Finding optimal decisions when a model’s rules can jump

Many optimization models describe a system that evolves through both deliberate decisions and random fluctuations. A controller chooses actions over time, seeking to minimize expected costs. Standard mathematical approaches often assume that the system’s coefficients—the functions describing its motion, uncertainty and costs—change continuously with its state. Abrupt regime changes can fall outside that framework.

In the preprint Stochastic Optimal Control with Measurable Coefficients and Applications, Filippo de Feo develops an approach that permits certain discontinuities. The work concerns infinite-horizon problems with discounted costs and establishes mathematical conditions for identifying optimal controls. Its advertising application illustrates the theory rather than testing an investment strategy against market data.

An equation for the best achievable cost

The system follows a stochastic differential equation. Its drift describes its systematic movement, while a diffusion term describes random fluctuations driven by Brownian motion. Both may depend on the current state and the chosen action. The objective is to minimize the expected total running cost, with future costs discounted.

The central object is the value function: the lowest achievable expected cost from each starting state. A Hamilton–Jacobi–Bellman, or HJB, equation connects this function to the available actions. It combines the running cost with terms involving the value function’s slope and curvature, then minimizes over controls.

For measurable coefficients, the usual continuous-coefficient viscosity-solution framework does not directly apply. De Feo instead uses an existing generalized notion called an Lp-viscosity solution. This replaces smooth test functions with a broader class of functions possessing integrable weak derivatives, allowing the equation to accommodate measurable ingredients.

From generalized solutions to optimal policies

Under the paper’s regularity assumptions, including local uniform ellipticity and local uniform continuity of the diffusion coefficient in the state, de Feo proves that these generalized solutions have locally integrable second derivatives. They consequently satisfy the HJB equation almost everywhere, making them strong solutions in the terminology used here. This does not mean they are twice continuously differentiable everywhere.

The next step is verification: proving that a candidate solution really represents the optimal cost. Using a version of Dynkin’s formula valid for this less-smooth class of functions, the paper establishes sufficient and necessary conditions for optimality, subject to the stated integrability and long-run conditions.

A key technical issue is that derivatives and the equation may be undefined on sets of zero volume. Under the relevant assumptions, controlled trajectories spend zero time in such sets, almost surely. That allows almost-everywhere identities to support the verification argument without requiring smoothness at every state.

The resulting feedback rule selects an action that minimizes the current-value Hamiltonian at the observed state. Constructing such a rule also requires a measurable selection and a solution of the resulting closed-loop stochastic equation. Under stronger boundedness and uniform ellipticity assumptions, and assuming the minimum is attained, the paper establishes optimal feedback controls. With the additional hypotheses of its uniqueness result, it identifies the value function as the unique Lp-viscosity solution among continuous functions of polynomial growth on the whole state space.

Advertising with discontinuous goodwill effects

The application models a product’s advertising goodwill. Investment raises its drift, while a bounded, non-positive measurable function represents image deterioration. That function can jump between regimes as goodwill changes. The utility associated with goodwill may also be discontinuous.

Advertising investment is restricted to a bounded interval. Its cost is continuous, while the baseline uncertainty is bounded, uniformly continuous and bounded away from zero. Investment can also increase the diffusion coefficient, so spending may affect both goodwill’s systematic movement and its uncertainty.

For this model, de Feo characterizes the value function through the HJB equation and constructs an optimal state-dependent investment policy using a weak solution of the closed-loop equation. The result is a mathematical characterization, not a numerical spending schedule.

The scope remains conditional. Measurability does not eliminate the need for assumptions on diffusion, boundedness, attainable minima and admissibility. In particular, the theory’s ellipticity requirements exclude settings where the relevant noise degenerates. The contribution is therefore a rigorous route to optimal control for a specified class of irregular models, rather than an unrestricted solution for discontinuous economic systems.