The Schrödinger Bridge (SB) problem offers a powerful framework for combining optimal transport and diffusion models. A promising recent approach to solve the SB problem is the Iterative Markovian Fitting (IMF) procedure, which alternates between Markovian and reciprocal projections of continuous-time stochastic processes. However, the model built by the IMF procedure has a long inference time due to using many steps of numerical solvers for stochastic differential equations.
To address this limitation, we propose a novel Discrete-time IMF (D-IMF) procedure in which learning of stochastic processes is replaced by learning just a few transition probabilities in discrete time. Its great advantage is that in practice it can be naturally implemented using the Denoising Diffusion GAN (DD-GAN), an already well-established adversarial generative modeling technique. We show that our D-IMF procedure can provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds.

In recent years, generative AI has profoundly impacted various fields through its ability to model large-scale datasets and synthesize new content. Among these innovations, diffusion models offer a robust framework by iteratively refining noise into data, significantly enhancing the generation of high-fidelity and diverse data samples. Despite its empirical success in different domains, the theoretical foundations and systematic design of diffusion models remain largely unexplored. Furthermore, the use of diffusion models to generate dynamic data with complex structures, such as those found in financial systems, is still in an early stage.

In this talk, we will discuss the mathematical foundations of diffusion models from both optimization and generalization perspectives. We will also demonstrate how diffusion models can be applied to generate high-dimensional asset returns, addressing the curse of dimensionality by leveraging structural properties. Finally, we will briefly talk about how to incorporate domain-specific knowledge, such as volatility clustering in financial time series, to enhance the authenticity of generated data.

A general diffusion semimartingale is a 1d continuous semimartingale that is also a regular strong Markov process. The class of general diffusion semimartingales is a natural generalization of the class of (weak) solutions to SDEs. A continuous semimartingale has the representation property if all local martingales w.r.t. its canonical filtration have an integral representation w.r.t. its continuous local martingale part. The representation property is closely related to market completeness. We show that the representation property holds for a general diffusion semimartingale if and only if its scale function is (locally) absolutely continuous in the interior of the state space. Surprisingly, this means that not all general diffusion semimartingales possess the representation property, which is in contrast to the SDE case. Furthermore, it follows that the laws of general diffusion semimartingales with absolutely continuous scale functions are extreme points of their semimartingale problems. We construct a general diffusion semimartingale whose law is not an extreme point of its semimartingale problem. This contributes to the solution of the problems posed by Jacod and Yor and by Stroock and Yor on the extremality of strong Markov solutions (to martingale problems). This is a joint work with David Criens.

Until recently, Mathematics, Physics and Computer Sciences were organized around algebraic structures such as Groups, Rings, and Fields, which are not well-suited to modeling and solving Non-Linear
problems. Tropical Mathematics find their roots in theoretical informatics and computer sciences, especially in operations research, and automation. It is still a productive and rich approach in pure and applied mathematics,
in theoretical and mathematical physics, informatics and engineering
sciences. They are constructed on Idempotent Semi-Rings and Dioids, and permit an extension of Linear methods to Non-Linear problems and provide powerful analyzing and computing tools in Theoretical Physics and Applied Mathematics. This new type of Mathematics is based on original algebraic frameworks which yield to define General Topologies, Analysis, and Functional Analysis. This allows to revisit definitions of Integration, Non-Linear Distributions, Legendre Transforms, and Wavelets Theory for example. Some applications to Partial Differential Equations, Images and Signals Processing, and Intervals Computing, will be exhibited in the seminar in order to illustrate the versatility and efficiency of Tropical Mathematics.

The talk is devoted to the study of time-varying linear stochastic control systems. In order to derive an optimal control law over an infinite time horizon, non-ergodic optimality criteria are introduced, extending the notions of long-run averages. We consider two special examples of linear time-varying control systems. A system with a time-varying diffusion matrix and a system involving discounting in the cost are examined. We also discuss issues related to convergence rates for non-ergodic criteria. The final part of the talk is devoted to the study of asymptotically autonomous LQG control systems. Such systems may arise due to linearization, running optimization algorithms, and estimation and identification tasks under cost uncertainty. We introduce the regret as the performance loss resulted from the employment of a simple nominally optimal time-invariant feedback instead of an optimal time-varying stable control law. The notion of regret is widely used in adaptive control, machine, and reinforcement learning. The reason comes from the fact that a truly optimal control law may not be available for implementation, but it’s still possible to design another control preserving some optimality when the planning or observation horizon increases. Thus, a natural task to estimate the regret seems relevant here. We provide corresponding asymptotic estimates in almost sure sense and on the average.

Signature methods represent a non-parametric way for extracting characteristic features from time series data which is essential in machine learning tasks, mathematical finance and risk assessment.
One focus of this talk lies on the use of signature as universal linear regression basis of certain continuous paths functionals in financial applications. In these applications key quantities that have to be computed efficiently are the expected signature or the characteristic function of the signature of some underlying stochastic process. Surprisingly this can be achieved for generic classes of diffusion processes, called signature SDEs, via techniques from affine and polynomial processes.
In terms of concrete applications we present recent contributions from signature based asset price models for joint VIX and SPX calibration and control problems in stochastic portfolio theory.
The talk is based on several joint works with Guido Gazzani, Janka Möller, Francesca Primavera, Sara-Svaluto Ferro and Josef Teichmann.

Synthetic data has emerged as a popular supplement for risk modelers who have grown frustrated with the limitations of historical data and desire access to a full range of stress scenarios. But this artificially created data merely replicates data patterns and may miss outliers. Presented methodology overcomes these flaws for a specific application to strategic planning through comprehensive forward-looking scenario generation enabling proactive risk mitigation. The model (1) automatically generates thousands of future scenarios including unprecedented ones; (2) links banks’ balance sheet and income statement segments to these scenarios; (3) calculates key performance indicators for each scenario at each point in time; (4) allows to analyze tail risks, their main contributors in banks’ business profiles, and early warning signals leading to them; (5) produces benchmarks across the universe of banks or peer groups.
The model overcomes known problems with the traditional regression analysis for stress testing and limitations of standard Monte Carlo simulations for scenario generation. It uses transparent machine learning (ML) techniques for scenario expansion and jump-diffusion model for scenario generation.

We study an optimal investment control problem for an insurance company having two business branches, life annuity insurance and non-life insurance. The company can invest its surplus into a risk free asset and a risky asset with the price dynamics given by a geometric Brownian motion. The optimization objective is to maximize the survival probability of the total portfolio on the infinite time interval. In the absence of investments, the portfolio surplus is described by a stochastic process involving two-sided jumps and a continuous drift. Downward jumps correspond to the claim sizes and upward jumps are interpreted as random gains that arise at the final moments of the life annuity contracts realizations (i.e. at the moments of death of policyholders) as a result of the release of unspent funds. The drift is determined by the difference between premiums in the non-life insurance contracts and the annuity payments. The solving to the optimization problem that yields the maximal survival probability, as well as the optimal strategy, is related to the classical solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation, if this solution exists. In the considered risk model, HJB includes integral operators of two types: classical Volterra operator and singular Volterra one, but their sum is not a Volterra-type operator. It makes the asymptotic analysis of the solution quite complicated. However, for the case of small jumps (when the jumps have exponential distributions), we obtained asymptotic representations of solutions for both small and large values of the initial surplus. Also, in the case of zero interest rate, we obtain a Lundberg-type bound for the survival probability corresponding to the optimal choice among strategies with constant amount of money invested to a risky asset. It turns out that this bound coincides with the asymptotic representation at infinity for the value function in the case of zero interest.

Discrete time control systems whose dynamics and observations are described by stochastic equations are common in engineering, operations research, health care, and economics. For example, stochastic filtering problems are usually defined via stochastic equations. This talk discusses sufficient conditions for solving such problems by value iterations, when the goal is to minimize expected total costs. The model is usually defined by a pair of stochastic equations and by a one-step cost function. Stochastic equations describe the state and observation processes, and these equations are defined by transition and observation functions. We formulate sufficient conditions on observation, transition, and one-step cost functions for convergence of value-iteration algorithms for problems with finite and infinite horizons. It is well-known that nonlinear and linear filtering problems can be presented as Partially Observable Markov Decision Processes (POMDPs). The paper applies contemporary results on convergence of value iterations for Markov Decision Processes (MDPs) and for POMDPs to filtering problems. It formulates conditions on observation and transition functions which imply weak continuity of the filter. Weak continuity of the filter means weak continuity of transition probabilities between belief states. The sufficient condition on one-step functions is their K-inf-compactness. Weak continuity of the filter and K-inf-compactness of costs imply the existence of optimal policies, validity of optimality equations, and convergence of value iterations. The sufficient conditions for weak continuity of filters described in the talk hold for broad classes of nonlinear filters and for Kalman filters. In a broader sense, this talk describes new methods for studying and optimizing discrete-time nonlinear filtering problems.

Blockchains are characterized by algorithms for synchronizing data between network nodes (consensus), methods of data storage, and the programs executed on them (smart contracts). The talk will examine each of these aspects using selected tasks as examples: fault-tolerant random number generation, trees for provable responses to user queries, and risk of a platform for cryptocurrency-backed lending.

References:

[1] Krasnoselskii, M., Melnikov, G., & Yanovich, Y. (2020). No-dealer: Byzantine fault-tolerant random number generator. In IEEE INFOCOM 2020-IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS).

[2] Krasnoselskii, M., Melnikov, G., & Yanovich, Y. (2021). DisCO: Peer-to-Peer Random Number Generator in Partial Synchronous Systems. In 2021 3rd Conference on Blockchain Research & Applications for Innovative Networks and Services (BRAINS).
[3] Chaleenutthawut, Y., Davydov, V., Evdokimov, M., Kasemsuk, S., Kruglik, S., Melnikov, G., & Yanovich, Y. (2024). Loan Portfolio Dataset From MakerDAO Blockchain Project. IEEE Access.

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 ambigous drift, and the problem of optimal divestment from a coal-fired power plant under transition scenario ambiguity.

In this paper we study optimal investment when the investor can peek some time units into the future, but cannot fully take advantage of this knowledge because of quadratic transaction costs. In the Bachelier setting with exponential utility, we give an explicit solution to this control problem with intrinsically infinite-dimensional memory. This is made possible by solving the dual problem where we make use of the theory of Gaussian Volterra integral equations. Joint work with Peter Bank and Miklos Rasonyi.

In this paper, we provide a general approach to reformulating any continuous-time stochastic Stackelberg differential game under closed-loop strategies as a single-level optimisation problem with target constraints. More precisely, we consider a Stackelberg game in which the leader and the follower can both control the drift and the volatility of a stochastic output process, in order to maximise their respective expected utility. The aim is to characterise the Stackelberg equilibrium when the players adopt "closed-loop strategies", i.e. their decisions are based solely on the historical information of the output process, excluding especially any direct dependence on the underlying driving noise, often unobservable in real-world applications. We first show that, by considering the-second-order-backward stochastic differential equation associated with the continuation utility of the follower as a controlled state variable for the leader, the latter’s unconventional optimisation problem can be reformulated as a more standard stochastic control problem with stochastic target constraints. Thereafter, adapting the methodology developed by Soner and Touzi or Bouchard, Élie, and Imbert, the optimal strategies, as well as the corresponding value of the Stackelberg equilibrium, can be characterised through the solution of a well-specified system of Hamilton-Jacobi-Bellman equations. For a more comprehensive insight, we illustrate our approach through a simple example, facilitating both theoretical and numerical detailed comparisons with the solutions under different information structures studied in the literature. This is a joint work with Camilo Hernández, Nicolás Hernández Santibáñez, and Emma Hubert.

Since the seminal work of Ho and Stoll, later refined by Avellaneda and Stoikov, algorithmic market-making models have evolved to incorporate increasingly realistic features such as trade sizes, complex price dynamics, tiering, externalization, and market impact. These models have been applied to a wide range of assets, from illiquid corporate bonds to highly liquid foreign exchange markets to cryptocurrencies (price-aware AMMs). This talk will provide an overview of the key developments of the past decade, highlighting both theoretical advancements and practical applications. It will then focus on recent applications of these models in the gold market by Barzykin, Bergault and I.

A recent approach to statistical inference is based on the concept of an e-variable: a nonnegative sample statistic whose expected value is at most one if a given null hypothesis is true. This approach has been found to produce strong statistical error bounds and high statistical power and is easily extendible to sequential, or online, settings. E-variables admit a natural interpretation as the payoff of a financial bet. In this talk I will discuss how classical ideas from mathematical finance, in particular the numeraire portfolio, enables an optimality theory for e-variables that significantly generalizes earlier results. Our results also lead to a duality theory which yields the so-called reverse information projection in complete generality. Our work showcases the power of financial methods in a setting where information-theoretic tools have traditionally been preferred. (Joint work with Aaditya Ramdas and Johannes Ruf.)

We investigate the convergence properties of sample-average approximations (SAA) for set-valued systemic risk measures. We assume that the systemic risk measure is defined using a general aggregation function with some continuity properties and value-at-risk applied as a monetary risk measure. Our focus is on the theoretical convergence of its SAA under Wijsman and Hausdorff topologies for closed sets. After building the general theory, we provide an in-depth study of an important special case where the aggregation function is defined based on the Eisenberg-Noe network model. In this case, we provide mixed-integer programming formulations for calculating the SAA sets via their weighted-sum and norm-minimizing scalarizations. To demonstrate the applicability of our findings, we conduct a comprehensive sensitivity analysis by generating a financial network based on the preferential attachment model and modeling the economic disruptions via a Pareto distribution.

Several models of partially observed diffusion processes depending on unknown parameters are presented. The proposed algorithms of adaptive Kalman-Bucy filters and parameter estimators have recurrent structure and the questions of their asymptotic optimality are discussed. The properties of the filters and estimators are studied in the asymptotics of small noise. For some nonlinear partially observed systems the construction and properties of the corresponding extended adaptive Kalman filters are discussed too.

For algorithms of machine learning, rate estimates are often provided in the Wasserstein metric or some variant thereof. There has been spectacular recent progress in the techniques for establishing such estimates.
At the same time, powerful methods have been developed in Malliavin calculus that enable to infer total variation convergence from weak convergence.
After presenting an overview of the developments above, we show some new results on total variation convergence that do not rely on Malliavin calculus nevertheless they are applicable to various stochastic systems.

We develop a continuous time framework for sequential goals-based wealth management.
A stochastic factor process drives asset price dynamics as well as the client’s goal amount and income. We prove the weak dynamic programming principle for the value function of our control problem, which we show to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We develop an equivalent and computationally efficient representation of the Hamiltonian, which yields the optimal portfolio within a factor-dependent opportunity set defined by the maximum and minimum variance hypersurfaces. Our analysis shows that it is optimal to fund an expiring goal up to the level where the marginal benefit of additional fundedness is exceeded by the opportunity cost of diverting wealth from future goals. An all-or-nothing investor is more risk averse towards an approaching goal deadline if she is well funded, but more risk seeking if she is not on track with upcoming goals, compared to an investor with flexible goals.