While the general principles involved in the formulation of random walk and Brownian motion equations (whether the random changes are directly on the position of a particle or individual, or on the velocity) are well-known, there are various situations considered in the literature involving the assumption of a constant speed (in magnitude). Thus the derivation by Goldstein (1951) of a one-dimensional wave-like equation involved the tacit assumption Ut = ± a, where Ut is the vector velocity dRt/dt,Rt being the (column) position vector (Bartlett (1957)). Biological models may involve the assumption of individuals moving at constant speed (cf. Kendall (1974)). Finally, the derivation of Schrodinger-type equations from Brownian motion models has sometimes involved the assumption U′tUt = c2, where c is the velocity of light (Cane (1967), (1975)).

The annual premium for a life assurance contract is obtained by equating the expected discounted value of the sum payable on death to the expected discounted value of the series of premiums, and loading the resultant net premium for expenses and contingencies. The approach is essentially deterministic. An adequate stochastic model of mortality is readily available, but it is of limited practical value. A life office writes a large number of policies on independent lives and, apart from the effects of a few very large policies, the overall mortality behaviour of its portfolio is effectively deterministic.

Let X(t) be a real-valued stochastic process with sample functions which are almost surely continuously differentiable. We say that X(t) has an upcrossing at level u at time t if X(t) = u, X′(t) > 0.

Suppose where S is a compact set (of say). Let Φ (mapping S into S) and Ψ be continuous on S and such that Ψ(xk) is monotone (non-decreasing, say) as k increases, and xk+1 = Φ(xk). Put Ψ∗ = lim (k → ∞)Ψ(xk); if {xk(i)}∞i=0 is a convergent subsequence of {xk}, with a limit-point α, then Ψ∗ = Ψ(α), and and