Dynamic Economics: Optimization by the Lagrange Method

By Gregory C. Chow | Go to book overview

CHAPTER SEVEN Models in Finance

7.1 Stochastic Differential Equations

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 sum

of 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.

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