures‐forward hedging differential is nontrivial, especially in risk‐return optimization. We also show that the covariances between interest rates and spot and futures prices explain the differential: the larger the covariances are, the larger the differential will be.

AbstractThe article develops a theorem which shows that the Lanchester linear war equations are not in general equal to the Kolmogorov linear war equations. The latter are time‐consuming to solve, and speed is important when a large number of simulations must be run to examine a large parameter space. Run times are provided, where time is a scarce factor in warfare. Four time efficient approximations are presented in the form of ordinary differential equations for the expected sizes and variances of each group, and the covariance, accounting for reinforcement and withdrawal of forces. The approximations are compared with "exact" Monte Carlo simulations and empirics from the WWII Ardennes campaign. The band spanned out by plus versus minus the incremented standard deviations captures some of the scatter in the empirics, but not all. With stochastically varying combat effectiveness coefficients, a substantial part of the scatter in the empirics is contained. The model is used to forecast possible futures. The implications of increasing the combat effectiveness coefficient governing the size of the Allied force, and injecting reinforcement to the German force during the Campaign, are evaluated, with variance assessments. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.

In this thesis, we study a class of coupled forward backward stochastic differential equations (FBSDEs), called singular FBSDEs, which were first introduced in 2013, to model the evolution of emissions and the price of emissions allowances in a carbon market such as the European Union Emissions Trading System. These FBSDEs have two key properties: the terminal condition of the backward equation is a discontinuous function of the terminal value of the forward equation, and the forward dynamics may not be strongly elliptic, not even in a neighbourhood of the singularities of the terminal condition. We first consider a model for an electricity market subject to a carbon market with a single compliance period. We show that the carbon pricing problem leads to a singular FBSDE. This type of model is then extended to a multiperiod emissions trading system in which cumulative emissions are compared with a cap at multiple compliance times. We show that the multi-period pricing problem is well-posed for various mechanisms linking the trading periods. We then introduce an infinite period model, for a carbon market with a sequence of compliance times and no end date. We show that, under appropriate conditions, the value function for the multi-period pricing problem converges, as the number of periods tends to infinity, to a value function for this infinite period model, and present a setting in which this occurs. Finally, we focus on numerical investigations. For the single period model for an electricity market with emissions trading, the processes and functions appearing in the pricing FBSDE are chosen to model the features of the UK energy market, using historical data. Numerical methods are used to solve the pricing FBSDE, and the results are interpreted. In the future, these could support policies seeking to mitigate the effects of climate change. ; Open Access