Global maximum principle for optimal control of stochastic Volterra equations with singular kernels: An infinite dimensional approach
Yushi Hamaguchi
TL;DR
The paper addresses optimal control of stochastic Volterra equations with memory effects and singular kernels, extending classical maximum principles to nonconvex control domains. It employs spike variations and an infinite dimensional lift of SVEs via a measure based decomposition of the kernels, yielding first and second order adjoint equations as backward stochastic evolution equations on weighted $L^2$ spaces. The main result is a global maximum principle that characterizes optimal controls through a Hamiltonian inequality augmented by a second order risk adjustment term, under structural kernel assumptions. A rigorous well posedness theory for the resulting BSEEs on weighted spaces is developed, providing a robust mathematical foundation for the adjoint framework and enabling tractable analysis beyond prior BSVIE/BSEE approaches.
Abstract
In this paper, we consider optimal control problems of stochastic Volterra equations (SVEs) with singular kernels, where the control domain is not necessarily convex. We establish a global maximum principle by means of the spike variation technique. To do so, we first show a Taylor type expansion of the controlled SVE with respect to the spike variation, where the convergence rates of the remainder terms are characterized by the singularity of the kernels. Next, assuming additional structure conditions for the kernels, we convert the variational SVEs appearing in the expansion to their infinite dimensional lifts. Then, we derive first and second order adjoint equations in form of infinite dimensional backward stochastic evolution equations (BSEEs) on weighted $L^2$ spaces. Moreover, we show the well-posedness of the new class of BSEEs on weighted $L^2$ spaces in a general setting.