This paper develops new bounds on the expectation of a convex-concave saddle function of a random vector with compact domains. The bounds are determined by replacing the underlying distribution by unique discrete distributions, constructed using second-order moment information. The results extend directly to new second moment lower bounds in closed-form for the expectation of a convex function. These lower bounds are better than Jensen's bound, the only previously known lower bound for the convex case, under limited moment information. Application of the second moment bounds to two-stage stochastic linear programming is reported. Computational experiments, using randomly generated stochastic programs, indicate that the new bounds may easily outperform the usual first-order bounds.
This paper develops upper and lower bounds on two-stage stochastic linear programs using limited moment information. The case considered is when both the right-hand side as well as the objective coefficients of the second stage problem are random. Random variables are allowed to have arbitrary multivariate probability distributions with bounded support. First, upper and lower bounds are obtained using first and cross moments, from which we develop bounds using only first moments. The bounds are shown to solve the respective general moment problems.
We previously obtained tight upper and lower bounds to the expectation of a saddle function of multivariate random variables using first and cross moments of the random variables without independence assumptions. These bounds are applicable when domains of the random vectors are compact sets in the euclidean space. In this paper, we extend the results to the case of unbounded domains, similar in spirit to the extensions by Birge and Wets in the pure convex case. The relationship of these bounds to a certain generalized moment problem is also investigated. Finally, for solving stochastic linear programs utilizing the above bounding procedures, a computationally more appealing order-cone decomposition scheme is proposed which behaves quadratically in the number of random variables. Moreover, the resulting upper and lower approximations are amenable to efficient solution techniques.
The spot price market for electricity is highly volatile. The time series of the daily average electricity price is characterised by seasonality, mean reversion, jumps, and regime-switching processes. In electricity markets, 'swing' contracts, which can provide some protection against the day-to-day price fluctuations, are used to incorporate flexibility in acquiring given quantities of electricity. We develop a lattice approach for the valuation of swing options by modelling the daily average price of electricity by a regime-switching process that utilises three regimes, consisting of Brownian motions and a mean-reverting process. Various numerical examples are presented to illustrate the methodology.Swing option, Regime-switching process, Mean-reverting process, Electricity market,
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