A probabilistic toolkit to study regularity of free boundaries in stochastic optimal control

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Imagine a spaceship that travels towards a planet and must reach it by a given date. After the launch, and in absence of further intervention, the change in relative position of the spaceship and the planet cannot be predicted with 100% accuracy. The trajectory must be constantly monitored and unforeseen variations must be accounted for, in order to reach the target. However, several constraints must be considered: for example availability of fuel and effectiveness of intervention (if the spaceship is too far off the target, late interventions will not bring it back to the desired trajectory). The question is therefore how to strike the right balance between costs and benefits in order to control the trajectory of the spaceship in the optimal way.This problem was formulated as one of "stochastic control" in the 60's by J. Bather and H. Chernoff. A quick search in the NASA Technical Reports Server shows that "stochastic control theory" is at the core of aerospace engineering (6,843 matching records). Interestingly this exciting branch of mathematics finds applications in many other real-world problems including physics, biology, energy and economics.To give an oversimplified idea of what a solution may look like in the problem above, we could say that it is optimal to make a "small" adjustment to the spaceship's trajectory each time that the offset between spaceship and planet exceeds a value that depends on the available fuel and on the time elapsed from launch. This critical value is called the "free boundary", in mathematics, and it is the key unknown quantity in most stochastic control problems. In practice, the shape and smoothness of the free boundary (as a function of time and fuel, in the example), are needed to enable efficient (numerical) evaluation of the spaceship's optimal trajectory (e.g. leading to minimal use of fuel). Associated to each control problem there is indeed a "cost function" which measures the performance of the control strategy. In the example this may be taken as proportional to the distance of the spaceship from the target, plus the cost of using fuel. The aim of the controller is to minimise the expected value of such cost. When the optimal control is implemented the resulting expected cost is called "value function". The value function is the other main unknown object in this context and, along with the free boundary, their study goes under the name of "free boundary problem" (FBP).FBPs are addressed in Analysis and in Probability theory. However it is often impossible to find a full analogy between results in the two fields. On the one hand, Probability can only explain very limited smoothness of the free boundary and of the value function, but is flexible in modelling randomness in the system. On the other hand, Analysis obtains fine regularity results but mostly under rather inflexible assumptions on the model. In our example above, PDE theory gives very accurate optimal controls if the spaceship's trajectory is described by a simple model. However, in practice engineers must deal with a wide class of random perturbations and need a versatile probabilistic approach. The latter must be supported by a refined probabilistic understanding of FBPs, which is the objective of this proposal. In this project I will develop a new framework for the study of free boundary problems that will hinge on properties of random noises drawn from the class of diffusion processes. My work will provide advanced tools that not only will unify the probabilistic and analytic approach to stochastic control but, more importantly, will remove some of the long-standing assumptions in both areas and allow for tractable solutions to a whole new class of applied problems. Moreover, I will obtain ODEs to accurately compute optimal strategies in multi-dimensional settings (see 2.5 in Case for Support). Non-linear integral equations, currently used, cannot be computed efficiently in dimension higher than two.