Global Seminar

Optimal investment and consumption for spread financial markets

In this talk we develop stochastic optimal control methods for spread financial markets defined by the Ornstein–Uhlenbeck (OU) processes. To this end, we study the Hamilton–Jacobi–Bellman (HJB) equation using the Feynman–Kac (FK) probability representation. We show an existence and uniqueness theorem for the classical solution of the HJB equation which is a quasi-linear partial derivative equation of parabolic type. Then we show a special verification theorem and, as a consequence, construct optimal consumption/investment strategies for power utility functions. Moreover, using fixed point tools we study the numeric approximation for the HJB solution and we establish the convergence rate which, as it turns out in this case, is super geometric, i.e., more rapid than any geometric one.

Reflected BSDEs arising from pricing and hedging in models under random horizon

We consider the model given by the quadruplet: complete probability space, a filtration F defined on this space and representing the flow of information available to all agents throughout time, and an arbitrary random time. This random time, which might not be observable via F, represents default time of a firm in credit risk and death time of an insured in life insurance where mortality and/or longevity risks pose serious challenges. Thus, the flow G that incorporates all the information, whether it is public or not, follows from the progressive enlargement of F with τ. In this setting, our analysis yields the family of reflected backward stochastic differential equations (RBSDE). We focus on answering the following questions:

1. 1. What are the sufficient minimal conditions on the data that guarantee the existence of the solution of the G-RBSDE in Lp (p > 1)?




2. 2. How this RBSDE can be explicitly connected to an RBSDE in F?




3. 3. How can we estimate –in norm– the solution in terms of the data-triplet? What are the adequate norms and adequate spaces for both the solution and the data-triplet?

In my talk, I will answer all these questions deeply and beyond. Importantly, I will prove that for any random time, having a positive Azéma supermartingale, there exists a positive discount factor.

Mean field stochastic control under sublinear expectation

The talk is devoted to the study of Pontryagin's stochastic maximum principle for a mean-field optimal control problem under Peng's sublinear G-expectation. The dynamics of the controlled state process is given by a SDE driven by a G-Brownian motion, whose coefficients depend on the control, the controlled state process but also on its law under the G-expectation. Also the associated cost functional is of mean-field type. Under the assumption of a convex control state space we study Pontryagin's stochastic maximum principle, which gives a necessary optimality condition for control process. Under additional convexity assumptions on the Hamiltonian it is shown that this necessary condition is also a sufficient one. The main difficulty which we had to overcome in our work consists in the differentiation of the G-expectation of parametrised random variables. As particularly delicate turns out to handle with the G-expectation of a function of the controlled state process inside the running cost of the cost functional. For this we had to study a measurable selection theorem for set-valued functions whose values are subsets of the representing set of probability measures for the G-expectation.

The talk is based on a recent joint work with Juan Li and Bowen He (Shandong University, Weihai, China).

Rogue Traders

Investing on behalf of a firm, a trader can feign personal skill by committing fraud that with high probability remains undetected and generates small gains, but that with low probability bankrupts the firm, offsetting ostensible gains. Honesty requires enough skin in the game: if two traders with isoelastic preferences operate in continuous-time and one of them is honest, the other is honest as long as the respective fraction of capital is above an endogenous fraud threshold that depends on the trader’s preferences and skill. If both traders can cheat, they reach a Nash equilibrium in which the fraud threshold of each of them is lower than if the other one were honest. More skill, higher risk aversion, longer horizons, and greater volatility all lead to honesty on a wider range of capital allocations between the traders.

Neural Optimal Transport

Solving optimal transport (OT) problems with neural networks has become widespread in machine learning. The majority of existing methods compute the OT cost and use it as the loss function to update the generator in generative models (Wasserstein GANs). In this presentation, I will discuss the absolutely different and recently appeared direction - methods to compute the OT plan (map) and use it as the generative model itself. Recent advances in this field demonstrate that they provide comparable performance to WGANs. At the same time, these methods have a wide range of superior theoretical and practical properties.

The presentation will be mainly based on our recent paper "Neural Optimal Transport" (ICLR, 2023). I am going to present a neural algorithm to compute OT plans (maps) for weak & strong transport costs. For this, I will discuss important theoretical properties of the duality of OT problems that make it possible to develop efficient practical learning algorithms. Besides, I will prove that neural networks actually can approximate transport maps between probability distributions arbitrarily well. Practically, I will demonstrate the performance of the algorithm on the problems of unpaired image-to-image style transfer and image super-resolution.

Related Work:
Neural Optimal Transport (ICLR, 2023)
Unpaired Image Super-Resolution with Optimal Transport Maps (preprint)
Generative Modeling with Optimal Transport Maps (ICLR, 2022)
Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark (NeurIPS, 2021) 

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