Conference Agenda

Optimal control problems involving Markovian stochastic differential equations have been extensively studied in the research literature; however, many real-world applications necessitate the consideration of path-dependent non-Markovian dynamics. In this talk, we consider an optimal control problem of (path-dependent) stochastic differential equations with delays in the state. To use the dynamic programming approach, we regain Markovianity by lifting the problem on a suitable Hilbert space. We characterize the value function $V$ of the problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully non-linear second-order partial differential equation on a Hilbert space with an unbounded operator. Since no regularity results are available for viscosity solutions of these kinds of HJB equations, via a new finite-dimensional reduction procedure and using the regularity theory for finite-dimensional PDEs, we prove partial $C^{1,\alpha}$-regularity of $V$. When the diffusion is independent of the control, this regularity result allows us to define a candidate optimal feedback control. However, due to the lack of $C^2$-regularity of $V$, we cannot prove a verification theorem using standard techniques based on Ito’s formula. Thus, using a technical double approximation procedure, we construct functions approximating $V$, which are supersolutions of perturbed HJB equations and regular enough to satisfy a non-smooth Ito’s formula. This allows us to prove a verification theorem and construct optimal feedback controls. We provide applications to optimal advertising and portfolio optimization. We discuss how these results extend to the case of delays in the control variable (also) and discuss connections with new results of $C^{1,1}$-regularity of the value function and optimal synthesis for optimal control problems of stochastic differential equations on Hilbert spaces via viscosity solutions.

The talk is based on the following manuscripts:

F. de Feo, S. Federico, A. Święch, "Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic models", SIAM J. Control Optim. 62 (2024), no. 3, 1490–1520.

F. de Feo, A. Święch, "Optimal control of stochastic delay differential equations: Optimal feedback controls", arXiv preprint arXiv:2309.05029 (2024).

F. de Feo, "Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and applications to optimal advertising and optimal investment problems", Decis. Econ. Finance (2024) 31 pp.

F. de Feo, A. Święch, L. Wessels, "Stochastic optimal control in Hilbert spaces: $C^{1,1}$-regularity of the value function and optimal synthesis via viscosity solutions", arXiv preprint, arXiv:2310.03181 (2024).

Focusing on gains & losses relative to a risk-free benchmark instead of terminal wealth, we consider an asset allocation problem to maximize time-consistently a mean-risk reward function with a general risk measure which is i) law-invariant, ii) cash- or shift-invariant, and iii) positively homogeneous, and possibly plugged into a general function. Examples include (relative) Value at Risk, coherent risk measures, variance, and generalized deviation risk measures. We model the market via a generalized version of the multi-dimensional Black-Scholes model using $\alpha$-stable Lévy processes and give supplementary results for the classical Black-Scholes model. The optimal solution to this problem is a Nash subgame equilibrium given by the solution of an extended Hamilton-Jacobi-Bellman equation. Moreover, we show that the optimal solution is deterministic under appropriate assumptions.

We consider an investment problem with incomplete information where an irreversible investment yields a flow of operating profits. These profits are determined by a geometric Brownian motion with unknown drift. Mathematically, this is an optimal stopping problem with incomplete information where the payoff function depends directly on the unknown parameter. We transform the problem into a two-dimensional optimal stopping problem with complete information. Thereby, we find that the optimal stopping time is a threshold time with respect to the underlying two-dimensional process. Then, we identify the optimal stopping set with a boundary function mapping the current posterior belief onto the minimal stopping trigger for the state of the geometric Brownian motion. Further, we use an elementary probabilistic way to show regularity and monotonicity of the value function and the boundary function. We finally derive a nonlinear integral equation for the boundary function.

Heat energy management plays an important role in the ongoing efforts to reduce carbon emissions. Due to the widespread use of solar collectors, many households are no longer just consumers of thermal energy but also producers (short: prosumers). A major challenge such prosumers face is the efficient use of the collected heat energy. Help is provided by technologies like long-term geothermal storage solutions and district heating networks. However, these more advanced systems raise questions about their optimal operation, for example, what amount of excess production should be stored long-term/sold vs. what amount should be kept on hand? This presentation aims to answer some of these questions by modeling prosumer heat storage with a Markov decision process, by deriving the structure of the optimal policy and by computing it via learning algorithms.

The talk is based on joint work with Nicole Bäuerle and funded by the Bundesministerium für Bildung und Forschung (BMBF).