Most models in finance are formulated in continuous time using stochastic differential equations rather than stochastic difference equations, following the work of Merton ( 1969). Instead of equation (2.13) of Chapter 2, the dynamic evolution of the vector of state variables is described by a stochastic differential equation written as follows
dx = f(x, u)dt + S(x, u)dz, (7.1)
in which x(t) is a p × 1 vector of state variables (the argument t being suppressed when understood), u(t) is a q × 1 vector of control variables, dx(t) = x(t + dt) - x(t) with dt denoting a small time interval, S is a p × n matrix, and z(t) is an n × 1 vector Wiener process.
A Wiener processz(t) has the property that a change from t to t + dt, dz(t) = z(t + dt) - z(t) is normally distributed with zero mean, independent of all past history up to z(t) and has a covariance matrix proportional to dt. One can write the covariance matrix cov(dz) of dz(t) as Φdt. The assumption that, for t > s, the covariance matrix of z(t) - z(s) is proportional to t - s is implied by the assumption that at each small step going from time s to time t, for example, s + dt, s + 2dt, . . ., s + ndt = t (the interval t - s being divided into n equal parts, each with length dt), the change in z is statistically independent of the change at any other step. Given the independence assumption, the variance of z(t) - z(s) is the sumof the variances of the changes z(s + kdt) - z(s + (k - 1)dt), k = 1, . . ., n. Doubling the time from s to t will double the variance of z(t) - z(s). In this sense, think of the residual dz of equation (7.1) as being serially independent. It I let Φ = I, the identity matrix, z(t) is said to follow a standard Wiener process. The matrix S(x, u) in (7.1) makes the covariance matrix of Sdz, or SΦSʹ dt ≡ Σdt, dependent on both the vector state variable x and the vector control variable u.