Multistage stochastic programming: Error analysis for the convex case

Abstract

Abstract:We consider convex stochastic multistage problems and present an approximation technique which allows to analyse the error with respect to time. The technique is based on barycentric approximation of conditional and marginal probability spaces and requires strict nonanticipativity for the constraint multifunction and the saddle property for the value functions.

23. Refinancing Mortgages in Switzerland

“…Both ψ t and t are bilinear functions since the integrand λ ν t (η t ) · τ µ t (ξ t ) in (23.12) and (23.13) is bilinear in (η t , ξ t ). It is shown in [13] that the following relation holds:…”

“…Then, the solution of the corresponding approximate problem based on the new partition P t +1 must be at least as good as the former bound. As t → ∞ and the sub-×-simplices become arbitrarily small with respect to diameter, the approximate value functions ψ t and t epi-converge to φ t [13].…”

“…As a consequence, the saddle property of the value function in T may not be "inherited'' to the previous stages due to the integration with respect to P t+1 (·|ω t ). However, if the corresponding distribution functions may be represented, e.g., in the form Q t+1 ω t+1 + H t+1 (ω t ) , where Q t+1 is independent of ω t and H t+1 is a linear mapping, then it can be shown that the relevant expectation functionals E t+1 φ t+1 (x t , ω t+1 ) are continuous saddle functions [13]. This leads to the following requirement:…”

“…To deal with the observed correlations between the risk factors, approximation schemes must take into account cross-moment information in a numerically efficient way. Such techniques have been developed, e.g., by Edirisinghe [6], Edirisinghe and Ziemba [8,9,10], and Frauendorfer [11,13,14]. See also [7] for a general survey on bound-based approximations.…”

Refinement Issues in Stochastic Multistage Linear Programming

“…If, in addition, the conditional probabilities P t ( j t?1 ; t?1 ) depend linearly on the observation in period t ? 1, the convexity of the stochastic multistage program (5) is preserved and the value function of stage t associated with (5) are saddle functions concave-convex in ( t ; t ) (see Frauendorfer 1994, 23]).…”