Mean field type control problems lead to Hamilton–Jacobi–Bellman (HJB) equations on the space of probability measures. In this talk, we will present a probabilistic approach to constructing classical solutions through forward-backward stochastic differential equations (FBSDEs), allowing for degenerate and unbounded diffusion coefficients. Starting with linear dynamics, we will explain how convexity ensures the global well-posedness of the associated FBSDE system, and how variational equations establish the regularity of the value function needed for the HJB equation. We will then extend the framework to jump-diffusion models, where the associated HJB equation is an integro-partial differential equation. Finally, we will consider the case where a backward equation forms part of the state.
Mathematical Finance Seminar
Date
Time
16:15
Location:
RUD 25; 1.115
Ziyu Huang (HU Berlin)
Classical Solutions to HJB Equations for Mean Field Type Control Problems: An FBSDE Approach
Mathematical Finance Seminar
Date
Time
16:15
Location:
HUB; RUD 25; 1.115
Frederico Cannerozzi (Bielefeld)
Singular Mean-Field Control via singular Mean-Field Games
Mean-field control (MFC) and mean-field games (MFGs) provide two different descriptions of large interacting populations: in MFC, a central planner optimizes the collective performance of the system, while in MFGs each representative player optimizes individually against a given population distribution, which is then determined through a consistency condition. In general, the corresponding optimal controls and equilibria need not coincide. A much tighter connection is known for potential mean-field games: an MFG equilibrium can be obtained by solving an associated auxiliary mean-field control problem. This naturally suggests the reverse question: can a mean-field control problem be solved by means of an auxiliary potential mean-field game? We discuss this approach in two settings of mean-field control with singular controls, where the associated potential MFG can provide a tractable route to the original MFC problem. In a stationary Ornstein-Uhlenbeck model with two-sided singular controls, we obtain a two-way correspondence between the MFC problem and an associated potential MFG. In a finite-horizon framework with general dependence on the state distribution, we show that equilibria of the associated potential MFG yield optimal controls for the MFC problem, and illustrate the approach through a mean-field monotone follower problem.
Workshop/Conference
Date
Time
9:00
Location:
TU Berlin; IMOS Building
Francois Delarue, Huyen Pham, Xunyu Zhou et al
CRC Day "Stochastic Control and Data Driven Learning"
Mathematical Finance Seminar
Date
Time
16:15
Location:
HUB; RUD 25; 1.115
Philipp Schmocker (ETH)
tba
Mathematical Finance Seminar
Date
Time
17:15
Location:
HUB; RUD 25; 1.115
Christoph Czichowsky (LSE)
tba
Mathematical Finance Seminar
Date
Time
17:15
Location:
HUB; RUD 25; 1.115
Federica Masiero (Bicocca)
tba
Mathematical Finance Seminar
Date
Time
17:15
Location:
HUB; RUD 25; 1.115
Purba Das (King's College)
tba
Mathematical Finance Seminar
Date
Time
16:15
Location:
HUB; RUD 25; 1.115
Giulia Liveri (LSE)
tba
Mathematical Finance Seminar
Date
Time
17:15
Location:
HUB; RUD 25; 1.115
Giuseppina Guatteri (Milano)