# Stochastics and Finance Workshop

Dr. Ari-Pekka Perkkiö, Dr. Alexander Kalinin and Georg Bollweg

Dr. Ari-Pekka Perkkiö, Dr. Alexander Kalinin and Georg Bollweg

**Date and Time**

- Friday, 8 November 2024

**Venue**

- Mathematical Insitut LMU, Theresienstraße 39, Room B349

Time | Speaker | Title |
---|---|---|

12:00 - 12:45 | Kostas Kardaras | Portfolio choice with watermark fees and market time constraint |

12:45 - 14:15 | Lunch Break | |

14:15 - 15:00 | Andrea Mazzon | Optimal stopping and divestment timing under scenario ambiguity and learning |

15:00 - 15:45 | Teemu Pennanen | TBA |

15:45 - 16:15 | Coffee Break | |

16:15 - 17:00 | Caroline Geiersbach | Optimality Conditions with Probabilistic State Constraints |

17:00 - 17:45 | Antoine Jacquier | TBA |

19:30 | Workshop Dinner |

Speaker | University |
---|---|

Kostas Kardaras | London School of Economics |

Andrea Mazzon | University of Verona |

Teemu Pennanen | King's College London |

Caroline Geiersbach | University of Hamburg |

Antoine Jacquier | Imperial College London |

We consider the problem of choosing an investment strategy that will maximise utility over distributions, under watermark fees (or, alternatively capital gains tax) and constraints on the expected liquidation date. We show that the problem can be decomposed in two separate ones. The first involves choosing an optimal target distribution, while the second involves optimally realising this distribution via an investment strategy and stopping time. The latter step may be regarded as a variant of the Skorokhod embedding problem. A solution is given very precisely in terms of the first time that the wealth of the growth optimal portfolio, properly reduced after watermark fees, crosses a moving stochastic (depending on its minimum-to-date) level. The suggested solution has the additional optimality property of stochastically minimising maximal losses over the investment period.

Aiming to analyze the impact of environmental transition on the value of assets and on asset stranding, we study optimal stopping and divestment timing decisions for an economic agent whose future revenues depend on the realization of a scenario from a given set of possible futures.

Since the future scenario is unknown and the probabilities of individual prospective scenarios are ambiguous, we adopt the smooth model of decision making under ambiguity aversion of Klibanoff et al (2005), framing the optimal divestment decision as an optimal stopping problem with learning under ambiguity aversion. We then prove a minimax result reducing this problem to a series of standard optimal stopping problems with learning. The theory is illustrated with two examples: the problem of optimally selling a stock with ambiguous drift, and the problem of optimal divestment from a coal-fired power plant under transition scenario ambiguity.

*TBA*

In this talk, we discuss optimization problems subject to random state constraints, where we distinguish between the chance-constrained case and the almost sure formulation. We highlight some of the difficulties in the infinite-dimensional setting, which is of interest in physics-based models where a control belonging to a Banach space acts on a system described by a partial differential equation (PDE) with random inputs or parameters. We study the setting in which the obtained state should be bounded uniformly over the physical domain with high probability, or even probability one. We apply our results to a model with a random elliptic PDE, where the randomness is induced by the right-hand side. For the chance-constrained setting, this structure allows us to obtain an explicit representation for the Clarke subdifferential of the probability function using the spherical radial decomposition of Gaussian random vectors. This representation is used for the numerical solution in a discretize-then-optimize approach. For the almost sure setting, we use a Moreau-Yosida regularization and solve a sequence of regularized problems in an optimize-then-discretize approach. The solutions are compared, providing insights for the development of further algorithms.

TBA