Firms shall be able to respond to their competitors’ strategies over time. Back and Paulsen (2009) thus advocate using closed-loop equilibria to analyze classic real-option exercise games but point out difficulties in defining closed-loop equilibria and characterizing the solution. We define closed-loop equilibria and derive a continuum of them in closed form. These equilibria feature either linear or nonlinear investment thresholds. In all closed-loop equilibria, firms invest faster than in the open-loop equilibrium of Grenadier (2002). We confirm Back and Paulsen (2009)’s conjecture that their closedloop equilibrium (with a perfectly competitive outcome) is the one with the fastest investment and in all other closed-loop equilibria firms earn strictly positive profits. This work is jointly with Zhaoli Jiang and Neng Wang. 

In this talk, we study a quadratic hedging problem in an affine Heston model with self-exiting jumps of Hawkes type. The hedging problem is set up for a variance swap and the strategies we consider are of semi-static type, that is, they consist of a dynamic part, based on the stock and continuously re-balanced, and of a static part, that is buy-and-hold positions in a given basket of European options. Semi-static strategies have the advantage that they reduce the hedging error in comparison to purely dynamic strategies. The model we present is new and combines features of continuous stochastic volatility models and of models with self-exciting jumps in the affine framework. Our results are based on Fourier methods and therefore the affine structure plays a central role for the set-up of the semi-static variance optimal strategy. In particular, we study the Laplace transform of our model and obtain semi-explicit expressions for the hedging strategy. This is a joint work with Giorgia Callegaro, Beatrice Ongarato and Carlo Sgarrra.

We propose a novel distributionally robust Q-learning algorithm for the non-tabular case accounting for continuous state spaces where the state transition of the underlying Markov decision process is subject to model uncertainty. The uncertainty is taken into account by considering the worst-case transition from a ball around a reference probability measure. To determine the optimal policy under the worst-case state transition, we solve the associated non-linear Bellman equation by dualising and regularising the Bellman operator with the Sinkhorn distance, which is then parameterized with deep neural networks. This approach allows us to modify the Deep Q-Network algorithm to optimise for the worst case state transition. We illustrate the tractability and effectiveness of our approach through several applications, including a portfolio optimisation task based on S&P 500 data
(This is joint work with Chung I Lu and Aijia Zhang)

This talk is devoted to the stochastic optimal control problem of infinite-dimensional differential systems allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases studied by Bayraktar and Keller [J. Funct. Anal. 275 (2018) 2096–2161], the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.

The Wasserstein distance Wp is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein distance AWp extends this theory to laws of discrete time stochastic processes in their natural filtrations, making it particularly well suited for analyzing time-dependent stochastic optimization problems. While the topological differences between AWp and Wp are well understood, their differences as metrics remain largely unexplored beyond the trivial bound Wp ≲ AWp. This paper closes this gap by providing upper bounds of AWp in terms of Wp through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of Wp, Eder’s modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on Wp automatically hold for AWp under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality AW1 ≤C√W1 on the set of measures that have Lipschitz kernels. Our work also reveals how smoothing of measures affects the adapted weak topology. In fact, we find that the topology induced by the smooth adapted Wasserstein distance exhibits a non-trivial interpolation property, which we characterize explicitly: it lies in between the adapted weak topology and the weak topology, and the inclusion is governed by the decay of the smoothing parameter. This talk is based on joint work with Jose Blanchet, Martin Larsson and Jonghwa Park.

Pension products and long-term insurance policies play a crucial role in our societies. This talk explores approaches for their cost-effective production through investments in financial markets. The key tool here is to link financial and insurance strategies to an appropriate fundamental theorem. To address the risks and uncertainties inherent in such investments, we draw on methods from financial mathematics and the framework of Knightian uncertainty. We will discuss recent developments in this field, highlighting their implications for the sustainable and resilient structuring of pension and insurance products.

A zero coupon bond is a contract where one party offers a fixed payment at a pre-specified

time point which is called its maturity. A forward rate curve is a theoretical function that encodes the

prices of all possible bonds with varying maturities at one given point of time. There are various models

that explain the behaviour of forward rate curves accross time. The most principle model in this direction

is the Heath Jarrow Morton (HJM)-model which models the forward rate curve directly. This model is

known to be free of arbitrage if and only if the HJM-drift condition holds.

We are interested in finite dimensional HJM-models which stay on one fixed given finite dimensional

manifold, roughly spoken this means that the model stays within a fixed finitely parametrised family

of curves. It is well known, that a curve valued process can only stay on a prescribed manifold if the

Stratonovich drift is tangential to the manifold at all time, or more simply, if we can instead find a

parameter process which selects the curve seen at a given time. From a statistical point of view it would

be desirable to leave the diffusion coefficient of the parameter process open for estimation, or in the

language of manifolds that means that any tangential diffusion coefficient should be left open as possible.

In this presentation, we find those finite dimensional manifolds where the diffusion coefficient remains

fully open for estimation while still allowing for the HJM-drift condition to be met. It turns out that the

resulting manifolds are nowhere locally affine. More so, they are nowhere affinely foliated as has been

suggested by earlier work (however under different assumptions).

We study mean-risk optimal portfolio problems where risk is measured by Recovery Average Value at Risk, a prominent example in the class of recovery risk measures. We establish existence results in the situation where the joint distribution of portfolio assets is known as well as in the situation where it is uncertain and only assumed to belong to a set of mixtures of benchmark distributions (mixture uncertainty) or to a cloud around a benchmark distribution (box uncertainty). The comparison with the classical Average Value at Risk shows that portfolio selection under its recovery version allows financial institutions to better control the recovery of liabilities while still allowing for tractable computations. The talk is based on joint work with Cosimo Munari, Justin Plückebaum and Lutz Wilhelmy.

Nonlinear Lévy processes, as established within the general framework by A. Neufeld and M. Nutz, offer a versatile foundation without restrictions on the characteristic triplets. Building on this foundational work, we focus specifically on G-Lévy processes, a concept introduced by S. Peng. Adopting Peng's approach, we construct the Itô integral with respect to G-Lévy processes and examine its associated properties. Alongside, we delve into results concerning the uniqueness of fully nonlinear integro-partial differential equations and briefly discuss the technical challenges.

We propose a continuous-time equilibrium model with a representative agent that is subject to stochastically fluctuating sentiments. Sentiments dynamically respond to past price movements and exhibit jumps, which occur more frequently when sentiments are disconnected from underlying fundamentals. We model feedback effects between asset prices and sentiment in both directions. Our analysis shows that in equilibrium, sentiments affect prices even though they have no direct impact on the asset’s fundamentals. Empirically, the equilibrium risk premia and risk-free rate respond to measurable shifts in sentiment in the direction predicted by the model.