For a random walk R_n≥0 given on a homogeneous irreducible finite MARKOV chain X_n≥0 the identity (8) is obtained. Generalizations (14)-(16) of WALD's Fundamental Identity and WALD's first and second equations for the two-dimensional process (R_n,X_n), n≥0 are proved. The Average Sample Number (21)-(22) and the Operating Characteristic Function (24)-(25) of a Sequential Probability Ratio Test follow. With this test a decision about two simple hypotheses on the unknown transition probability matrix of X_n, n≥0 and the unknown parameters of the probability distributions for the increments of X_n, n≥0 can be made. For a special case these results were proved by PHATARFOD [6] and KCHLER [3] with other methods.