Abstracts

Álvaro Cartea University of Oxford

TBA

Alexander Schied University of Waterloo

Exploring Roughness in Stochastic Processes: From Weierstrass Bridges to Volatility Estimation

Motivated by the recent success of rough volatility models, we introduce the notion of a roughness exponent to quantify the roughness of trajectories. It can be computed in a straightforward manner for many stochastic processes and fractal functions. It also inspired the introduction of a new class of stochastic processes, the so-called Weierstrass bridges. After taking a look at Weierstrass bridges and their sample path properties, we discuss the relations between the roughness exponent and other roughness measures. We show furthermore that the roughness exponent can be statistically estimated in a model-free manner from direct observations of a trajectory but also from discrete observations of an antiderivative — a situation that corresponds to estimating the roughness of volatility from observations of the integrated variance, which in turn can be approximated by the realized variance. This is joint work with Xiyue Han and Zhenyuan Zhang.

Antoine Debouchage Paris Dauphine University

Transfer learning of entropic optimal transport

In many applications of (entropic) optimal transport, the transportation cost is unknown and cannot be specified a priori. In this talk, we present a strategy to address this challenge in a particular setting. We aim to transport a distribution \(\mu\) to \(\nu\), and we assume access to samples from a coupling between \(\mu'\) and \(\nu'\) that is optimal for the same unknown cost. Leveraging flow matching and structural properties of entropic optimal transport, we develop an algorithm that transfers this information to learn the corresponding Schrödinger bridge between \(\mu\) and \(\nu\).

Ariel Neufeld Nanyang Technological University

Deep Learning algorithm for solving high-dimensional nonlinear PIDEs in finance

We present a (random) neural networks based algorithm which can efficiently solve high-dimensional nonlinear partial integro-differential equations (PIDEs) and apply this algorithm to price high-dimensional financial derivatives under default risk. We provide a full error analysis of our algorithm as well as empirically demonstrate that our algorithm can approximately solve nonlinear PIDEs in 10'000 dimensions within seconds.

Christian Bayer Weierstrass Institute Berlin

Global and local regression: a signature approach with applications

The path signature is a powerful tool for solving regression problems on path space, i.e., for computing conditional expectations \(\mathbb{E}[Y | X]\) when the random variable \(X\) is a stochastic process — or a time-series. We provide new theoretical convergence guarantees for two different, complementary approaches to regression using signature methods. In the context of global regression, we show that linear functionals of the robust signature are universal in the \(L^p\) sense in a wide class of examples. In addition, we present a local regression method based on signature semi-metrics, and show universality as well as rates of convergence. Based on joint works with Davit Gogolashvili, Luca Pelizzari, and John Schoenmakers.

Chen Yang The Chinese University of Hong Kong

Arbitrage on Decentralized Exchanges

Decentralized exchanges using automated market makers create arbitrage opportunities with centralized exchanges, where gas fees and transaction ordering are critical. Existing models largely overlook competition among arbitrageurs, despite price discrepancies being public information. We develop the first equilibrium model of gas fee competition between two arbitrageurs under three transaction reversion settings: no-revert, auto-revert, and selectable-revert. We show that pure symmetric equilibria do not exist, but unique mixed equilibria can be characterized. Comparative analysis reveals that under low inventory risk, the no-revert setting favors arbitrageurs in terms of profit, while auto-revert and selectable-revert settings enhance market efficiency. Under high inventory risk, the no-revert and selectable-revert settings dominate the auto-revert setting in both profitability and efficiency. Using data from Binance and Uniswap V2, we empirically confirm that arbitrageurs face positive inventory risk and validate our model's implications: gas fees increase with price discrepancies and liquidity, while trading amounts rise with both price discrepancies and gas fees.

Chenghu Ma Fudan University

Disentangling the quantitative measure of risk and uncertainty

This paper employs the framework of location-scale (LS) family with i.i.d. Gaussian seeds to measure Knightian uncertainty in asset returns. We first prove impossibility theorems showing that no single measure is universally agreeable among investors with general risk-and-uncertainty-averse preferences. Nevertheless, we prove that investors can always rank uncertainty embedded in any two return processes by examining their mean–average characteristic functions. Under a rectangular LS-range specification, we show that uncertainty—measured by the ranges of LS coefficients—can be fully identified from the first, second, fourth, and sixth moments of returns. We further propose a composite uncertainty measure aligned with Shannon's (1948) relative entropy. Applying these measures to daily and high-frequency returns of U.S. and Chinese stock indices, we find that uncertainty is persistent and does not collapse to risk. These results underscore the importance of separating uncertainty from risk in asset pricing and portfolio management.

Christoph Czichowsky London School of Economics

TBA

Damir Filipović EPFL

TBA

Dirk Becherer Humboldt University Berlin

Limiting Mean-Field Games and Structural Decomposition of Equilibria for Portfolio Games of Optimal Hedging

We present new results on mean-field games of optimal investment and hedging with relative performance concerns under CARA preferences. Equilibria are fully characterized by McKean-Vlasov BSDEs which are well-posed. Our proof yields a constructive description of the mean-field equilibrium: one first solves a classical single-agent control problem, then a linear projection on the common-noise filtration. This relies on our structural decomposition, revealing how interaction, investment, and hedging combine in equilibrium. We further identify two limiting mean-field games arising as absolute risk aversion tends to zero and to infinity, yielding novel types of portfolio games. (Partially based on arXiv:2408.01175.)

Emma Hubert Paris Dauphine University

Revisiting contract theory with volatility control

In this talk, we revisit the resolution of continuous-time principal–agent problems with drift and volatility control, originally addressed by Cvitanic, Possamai, and Touzi (2018) through dynamic programming and second-order backward stochastic differential equations (2BSDEs), and develop new results in this framework. We begin by introducing an alternative problem in which the principal is allowed to directly control the quadratic variation of the output process. On the one hand, the resolution of this contractible-volatility problem follows the classical methodology of Sannikov (2008), thus relying on standard (first-order) BSDEs only. On the other hand, we introduce a new form of contracts allowing the principal to achieve her contractible-volatility value, thereby ensuring both the optimality of this contract form and the equivalence between the original and the alternative problems. At the same time, this alternative approach reveals that the optimality of the original contract form introduced in CPT implicitly relies on an additional duality assumption, which was not identified before. This observation motivates the construction of new families of contracts that remain optimal even when the duality assumption fails. Altogether, this line of work both simplifies and strengthens the existing theory of continuous-time principal–agent problems with volatility control, and opens new directions for further extensions and applications in economics and finance. Talk based on joint works with Alessandro Chiusolo, Dylan Possamaï, and Nizar Touzi.

Gu Wang Worcester Polytechnic Institute

Continuous Policy and Value Iteration for Stochastic Control Problem and Its Convergence

We propose a policy iteration algorithm, in which the approximations of value function and the control are updated simultaneously, for both the entropy regularized relaxed control problem and the classical control problem, with infinite horizon. We show the policy improvement and the convergence to the optimal control. Since both the value function and the control are updated according to differential equations in a continuous manner, we also confirm the convergence rate of the proposed algorithm.

Hans Buehler Various

TBA

Harry Zheng Imperial College London

Reinforcement Learning for Speculative Trading under Exploratory Framework

We study a speculative trading problem within the exploratory reinforcement learning (RL) framework of Wang et al. (2020). The problem is formulated as a sequential optimal stopping problem over entry and exit times under general utility function and price process. We first consider a relaxed version of the problem in which the stopping times are modelled by the jump times of Cox processes driven by bounded, non-randomized intensity controls. Under the exploratory formulation, the agent's randomized control is characterized via the probability measure over the jump intensities, and their objective function is regularized by Shannon's differential entropy. This yields a system of the exploratory HJB equations and a Gibbs-Boltzmann distribution in closed-form as the optimal policy. Error estimates and convergence of the RL objective to the value function of the original problem are established. Finally, an RL algorithm is designed, and its implementation is showcased in a pairs-trading application. (Joint work with Alex Tse and Yun Zhao)

Hélène Halconruy Télécom SudParis

Local transfer learning for nonparametric regression: An application to stock prediction

Transfer learning can improve prediction by leveraging related source tasks, but it may also suffer from negative transfer when similarities vary across the covariate space. In this work, we propose a nonparametric framework based on a local transfer assumption, where the target function is approximated locally by a simple transformation of the source. This allows similarity to vary across the space, encouraging useful transfer while avoiding harmful information sharing. In this talk, I will present the main theoretical results, including minimax rates showing that transfer can alleviate the curse of dimensionality, as well as fully data-driven procedures to estimate both the transfer and the target functions. I will conclude with an illustration on stock return prediction using signature-based features. This is joint work with Benjamin Bobbia and Paul Lejamtel.

Hoi Ying Wong The Chinese University of Hong Kong

Duality and DeepMartingale for High-Dimensional Optimal Switching: Computable Upper Bounds and Approximation-Expressivity Guarantees

Optimal switching problems (OSP) generalize optimal (multiple) stopping problems by allowing reversible decisions among multiple operational regimes. We study finite-horizon optimal switching with discrete intervention dates under a general filtration, permitting continuous-time observations between decision times. We develop a deep-learning-based dual framework that provides computable upper bounds. A dual representation is derived via a family of martingale penalties; due to reversibility and regime interactions, the duality structure is substantially more intricate than in standard stopping problems. The minimal penalty is characterized by Doob martingales of continuation values, yielding fully computable upper bounds. We extend the DeepMartingale approach from optimal stopping to optimal switching and establish convergence under both the upper-bound loss and a surrogate loss. An expressivity analysis leads to practical scaling guidelines for neural network architectures to mitigate the curse of dimensionality. Combining the dual method with a deep policy-based primal solver produces tight lower and upper bounds in practice. Numerical experiments on Brownian and Brownian–Poisson models demonstrate small primal–dual gaps and strong high-dimensional performance, and the learned dual martingale further induces a practical delta-hedging strategy. (Coauthor: Junyan Ye)

Huyên Pham École Polytechnique

TBA

Jiacheng Zhang The Chinese University of Hong Kong

Major-Minor Mean Field Game of Stopping: An Entropy Regularization Approach

This paper studies a discrete-time major-minor mean field game of stopping where the major player can choose either an optimal control or stopping time. We look for the relaxed equilibrium as a randomized stopping policy, which is formulated as a fixed point of a set-valued mapping, whose existence is challenging by direct arguments. To overcome the difficulties caused by the presence of a major player, we propose to study an auxiliary problem by considering entropy regularization in the major player's problem while formulating the minor players' optimal stopping problems as linear programming over occupation measures. We first show the existence of regularized equilibria as fixed points of some simplified set-valued operator using the Kakutani-Fan-Glicksberg fixed-point theorem. Next, we prove that the regularized equilibrium converges as the regularization parameter \(\lambda\) tends to 0, and the limit corresponds to a fixed point of the original operator, thereby confirming the existence of a relaxed equilibrium in the original mean field game problem. We also extend this entropy regularization method to the mean-field game problem where the minor players choose optimal controls.

Jingtang Ma Southwestern University of Finance and Economics

Optimal Investment Under Non-Markovian Models via BSPDEs and Deep Learning

We study optimal investment (utility maximization problems) in non-Markovian settings, where the dynamic programming principle (DPP) fails and Hamilton-Jacobi-Bellman (HJB) equations are inapplicable. Instead, backward stochastic partial differential equations (BSPDEs) can characterize the random field values of such problems. We propose iterative deep learning algorithms to solve these fully nonlinear BSPDEs and establish their convergence. Numerical experiments on rough volatility models validate the theoretical convergence of our methods.

Joseph Teichman ETH Zurich

TBA

Juan Li Shandong University

Controllability Concepts for Mean-Field Dynamics With Reduced-Rank Coefficients

In this talk, we consider different novel notions of exact controllability of mean-field linear controlled stochastic differential equations (SDEs). The key feature is that the noise coefficient is not required to be of full rank. We begin with showing that classical exact controllability with \(L^2\)-controls necessarily requires both rank conditions on noise. When this condition fails, we turn to similar rank requirements on drift allowing exact controllability by lowering the regularity of controls. When both the aforementioned rank conditions fail, we introduce and characterize a new notion of exact terminal controllability to normal laws (ETCNL). We also investigate a new class of Wasserstein-set-valued backward SDEs naturally associated to ETCNL. Based on a joint work with Dan Goreac (SDU & University of Laval, Canada), Xinru Zhang (SDU).

Julien Claisse Paris Dauphine University

TBA

Kyunghyun Park Nanyang Technological University

Robust Q-learning Algorithm for Markov Decision Processes under Wasserstein Uncertainty

TBA

Lingfei Li The Chinese University of Hong Kong

Deep learning of derivatives pricing functions in jump-diffusion models

We study the pricing of financial derivatives as functions of time, state variables, and parameters in the model and contract. Traditional computational methods evaluate the derivative price only for fixed parameters, which becomes prohibitively time-consuming when a large number of valuations are required. To enhance computational efficiency, we approximate the derivative pricing function by deep neural networks (DNNs), allowing some components of the model to vary as time-dependent functions. We train the DNN via a loss function derived from the martingale property of the discounted derivative price process, providing a unified training framework for different types of derivatives. For European options under jump-diffusion models, we theoretically prove that DNNs can overcome the curse of dimensionality in learning the pricing function. Numerical experiments on European, path-dependent, and Bermudan options confirm the effectiveness of our approach.

Marko Weber National University of Singapore

We study how social comparisons shape asset prices in an incomplete-market economy with uninsurable income risk. Agents are heterogeneous in preferences, endowments, and in the weights they place on the consumption of others. Using a mean-field game framework with a directed network described by a finite-rank kernel, we derive in closed form the equilibrium interest rate, stock price, and individual consumption and portfolio policies. A social multiplier matrix captures how network interactions reallocate risk across agents. Social comparisons reduce the market price of hedgeable risk by raising the effective risk tolerance of the economy, as consumption and social reference benchmarks co-move with common shocks. The network structure governing social comparisons determines whether the precautionary-savings motive induced by unhedgeable risk is amplified or mitigated. If all agents exert the same influence on others and are equally sensitive to the rest of the system, as in a doubly stochastic comparison network, the specific network topology does not affect equilibrium prices.

Mathieu Rosenbaum Paris Dauphine University

A unified theory of order flow, market impact, and volatility

We propose a microstructural model for the order flow in financial markets that distinguishes between core orders and reaction flow, both modeled as Hawkes processes. This model has a natural scaling limit that reconciles a number of salient empirical properties: persistent signed order flow, rough trading volume and volatility, and power-law market impact. In our framework, all these quantities are pinned down by a single statistic \(H_0\), which measures the persistence of the core flow. Specifically, the signed flow converges to the sum of a fractional process with Hurst index \(H_0\) and a martingale, while the limiting traded volume is a rough process with Hurst index \(H_0-1/2\). No-arbitrage constraints imply that volatility is rough, with Hurst parameter \(2H_0-3/2\), and that the price impact of trades follows a power law with exponent \(2-2H_0\). The analysis of signed order flow data yields an estimate \(H_0\) close to \(3/4\). This is not only consistent with the square-root law of market impact, but also turns out to match estimates for the roughness of traded volumes and volatilities remarkably well. This is joint work with Johannes Muhle-Karbe, Youssef Ouazzani-Chahdi and Grégoire Szymanski.

Mehdi Talbi Paris Diderot University

Deep Learning for the Multiple Optimal Stopping Problem

This talk presents a novel deep learning framework for solving multiple optimal stopping problems in high dimensions. While deep learning has recently shown promise for single stopping problems, the multiple exercise case involves complex recursive dependencies that remain challenging. We address this by combining the Dynamic Programming Principle with neural network approximation of the value function. Unlike policy-search methods, our algorithm explicitly learns the value surface. We first consider the discrete-time problem and analyze neural network training error. We then turn to continuous problems and analyze the additional error due to the discretization of the underlying stochastic processes. Numerical experiments on high-dimensional American basket options and nonlinear utility maximization demonstrate that our method provides an efficient and scalable method for the multiple optimal stopping problem. This is a joint work with Mathieu Laurière.

Min Dai The Hong Kong Polytechnic University

Connecting Singular Control and Optimal Stopping in Multi-Asset Portfolio Selection with Transaction Costs

It is well known that certain singular control problems can be reformulated as optimal stopping problems. For portfolio selection with transaction costs and a single risky asset, this equivalence was first established by Dai and Yi (2009). In this talk, we extend the equivalence to the multi-asset case, which allows us to develop efficient reinforcement learning algorithms.

Na Li Dalian University of Technology

Motivated by the multi-scheme supply chain problem, a linear-quadratic generalized Stackelberg game for time-delay is studied, in which the multi-level hierarchy structure with delay is involved. With the help of the continuity method, we first establish the unique solvability of nonlinear anticipated forward–backward stochastic delayed differential equations with a multi-level self-similar domination-monotonicity structure. Based on it, we derive the Stackelberg equilibrium in this framework. By the theoretical results, a corporate social responsibility problem is studied in the view of a multi-scheme supply chain problem, some simulations are also presented to illustrate the Stackelberg equilibrium in a special case.

Nan Chen The Chinese University of Hong Kong

Optimal Bonding Function for Decentralized Exchange Design: A Convex Analysis Approach

Decentralized exchanges rely on automated market makers to provide liquidity passively. This exposes liquidity providers to the risk of impermanent losses caused by both stale price arbitrage and reverse trade arbitrage. In this paper, we combine tools from the calculus of variations and convex optimization to study the optimal design of bonding functions, the core mechanism of AMMs, in the presence of these two forms of arbitrage. We also discuss the economic implications of our findings. This is a joint work with Agostino Capponi (Columbia), Ling Qin (Shanghai Tech University), and Zhou Yang (South China Normal University).

Nan Li Shanghai Jiao Tong University, SAIF

The Training Set Delusion: Machine Learning Overfitting in Bitcoin Return and Volatility Prediction

This paper compares machine learning methods with classical econometrics models in forecasting returns and volatility, using Bitcoin as a testing asset whose price dynamics are driven largely by investor behavior and feature an extremely low signal-to-noise ratio. Under a strict out-of-sample framework, almost no machine learning method can reliably forecast Bitcoin returns or volatility, and the seemingly strong results reported in prior studies appear largely in the training set. Our findings underscore the importance of reporting performance in the training set, validation set and testing set when necessary, and incorporating practically relevant metrics when applying ML methods in asset pricing.

Paolo Guasoni University College Dublin

A Variational Approach to Portfolio Choice

We develop a variational approach to solve consumption-investment problems for infinite-horizon isoelastic investors with any number of assets and stochastic investment opportunities driven by a scalar diffusion. We identify a strictly convex functional whose unique minimizer coincides with the optimal wealth-consumption ratio and yields the optimal portfolio explicitly. We demonstrate the method's effectiveness by computing optimal consumption-investment policies through gradient-descent in models with nonlinear predictability, stochastic volatility, and interest-rate risk. The method combines verification with numerical tractability and yields accurate solutions in analytically intractable models.

Peter Bank TU Berlin

TBA

Peter Friz TU Berlin

TBA

Ping Li Beihang University

A Mean Field Game Approach to Equilibrium Pricing and Hedging Effect of Climate Derivatives

This paper discusses the equilibrium pricing of climate derivatives using the mean field game method, especially under the reference price effect. We first consider the concentration of investors on a single asset class, and then examine the approximate Nash equilibrium in finite population games under reference price effect when the investor is sufficiently large. The solution and results of equilibrium pricing are verified by numerical simulation. We also use stock indexes and extreme climate data of several developed countries and several industries in China to calculate simulated prices for climate derivatives and discuss their hedging effects, as well as a comparative analysis with the Ornstein-Uhlenbeck model.

Ruimeng Hu University of California Santa Barbara

Learning Mean Field Games via Mean Field Actor Critic Flow

We introduce the Mean-Field Actor-Critic (MFAC) flow, a continuous-time learning dynamics for solving mean-field games (MFGs), drawing on ideas from reinforcement learning, generative modeling, and optimal transport. The MFAC framework jointly evolves the actor, critic, and distribution through gradient-based updates, with the distribution governed by a novel Optimal Transport Geodesic Picard (OTGP) flow. The OTGP flow drives the distribution toward equilibrium along Wasserstein-2 geodesics. We rigorously analyze the MFAC flow using Lyapunov functionals and establish global exponential convergence under suitable time scales. The analysis highlights the coupled structure of the algorithm and offers practical guidelines for choosing learning rates. Numerical results further support the theory and demonstrate the effectiveness of the proposed approach. This is joint work with Mo Zhou (UCLA) and Haosheng Zhou (UCSB).

Ruixun Zhang Peking University

DORADO: Dynamic Optimization of R&D Options

We propose a dynamic strategy to optimize a portfolio of research and development (R&D) projects with correlated binary outcomes, which we call dynamic optimization of R&D options (DORADO). DORADO utilizes project correlation as a knowledge discovery mechanism in order to exercise real options with the information learned from previous outcomes to dynamically adjust future R&D plans. We focus on a class of R&D projects, known as long shots, that are difficult to manage using standard approaches. We derive the DORADO-optimal strategy and its corresponding financial value for these portfolios under batch testing, block correlation, and risk-sensitive objectives. We further extend DORADO with a reinforcement learning policy with offline model estimation to accommodate R&D projects with heterogeneous characteristics at scale. DORADO generates an improved efficient frontier with a higher expected payoff and lower volatility than the comparable parallel development of these portfolios. Depending on investor preferences, the financial value of such an investment portfolio is maximized when the correlation between projects is either very low or very high, which reflects the tradeoff between the diversification gains at low correlations and the information value gained at high correlations. We provide an empirical example in the context of curative treatments for glioblastoma. Joint work with Zixi Chen (PKU), Leonid Kogan (MIT), Andrew W. Lo (MIT), and Qingyang Xu (MIT).

Sebastian Jaimungal University of Toronto

Deep Learning and Elicitability for McKean-Vlasov FBSDEs with common noise

We propose a deep learning–based numerical framework for solving McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) with common noise, combining global Picard iterations with the concept of elicitability to construct tractable, pathwise loss functions. The approach parameterizes the mean-field interaction via recurrent neural networks and the backward component through a neural approximation of the decoupling field, thereby avoiding computationally prohibitive nested Monte Carlo methods while accommodating non-Markovian dependencies and general (e.g., quantile-based) interactions. We also analyse the convergence for the resulting approximate Picard scheme, where each iteration involves learned conditional expectations. We explicitly quantify how errors arising from function approximation, stochastic optimization, and sampling propagate through the forward-backward system, and show that, under some technical conditions, the scheme remains contractive and converges to a neighborhood of the true solution. We also conduct numerical experiments in a systemic risk banking model and an economic growth model. [ this is based on joint work with Felipe J. P. Antunes and Yuri F. Saporito ]

Shi Jin Shanghai Jiao Tong University

TBA

Shuoqing Deng HKUST

Distribution constrained optimal stopping: beyond the Root-type solution

We give an explicit construction of the boundary which solves the distribution constrained optimal stopping when the cost function is not of Root-type. The boundary can be characterised analytically and probabilistically. From the analytical perspective, it is characterised by the viscosity solution of a variational inequality with Wentzell type boundary condition. From the probabilistic perspective, it can be characterised by the optimal stopping of a Sticky Brownian motion without distribution constraint.

Stéphane Crépey Paris Cité University

Comparison of Tax and Cap-and-Trade Carbon Pricing Schemes

Carbon pricing has become a central pillar of modern climate policy, with carbon taxes and emissions trading systems (ETS) serving as the two dominant approaches. Although economic theory suggests these instruments are equivalent under idealized assumptions, their performance diverges in practice due to real-world market imperfections. A particularly less explored dimension of this divergence concerns the role of financial intermediaries in emissions trading markets. This paper develops a unified framework to compare the economic and environmental performance of tax- and market-based schemes, explicitly incorporating the involvement of intermediaries. By calibrating both instruments to deliver identical emissions reductions, we assess their economic performance across alternative market structures. Our results suggest that, although the two schemes are equivalent under perfect competition, the presence of intermediaries in ETS reduces regulatory revenues and the aggregate profits of financial actors relative to carbon taxation. These effects arise from intermediaries' influence on price formation and their appropriation of part of the revenue stream. The findings underscore the importance of accounting for intermediaries' behavior in the design of carbon markets and highlight the need for further empirical research on the evolving institutional structure of emissions trading systems.

Tianyang Nie Shandong University

TBA

Tianyi Wang University of International Business and Economics

Regime switching, time-varying volatility, and discrete-time option pricing

The traditional discrete-time option pricing framework often assumes that volatility has only one regime, which is inconsistent with the characteristic of multiple regime switching pointed out in the volatility literature. To address this shortcoming, we propose a dual MS-Heston-Nandi GARCH pricing framework that considers regime switching and switching premiums, based on the structure of the classic Heston-Nandi GARCH model. On this basis, our model decomposes the risks faced by investors into "equity risk" and a new "switching risk", and develops a unified risk-neutral approach. Compared to existing pricing methods involving regime switching, this framework is simpler and more intuitive in risk-neutral processing, and the theoretical results are more in line with economic intuition. Empirical results based on SSE 50ETF options show that the dual MS-Heston-Nandi GARCH model can well describe the price changes of options and outperforms traditional pricing benchmarks in both in-sample and out-of-sample tests.

Wenjing Cao The Chinese University of Hong Kong

Quantitative weak propagation of chaos for McKean–Vlasov branching diffusion processes

We study the weak propagation of chaos for McKean–Vlasov diffusions with branching, whose induced marginal measures are nonnegative finite measures but not necessarily probability measures. The flow of marginal measures satisfies a non-linear Fokker–Planck equation, along which we provide a functional Ito's formula. We then consider a functional of the terminal marginal measure of the branching process, whose conditional value is solution to a Kolmogorov backward master equation. By using Ito's formula and based on the estimates of second-order linear and intrinsic functional derivatives of the value function, we finally derive a quantitative weak convergence rate for the empirical measures of the branching diffusion processes with finite population.

Xianhua Peng Peking University (Shenzhen)

A Risk Sensitive Contract-unified Reinforcement Learning Approach for Option Hedging

We propose a new risk sensitive reinforcement learning approach for the dynamic hedging of options. The approach focuses on the minimization of the tail risk of the final P&L of the seller of an option. Different from most existing reinforcement learning approaches that require a parametric model of the underlying asset, our approach can learn the optimal hedging strategy directly from the historical market data without specifying a parametric model; in addition, the learned optimal hedging strategy is contract-unified, i.e., it applies to different options contracts with different initial underlying prices, strike prices, and maturities. Our approach extends existing reinforcement learning methods by learning the tail risk measures of the final hedging P&L and the optimal hedging strategy at the same time. We carry out comprehensive empirical study to show that, in the out-of-sample tests, the proposed reinforcement learning hedging strategy can obtain statistically significantly lower tail risk and higher mean of the final P&L than delta hedging methods. This is a joint work with Xiang Zhou, Bo Xiao, and Yi Wu.

Xiaofei Shi University of Toronto

A Dynamic Equilibrium Model of Liquidity Risk

We present a framework for analyzing the equilibrium implications of liquidity risk dynamics on asset prices. Our model features two risk-averse agents who continuously trade a security to hedge nontraded risks, while facing stochastic transaction costs correlated with their trading needs. We derive explicit solutions for equilibrium prices and traded quantities under small transaction costs, showing that the illiquidity discount increases with the correlation between transaction costs and trading needs. Calibrating the model using NYSE and AMEX data, we find that liquid portfolios recover faster from liquidity shocks and exhibit smaller fluctuations, whereas illiquid portfolios are highly sensitive to trading-cost dynamics. For the most illiquid portfolio, the illiquidity discount increases by 12% when transaction costs and high-frequency trading needs are fully correlated, compared to the uncorrelated case.

Xiaolu Tan The Chinese University of Hong Kong

On the regularity of solutions to a class of path-dependent HJB equations

We study a class of path-dependent Hamilton-Jacobi-Bellman (HJB) equations arising from a path-dependent optimal control problem in which only the drift coefficient is controlled. Under the assumption of strict concavity of the Hamiltonian, we establish the existence and uniqueness of the optimal control for the corresponding control problem. Based on this, we first establish the \(C^1\)-regularity of the value function and hence the existence of a continuous feedback optimal control. Next, by using our previous results on the regularity of linear path-dependent PDE, we show that the value function is in fact a classical solution to the path-dependent HJB equation. In particular, the optimal feedback control is shown to be (locally) Lipschitz in the state variable.

Xiaozhen Wang Paris Dauphine University

Entropic Optimal Transport Problem with Convex Functional Cost

We study an entropic optimal transport problem with an additional nonlinear convex penalty on the coupling. We prove existence, uniqueness, and uniform a priori bounds for the minimizer, which satisfies a fixed-point first-order optimality system via an exponentially tilted reference measure. Leveraging this variational structure, we propose a Sinkhorn–Frank–Wolfe flow, establish global well-posedness, and derive an energy–dissipation inequality implying exponential convergence to the unique optimum. We apply the resulting SFW algorithm to UAV routing with congestion aversion. Joint work with Anna Kazeykina, Zhenjie Ren, and Yufei Zhang.

Xin Zhang New York University

Optimization of win martingales

Prediction market is a market where people can trade based on outcomes of future events. It is widely used in sports games, elections, and pricing of digital options. In math finance, prediction markets can be modeled by the so-called win martingales, which are continuous time martingales that end up with Bernoulli distributions. In this talk, choosing different divergences as objective functionals, we will solve a class of optimal win martingales. In some cases, we will get explicit formulas of optimizers, and make connections to Schrödinger, filtering problems, Wright-Fisher diffusion, and the problem of identifying most exciting games.

Xuefeng Gao The Chinese University of Hong Kong

Generating solution paths of Markovian stochastic differential equations using diffusion models

This paper introduces a new approach to generating sample paths of unknown Markovian stochastic differential equations (SDEs) using diffusion models, a class of generative AI methods commonly employed in image and video applications. Unlike the traditional Monte Carlo methods for simulating SDEs, which require explicit specifications of the drift and diffusion coefficients, ours takes a model-free, data-driven approach. Given a finite set of sample paths from an SDE, we utilize conditional diffusion models to generate new, synthetic paths of the same SDE. Numerical experiments show that our method consistently outperforms two alternative methods in terms of the Kullback–Leibler (KL) divergence between the distributions of the target SDE paths and the generated ones. Moreover, we present a theoretical error analysis deriving an explicit bound on the said KL divergence. Finally, in simulation and empirical studies, we leverage these synthetically generated sample paths to boost the performance of reinforcement learning algorithms for continuous-time mean-variance portfolio selection, hinting promising applications of our study in financial analysis and decision-making.

Yuchen Sun Humboldt-Universität zu Berlin

Feynman-Kac for Singular HJB-PDE and FBSDE with Application to KPZ

We prove well-posedness of a class of singular forward–backward stochastic differential equations, where the forward equation is driven by a distributional drift and the backward component has quadratic growth. We show that their solutions provide a Feynman–Kac-type representation for a class of singular HJB-type equations in a newly developed viscosity sense, which coincides with the paracontrolled solution if the latter exists. As an application, we obtain estimates for solutions to Kardar–Parisi–Zhang (KPZ) equations, which model stochastic interface growth and surface roughening phenomena and are of central interest in the theory of singular stochastic partial differential equations (SPDE). (Joint work with Nicolas Perkowski and Dirk Becherer, FU + HU Berlin)

Yuchao Dong Tongji University

A Two-fold Randomization Framework for Impulse Control Problems

We propose and analyze a randomization scheme for a general class of impulse control problems. The solution to this randomized problem is characterized as the fixed point of a compound operator which consists of a regularized nonlocal operator and a regularized stopping operator. This approach allows us to derive a semi-linear Hamilton-Jacobi-Bellman (HJB) equation. Through an equivalent randomization scheme with a Poisson compound measure, we establish a verification theorem that implies the uniqueness of the solution. Via an iterative approach, we prove the existence of the solution. The existence-and-uniqueness result ensures the randomized problem is well-defined. We then demonstrate that our randomized impulse control problem converges to its classical counterpart as the randomization parameter \(\lambda\) vanishes. This convergence, combined with the value function's \(\mathcal{C}^{2,\alpha}_{loc}\) regularity, confirms our framework provides a robust approximation and a foundation for developing learning algorithms. Under this framework, we propose an offline reinforcement learning (RL) algorithm. Its policy improvement step is naturally derived from the iterative approach from the existence proof, which enjoys a geometric convergence rate. We numerically demonstrate the effectiveness of our algorithm using a widely-studied example.

Yufeng Shi Shandong University

BSDE Deep learning approaches for option pricing

Deep learning can effectively overcome dimensional disasters in the numerical methods of nonlinear partial differential equations and backward stochastic differential equations, and has become an important research direction of numerical computation in recent years. We study the numerical calculation methods of high-dimensional backward doubly stochastic differential equations and high-dimensional mean field backward doubly stochastic differential equations, in which the deep neural network is introduced as the key step to achieve numerical solution. Based on the numerical method of backward stochastic differential equations, this paper studies the option data driven \(g\)-pricing modeling method, and verifies the model performance with SPX options.

Yupeng Bai Paris Dauphine University (Evry)

Self-fictitious-play for Potential Monotone Ergodic Mean-field Games

We study long-time learning in ergodic, potential, monotone mean-field games (MFG) via a self-fictitious-play (SFP) dynamics that couples an optimally controlled diffusion with a slowly updated belief. At each time, the state follows the optimal feedback for the current belief; the belief itself tracks the running occupation measure. For monotone potential MFGs on the torus, we show that the SFP system is contractive and admits a unique stationary law. We then prove that this stationary distribution is quantitatively close to the MFG Nash equilibrium: after an exponentially fast transient, the gap scales on the order of the square root of the belief-update rate. A linear–quadratic case confirms sharpness of this rate. The analysis combines uniform regularity for the ergodic Hamilton–Jacobi–Bellman equation, a Lipschitz bound on the induced optimal drift, and a reflection-coupling argument that transports the Lasry–Lions divergence into Wasserstein-distance control. Together, these results provide the first rigorous convergence guarantees for SFP in this setting and quantify how slow belief updates ensure accurate approximation of the mean-field equilibrium.

Youssef Ouazzani Chahdi Paris Dauphine University

Trading with market resistance and concave price impact

We consider an optimal trading problem under a market impact model with endogenous market resistance generated by a sophisticated trader who (partially) detects metaorders and trades against them to exploit price overreactions induced by the order flow. The model features a concave transient impact driven by a power-law propagator with a resistance term responding to the trader's rate via a fixed-point equation involving a general resistance function. We derive a (non)linear stochastic Fredholm equation as the first-order optimality condition satisfied by optimal trading strategies. Existence and uniqueness of the optimal control are established when the resistance function is linear, and an existence result is obtained when it is strictly convex using coercivity and weak lower semicontinuity of the associated profit-and-loss functional. We also propose an iterative scheme to solve the nonlinear stochastic Fredholm equation and prove an exponential convergence rate. Numerical experiments confirm this behavior and illustrate optimal round-trip strategies under "buy" signals with various decay profiles and different market resistance specifications.

Zengjing Chen Shandong University

TBA

Zhenjie Ren Université d'Évry

Specific Entropic Martingale Optimal Transport

In this talk, we review the notion of specific entropy, introduced by Nina Gantert, and the recently developed framework of Specific Entropic Martingale Optimal Transport (SEMOT) built upon it. We show that this entropy is genuinely specific, in the sense that it appears as a particular scaling limit of the Kullback–Leibler divergence when a continuous martingale diffusion is approximated by a compound Poisson process. Focusing on the trace-normalized specific entropy, we establish the associated duality for the SEMOT problem. We conclude by introducing an intuitive Sinkhorn-type algorithm for its numerical solution.

Zhenhua Wang Shandong University

Existence of Equilibrium for Time-Inconsistent Control Problems by Vanishing Entropy Regularization

We develop a general framework for solving Equilibrium HJB system in time-inconsistent control problems via an entropy-regularization approach. Since standard equilibrium strategies may fail to exist, we consider relaxed equilibria. By introducing a weighted entropy regularization, we establish the existence of equilibria for the regularized problem and prove global Hölder regularity of the associated value functions. We then show, as the entropy parameter vanishes, the regularized equilibria converge to an equilibrium of the original control problem. This talk is based on joint work with Xiang Yu, Jingjie Zhang, and Zhou Zhou.

Zuoquan Xu The Hong Kong Polytechnic University

Learning to optimally stop diffusion processes, with financial applications

We study optimal stopping for diffusion processes with unknown model primitives within the continuous-time reinforcement learning (RL) framework developed by Wang et al. (2020), and present applications to option pricing and portfolio choice. By penalizing the corresponding variational inequality formulation, we transform the stopping problem into a stochastic optimal control problem with two actions. We then randomise controls into Bernoulli distributions and add an entropy regulariser to encourage exploration. We derive a semi-analytical optimal Bernoulli distribution, based on which we devise RL algorithms using the martingale approach established in Jia and Zhou (2022a). We establish a policy improvement theorem and prove the fast convergence of the resulting policy iterations. We demonstrate the effectiveness of the algorithms in pricing finite-horizon American put options, solving Merton's problem with transaction costs, and scaling to high-dimensional optimal stopping problems. In particular, we show that both the offline and online algorithms achieve high accuracy in learning the value functions and characterising the associated free boundaries. This is a joint work with Min Dai, Yu Sun, and Xun Yu Zhou.