Optimization problems for a class of diffusion processes in semimarkov environment.

*(Bulgarian. English summary)*Zbl 0729.60084
Mathematics and education in mathematics, Proc. 19th Spring Conf., Sunny Beach/Bulg. 1990, 315-320 (1990).

Summary: [For the entire collection see Zbl 0713.00003.]

Let Y be a random process determined by a class of diffusion processes and a semi-Markov process (SMP) \(X_ 0\) with a phase space consisting of a single class of stable states. We say the process \(Y_ 0\) is controlled by the SMP \(X_ 0\). A process \(Y^{\epsilon}\) controlled by the perturbed SMP \(X^{\epsilon}\) with a phase space consisting of a single class of stable states and one absorbing state is considered. If \(\epsilon\to 0\), the process \(Y^{\epsilon}\) converges to the random process Y controlled by the limit phase aggregated Markov process X. The present work deals with some optimization problems where the relations between the moment of first reaching of given level by the process Y and the absorbing time of the process X are studied.

Let Y be a random process determined by a class of diffusion processes and a semi-Markov process (SMP) \(X_ 0\) with a phase space consisting of a single class of stable states. We say the process \(Y_ 0\) is controlled by the SMP \(X_ 0\). A process \(Y^{\epsilon}\) controlled by the perturbed SMP \(X^{\epsilon}\) with a phase space consisting of a single class of stable states and one absorbing state is considered. If \(\epsilon\to 0\), the process \(Y^{\epsilon}\) converges to the random process Y controlled by the limit phase aggregated Markov process X. The present work deals with some optimization problems where the relations between the moment of first reaching of given level by the process Y and the absorbing time of the process X are studied.