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The two-dimensional stochastic process (x(t), y(t)), where y(t) dx(t)/dt is a Wiener process (Brownian motion) is considered. The value of x(t) when y(t) first hits a line in the plane is of interest. The joint moment generating function (m.g.f.) of the first hitting time and place, as well as the probability density function (p.d.f.) of the first hitting place, in a special case, are obtained by solving the appropriate Kolmogorov backward equation. In the last section of the paper, the joint m.g.f, of the first hitting time and place is used to obtain the optimal control of the process in the first quadrant.

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A Markov chain with state space{0,…,N}and transition probabilities depending on the current state is studied. The chain can be considered as a discrete Ornstein-Uhlenbeck process. The probability that the process hitsNbefore 0 is computed explicitly. Similarly, the probability that the process hitsNbefore−Mis computed in the case when the state space is{−M,…,0,…,N}and the transition probabilitiespi,i+1are not necessarily the same wheniis positive andiis negative.

No abstract