Controlled diffusions and admissible controls

Fix a finite horizon, a complete filtered probability space with a right-continuous filtration, and a Brownian motion relative to that filtration. A control takes values in a specified Borel action set and is progressively measurable. Progressive measurability ensures that its restriction up to each time is measurable using only information then available; together with suitable moment bounds it permits the time and stochastic integrals. A controller cannot select an action at time zero from the realized terminal Brownian increment.

A feedback rule can use the current observed state, while an open-loop admissible control may depend on the available history. Neither label alone permits anticipation. Under partial observation, admissibility is relative to the observation filtration rather than the full state filtration. Enlarging information may lower a minimizing value, so the filtration is a mathematical component of the objective. A comparison of two policies is meaningful only when both belong to the same specified information class.

A useful sufficient setting has coefficients measurable in time and action, uniformly Lipschitz in the state, and at most linear growth in the state and action. For square-integrable initial data and controls with finite expected integrated squared norm, standard SDE estimates give a unique strong state solution and a finite second moment of its time supremum. Compact bounded action sets simplify the growth condition, but compactness alone cannot repair a nonmeasurable coefficient or a non-Lipschitz state equation.

Define the expected running-plus-terminal cost only after verifying its integrability. Quadratic-growth costs are controlled by the second-moment estimates in this setting. More singular costs or exponential criteria need stronger assumptions, and state constraints may remove controls whose otherwise well-defined states leave the permitted region. One may instead optimize over weak formulations where the probability space also varies, but that is a different admissible class. Here we keep the strong formulation and its information fixed.

The value is the infimum over all admissible controls, so optimizing a smaller subclass produces an upper bound on a minimizing value. It need not find the globally optimal feedback policy. For additive noise and a deterministic constant action, the terminal mean and variance can be calculated separately. This makes a transparent benchmark: the control shifts the mean, the diffusion contributes an irreducible variance within that subclass, and an effort penalty discourages overly large shifts.

For an adapted action that changes with the observed noise, the integral of the action and the terminal noise need not be independent. Reusing the constant-control variance formula for every feedback policy would therefore be wrong. The stochastic integral remains nonanticipating, but dependence is precisely what allows a feedback controller to respond to disturbances. Later lessons derive dynamic programming equations that account for that response rather than assuming it away in a static mean calculation.