Three expressions are provided for the first hitting time density of an Ornstein-Uhlenbeck process to reach a fixed level. The first hinges on an eigenvalue expansion involving zeros of the parabolic cylinder functions. The second is an integral representation involving some special functions whereas the third is given in terms of a functional of a 3-dimensional Bessel bridge. The expressions are used for approximating the density.

Assuming that the wealth process X u is generated self-financially from the given initial wealth by holding its fraction u in a risky stock (whose price follows a geometric Brownian motion with drift μ ∈ R and volatility σ > 0) and its remaining fraction 1−u in a riskless bond (whose price compounds exponentially with interest rate r ∈ R), and letting P t,x denote a probability measure under which X u takes value x at time t, we study the dynamic version of the nonlinear mean-variance optimal control problemwhere t runs from 0 to the given terminal time T > 0, the supremum is taken over admissible controls u, and c > 0 is a given constant. By employing the method of Lagrange multipliers we show that the nonlinear problem can be reduced to a family of linear problems. Solving the latter using a classic Hamilton-Jacobi-Bellman approach we find that the optimal dynamic control is given bywhere δ = (μ−r )/σ . The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself.

Assuming that the stock price X follows a geometric Brownian motion with drift µ ∈ IR and volatility σ > 0 , and letting P x denote a probability measure under which X starts at x > 0 , we study the dynamic version of the nonlinear mean-variance optimal stopping problemwhere t runs from 0 onwards, the supremum is taken over stopping times τ of X , and c > 0 is a given and fixed constant. Using direct martingale arguments we first show that when µ ≤ 0 it is optimal to stop at once and when µ ≥ σ 2 /2 it is optimal not to stop at all. By employing the method of Lagrange multipliers we then show that the nonlinear problem for 0 < µ < σ 2 /2 can be reduced to a family of linear problems. Solving the latter using a free-boundary approach we find that the optimal stopping time is given bywhere γ = µ/(σ 2 /2) . The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself.

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.