Index of exponential growth for a class of stochastic semigroups

We consider finite-dimensional homogeneous stochastic semigroups X_s^t, 0 s t < assuming values in the space of real square matrices. For stochastic semigroups assuming values in the class of upper triangular matrices we compute the index of exponential growth, where · is the operator norm of a matrix. The answer is given in terms of the characteristic Y_t of the generating process Y_t of the semigroup X_s^t:x=–(1/2), where is the smallest eigenvalue of the matrix B which defines the characteristic Y_t=Bt.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 78–84, 1988.