Infinite Quasimonotone Systems of Ordinary Differential Equations with Applications to Stochastic Processes

We deal with the initial value problem for countably infinite linear systems of ordinary differential equations of the form y '( t ) = A ( t ) y ( t ) where A ( t ) = ( a ij ( t ): i , j S 1) is a measurable, infinite and essentially positive matrix, i.e., a ij ( t ) S 0 for i p j . The main novelty of our approach is the systematic use of a classical comparison theorem for finite linear systems which leads easily to the existence of a nonnegative minimal solution and its properties. Application to generalized stochastic birth and death processes produces criteria for honest and dishonest probability distributions. A short proof of the Kolmogorov and Chapman-Kolmogorov equations for stochastic processes follows. The results hold for L 1 -coefficients. Our method extends to nonlinear infinite systems of quasimonotone type and can be used for numerical procedures that yield exact results; cf. the Addendum.