A subsidy-surplus model and the Skorokhod problem in an orthant

Abstract

We consider the deterministic Skorokhod problem in an orthant of the form Zw(t)=w(t)+∫₀^tb(u,Yw(u),Zw(u))du+∫₀^tR(u,Yw(u−),Zw(u−))dYw(u) with (Yw)_i(·) nondecreasing, and (·) not increasing while (Zw)_i(·)>0 This can be viewed as a subsidy-surplus model in an interdependent economy. Existence of a unique solution is established under fairly general conditions (viz. with R(.,.,.,.,) satisfying a uniform spectral radius condition). Comparison result for (SP) vis-a-vis the usual partial order on the orthant is studied; we show that the more "inward looking" the reflection vectors and the drift, the larger the values of Yw will be but the values of Zw will be smaller. In addition to showing that the Leontief-type output is a feasible subsidy, connection between (SP) and "minimality" of feasible subsidies is discussed (consequently it is suggested that (SP) may be taken as a continuous time feedback-form analogue of open Leontief model). In the stochastic case, (Y(t), Z(t)) turns out to be a strong Markov process if w(·) arises from aLevy process. Relevance of the comparison result to recurrence/positive recurrence of Z(·) process is pointed out.