A maximum principle for stochastic control systems

Abstract Let ξn be a sequence of independent vector-valued random variables. Let xn be a vector with components x0(n), …, xr(n), and let θn be a vector control. The maximum principle and canonical equations of Pontryagin are derived for the vector system xn = F(xn − 1, θn, ξn) with loss function E x0(N), where N, the control time, is fixed. In the continuous case, it is derived for the form x = ƒ(x, θ) + σξ , and loss E x0(T), where σ is a nonnegative definite matrix and ξ is vector-valued white Gaussian noise. Under suitable smoothness assumptions, the expectation (conditioned upon xn) of the adjoint variables are the derivatives of the loss (min E x0(N) conditioned upon xn).