Weak second-order conditions of Runge-Kutta method for stochastic optimal control problems (Q6596347)

The authors consider the optimal control problem: \(\text{minimize}_{u\in \mathcal{A} }J(u)=\mathbb{E}[\phi (y_{T})]+\int_{t_{0}}^{T}\mathbb{E}[g(y_{t},u_{t})]dt\), subject to \(dy_{t}=f(y_{t},u_{t})dt+h(y_{t})dW_{t}\), \(t\in \lbrack 0,T]\), \( y(t_{0})=y^{0}\), where \((W_{t})_{t}\), \(0\leq t\leq T\) is a 1D Brownian motion on a filtered probability space \((\Omega ,\mathcal{F},(\mathcal{F} _{t})_{t\in \lbrack t_{0},T]},\mathbb{P})\), with \(T>0\) and \(\Omega =[t_{0},T] \) a fixed finite interval, \(y^{0}\in \mathbb{R}\), \(f,h,\phi ,g\) are continuously differentiable functions, \(f\) and \(h\) satisfying both Lipschitz and linear growth bound conditions in \(y\), and the control \(u=(u_{t})_{t\in \lbrack t_{0},T]}\) is a process in a closed convex set \(\mathcal{A}\) in the control space \(L^{2}(t_{0},T)\). This problem has a unique solution. The authors recall the generalized Hamiltonian function of this optimal control problem which involves a coupled process \((p_{t},q_{t})\) adapted with respect to \((\mathcal{F}_{t})_{t\in \lbrack t_{0},T]}\) and they write the associated continuous first-order optimality system, that they write in the form \(dX_{t}=\mathbf{F}(y_{t},u_{t},p_{t},q_{t})dt+H(y_{t},q_{t})dW_{t}\) (\( t\in \lbrack t_{0},T]\)), or as an Itô stochastic differential equation. \N\NThe purpose of the paper is to propose an \(s\)-stage Runge-Kutta method for such stochastic optimal control problems, starting from a discretization \( t_{0}<t_{1}<\ldots <t_{N}=T\) of the time interval \([t_{0},T]\) and to derive the associated weak order-1 and order-2 conditions. The authors first write the discrete first-order optimality conditions and they proceed with direct computations. The paper ends with two numerical examples (Black-Scholes type of optimal control problems), for which exact solutions are known. The authors compute the convergence rate of the Runge-Kutta method for different values of a parameter involved in the problem.