Stochastic Optimal Control for Systems with Drifts of Bounded Variation: A Maximum Principle Approach

We study a stochastic control problem for nonlinear systems governed by stochastic differential equations with irregular drift. The drift coefficient is assumed to decompose as $b(t,x,a)=b_1(t,x)+b_2(x)b_3(t,a)$, where $b_1$ is bounded and Borel measurable, $b_2$ has bounded variation, and $b_3$ is bounded and smooth. Under these minimal regularity assumptions, we establish a Pontryagin-type stochastic maximum principle. The analysis relies on new results for SDEs with random drift of bounded variation, including existence, uniqueness, and Malliavin-Sobolev differentiability of the state process. A key ingredient is an explicit representation of the first variation process obtained via integration with respect to the space-time local time of bounded variation processes. By combining a suitable approximation scheme with Ekeland’s variational principle, and using a Garcia-Rodemich-Rumsey inequality to obtain a uniform control of the first variation, we derive the maximum principle. As an application, we derive an optimal corridor-type capital adjustment policy for an insurance surplus model.

💡 Research Summary

This paper establishes a Pontryagin-type stochastic maximum principle (SMP) for optimal control problems governed by stochastic differential equations (SDEs) with irregular drift coefficients. The authors consider a nonlinear system where the state process X^α evolves according to an SDE whose drift function b(t, x, a) has minimal regularity. Specifically, the drift is assumed to decompose as b(t, x, a) = b1(t, x) + b2(x)b3(t, a), where b1 is bounded and Borel measurable, b2 is a function of bounded variation (and thus not necessarily Lipschitz continuous), and b3 is bounded and smooth in the control variable. This structure significantly relaxes the standard differentiability assumptions required in classical SMP theory.

The primary challenge arises from the non-smoothness of b2. To overcome this, the authors develop several novel technical tools:

1. Foundational SDE Results: They first prove existence, uniqueness, and regularity properties for SDEs with random drift of the form b(t, ω, x) = b1(t, x) + b2(x)b3(t, ω). Key results include the Malliavin differentiability and Sobolev differentiability (with respect to the initial condition) of the solution.
2. Integration with Respect to Local Time: A central innovation is the use of integration with respect to the space-time local time of bounded variation processes. This allows for an explicit representation of the first variation process Φ of the controlled state with respect to its initial condition, which is crucial for deriving the adjoint equation in the SMP. The representation is given by Φ_{t,s} = exp(-∫_t^s ∫_R b(u, z, α_u) L^{X^x}(du, dz)).
3. Approximation Scheme and Uniform Estimates: The proof of the main theorem employs a sophisticated approximation argument. A sequence of optimal control problems with smoothed coefficients is constructed. Ekeland’s variational principle is then applied to these approximate problems to obtain a sequence of “almost optimal” controls and corresponding adjoint processes. A major difficulty is controlling the limit of the derivative of the approximate Hamiltonian. The authors resolve this by using a Garcia-Rodemich-Rumsey-type inequality to establish a uniform bound on the supremum norm of the difference between the approximate and the true state processes, which ensures the necessary convergence.

The main result (Theorem 3.2) states that if ˆα is an optimal control, then the derivative of the Hamiltonian H(t, x, y, a) = f(t, x, a) + b(t, x, a)y with respect to the control variable a must satisfy a variational inequality almost everywhere. The adjoint process Y is characterized by a backward representation involving the first variation process Φ. Under additional convexity conditions, this necessary condition is also sufficient.

As a concrete application, the authors analyze an optimal capital adjustment problem for an insurance surplus model. The surplus process is modeled to evolve within a target corridor, and the drift contains a discontinuous component activated when the surplus exits the corridor. The derived SMP is used to characterize the optimal policy for adjusting capital or withdrawing dividends to minimize the expected squared deviation of the terminal surplus.

In summary, this work provides a significant extension of the stochastic maximum principle to systems with drifts of bounded variation. It successfully bridges advanced probabilistic techniques—local time integration and Malliavin calculus—with control theory, offering a new framework for solving optimization problems arising in fields like insurance and finance where non-smooth dynamics are inherent.