this paper the value of the option (i.e. the supremum in (1.2)) will be found exactly, and in particular it will be shown that the maximum in (1.2) is finite if and only if r ? ¯ : (1.4) Assuming (1.4), an explicit formula is given for both the maximal expected present value and the optimal stopping rule in (2.4), which is not a fixed time rule but depends heavily on the observed values of X t and S t . We call the financial option described above a "Russian option" for two reasons. First, this name serves to (facetiously) differentiate it from American and European options, which have been extensively studied in financial economics, especially with the new interest in market economics in Russia. Second, our solution of the stopping problem (1.2) is derived by the so-called principle of smooth fit, first enunciated by the great Russian mathematician, A. N. Kolmogorov, cf. [4, 5]. The Russian option is characterized by "reduced regret" because the owner is paid the maximum stock price up to the time of exercise and hence feels less remorse at not having exercised at the maximum. For purposes of comparison and to emphasize the mathematical nature of the contribution here, we conclude the paper by analyzing an optimal stopping problem for the Russian option based on Bachelier's (1900) original linear model of stock price fluctuations, X

.<F3.838e+05> This paper is concerned with long-run average risk-sensitive control of production planning in a manufacturing system with machines that are subject to breakdown and repair. By using a logarithmic transformation, it is shown that the associated Hamilton--Jacobi--Bellman equation has a viscosity solution. The risk-sensitive control problem has a dynamic stochastic game interpretation. Finally, a limiting problem is obtained when the rates of machine breakdown and repair go to infinity.<F4.005e+05> Key words.<F3.838e+05> risk-sensitive control, production planning, logarithmic transformation, irreducible Markov chain<F4.005e+05> AMS subject classifications.<F3.838e+05> 93E20, 93B35, 90B30<F4.005e+05> PII.<F3.838e+05> S036301299631034X<F4.817e+05> 1. Introduction.<F4.461e+05> In this paper we consider a manufacturing system which consists of machines that are subject to breakdown and repair. The objective of the problem is to choose a production planning to minimize a risk-s...