# Hamilton-Jacobi-Bellman-Isaacs equations in Hilbert spaces with applications in decision making under uncertainty --- Prof. Dr. A. N. Yannacopoulos

**Introduction and fundamental aims**

**Fundamental questions:**

How do we make decisions in stochastic models under uncertainty concerning the exact stochastic model governing the system?

If the state of the system is distributed in “space” (e.g. as in systems related to spatial economics, resource management or other problems related to the physical world) how can be formulate spatial decision rules which are robust under model uncertainty?

How can the theory of nonlinear PDEs in infinite dimensional spaces interact with stochastic analysis and provide us with a concrete framework for the treatment of such problems?

We will present a general framework which allows us to express spatially dependent decision problems under model uncertainty as an infinite dimensional stochastic differential game.

Using dynamic programming techniques we will show that the value function of the game satisfies a nonlinear PDE on an infinite dimensional Hilbert space, the solution of which if it exists, will provide the value function under various scenarios concerning the initial state of the system.

We will then show existence of a weak type of solutions for this equation (mild solutions) and connect them with the construction of robust optimal controls for the system.

These solutions allow us to obtain important information concerning spatial variability and uncertainty.

link to material: https://www.ifors.org/wp-content/uploads/2020/10/hjbi-hilbert-malaysia-2020.pdf