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Erhan Bayraktar

Publications and source records attributed to Erhan Bayraktar.

At least 55 records · Page 3Linked to original sources
arXiv2026

The McCormick martingale optimal transport

Martingale optimal transport (MOT) often yields broad price bounds for options, constraining their practical applicability. In this study, we extend MOT by incorporating causality constraints among assets, inspired by the nonanticipativity condition of stochastic processes. This, however, introduces a computationally challenging bilinear program. To tackle this issue, we propose McCormick relaxations to ease the bicausal formulation and refer to it as McCormick MOT. The primal attainment and strong duality of McCormick MOT are established under standard assumptions. Empirically, we apply McCormick MOT to basket and digital options. With natural bounds on probability masses, the average price reduction for basket options is approximately 1.08% to 3.90%. When tighter probability bounds are available, the reduction increases to 12.26%, compared to the classic MOT, which also incorporates tighter bounds. For most dates considered, there are basket options with suitable payoffs, where the price reduction exceeds 10.00%. For digital options, McCormick MOT results in an average price reduction of over 20.00%, with the best case exceeding 99.00%.

q-fin.MF↗
arXiv2026

Quantitative convergence rates for extended mean field games with volatility control

We investigate the convergence of symmetric stochastic differential games with interactions via control, where the volatility terms of both idiosyncratic and common noises are controlled. We apply the stochastic maximum principle, following the approach of Laurière and Tangpi, to reduce the convergence analysis to the study of forward-backward propagation of chaos. Under the standard monotonicity conditions, we derive quantitative convergence rates for open-loop Nash equilibria of $N$-player stochastic differential games toward the corresponding mean field equilibrium. As a prerequisite, we also establish the well-posedness of the conditional McKean--Vlasov forward-backward stochastic differential equations by the method of continuation. Moreover, we analyze a specific class of linear-quadratic settings to demonstrate the applicability of our main result.

math.PR↗
arXiv2026

Uniform-in-time propagation of chaos for consensus-based minimax algorithm

We study the large-population convergence of a consensus-based algorithm for the saddle point problem proposed by ArXiv: 2212.12334, establishing the uniform-in-time propagation of chaos using a coupling method. Our work shows that the $L^2$-deviation has order $O(N_1^{-1} + N_2^{-1})$ uniformly in time, where $N_1$ and $N_2$ denote the numbers of particles corresponding to the two competing players. It demonstrates the convergence of the particles to some location near a saddle point of the given objective function, which confirms the computational feasibility of the algorithm. The main idea behind the proofs is the exponential decay and the concentration of the variances of the particle system.

math.PR↗
arXiv2026

Two-fund separation under hyperbolically distributed returns and concave utility functions

Portfolio selection problems that optimize expected utility are usually difficult to solve. If the number of assets in the portfolio is large, such expected utility maximization problems become even harder to solve numerically. Therefore, analytical expressions for optimal portfolios are always preferred. In our work, we study portfolio optimization problems under the expected utility criterion for a wide range of utility functions, assuming return vectors follow hyperbolic distributions. Our main result demonstrates that under this setup, the two-fund monetary separation holds. Specifically, an individual with any utility function from this broad class will always choose to hold the same portfolio of risky assets, only adjusting the mix between this portfolio and a riskless asset based on their initial wealth and the specific utility function used for decision making. We provide explicit expressions for this mutual fund of risky assets. As a result, in our economic model, an individual's optimal portfolio is expressed in closed form as a linear combination of the riskless asset and the mutual fund of risky assets. Additionally, we discuss expected utility maximization problems under exponential utility functions over any domain of the portfolio set. In this part of our work, we show that the optimal portfolio in any given convex domain of the portfolio set either lies on the boundary of the domain or is the unique globally optimal portfolio within the entire domain.

q-fin.PM↗
arXiv2026

On the Mean-Field limit of diffusive games through the master equation: $L^{\infty}$ estimates and extreme value behavior

We consider an $N$-player game where the states of the players evolve with time as Stochastic Differential Equations (SDEs) with interaction only in the drift terms. Each player controls the drift of the SDE satisfied by her state process, aiming to minimize the expected value of a cost that depends on the paths of the player's state and the empirical measure of the states of all the players until a terminal time. When $N \to \infty$, previous works have established Central Limit Theorems and Large Deviation Principles for the state processes when the game is in Nash Equilibrium (the Nash states), by using the Master Equation to construct approximations of those processes that evolve with time as SDEs with classical Mean-Field interaction. Staying in this framework, we improve an existing $L^{1}$ estimate for the total error of approximating all the Nash states to an $L^{\infty}$ one, and we also establish the $N \to \infty$ asymptotic behavior of the upper order statistics of the Nash states. The latter initiates the development of an Extreme Value Theory for Stochastic Differential Games.

math.PR↗
arXiv2026

The Learning Approach to Games

This work introduces a unified framework for analyzing games in greater depth. In the existing literature, players' strategies are typically assigned scalar values, and equilibrium concepts are used to identify compatible choices. However, this approach neglects the internal structure of players, thereby failing to accurately model observed behaviors. To address this limitation, we propose an abstract definition of a player, consistent with constructions in reinforcement learning. Instead of defining games as external settings, our framework defines them in terms of the players themselves. This offers a language that enables a deeper connection between games and learning. To illustrate the need for this generality, we study a simple two-player game and show that even in basic settings, a sophisticated player may adopt dynamic strategies that cannot be captured by simpler models or compatibility analysis. For a general definition of a player, we discuss natural conditions on its components and define competition through their behavior. In the discrete setting, we consider players whose estimates largely follow the standard framework from the literature. We explore connections to correlated equilibrium and highlight that dynamic programming naturally applies to all estimates. In the mean-field setting, we exploit symmetry to construct explicit examples of equilibria. Finally, we conclude by examining relations to reinforcement learning.

cs.GT↗
arXiv2026

Entropy-Based Dimension-Free Convergence and Loss-Adaptive Schedules for Diffusion Models

Diffusion generative models synthesize samples by discretizing reverse-time dynamics driven by a learned score (or denoiser). Existing convergence analyses of diffusion models typically scale at least linearly with the ambient dimension, and sharper rates often depend on intrinsic-dimension assumptions or other geometric restrictions on the target distribution. We develop an alternative, information-theoretic approach to dimension-free convergence that avoids any geometric assumptions. Under mild assumptions on the target distribution, we bound KL divergence between the target and generated distributions by $O(H^2/K)$ (up to endpoint factors), where $H$ is the Shannon entropy and $K$ is the number of sampling steps. Moreover, using a reformulation of the KL divergence, we propose a Loss-Adaptive Schedule (LAS) for efficient discretization of reverse SDE which is lightweight and relies only on the training loss, requiring no post-training heavy computation. Empirically, LAS improves sampling quality over common heuristic schedules.

cs.LG↗
arXiv2025

Comparison for semi-continuous viscosity solutions for second order PDEs on the Wasserstein space

In this paper, we prove a comparison result for semi-continuous viscosity solutions of a class of second-order PDEs in the Wasserstein space. This allows us to remove the Lipschitz continuity assumption with respect to the Fourier-Wasserstein distance in AriX: 2309.05040 and obtain uniqueness by directly working in the Wasserstein space. In terms of its application, we characterize the value function of a stochastic control problem with partial observation as the unique viscosity solution to its corresponding HJB equation. Additionally, we present an application to a prediction problem under partial monitoring, where we establish an upper bound on the limit of regret using our comparison principle for degenerate dynamics.

math.AP↗
arXiv2025

Deep Neural Operator Learning for Probabilistic Models

We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining.

cs.LG↗
arXiv2025

Goal-based portfolio selection with fixed transaction costs

We study a goal-based portfolio selection problem in which an investor aims to meet multiple financial goals, each with a specific deadline and target amount. Trading the stock incurs a strictly positive transaction cost. Using the stochastic Perron's method, we show that the value function is the unique viscosity solution to a system of quasi-variational inequalities. The existence of an optimal trading strategy and goal funding scheme is established. Numerical results reveal complex optimal trading regions and show that the optimal investment strategy differs substantially from the V-shaped strategy observed in the frictionless case.

math.OC↗
arXiv2025

Fitted value iteration methods for bicausal optimal transport

We develop a fitted value iteration (FVI) method to compute bicausal optimal transport (OT) where couplings have an adapted structure. Based on the dynamic programming formulation, FVI adopts a function class to approximate the value functions in bicausal OT. Under the concentrability condition and approximate completeness assumption, we prove the sample complexity using (local) Rademacher complexity. Furthermore, we demonstrate that multilayer neural networks with appropriate structures satisfy the crucial assumptions required in sample complexity proofs. Numerical experiments reveal that FVI outperforms linear programming and adapted Sinkhorn methods in scalability as the time horizon increases, while still maintaining acceptable accuracy.

stat.ML↗
arXiv2025

Learning with Linear Function Approximations in Mean-Field Control

The paper focuses on mean-field type multi-agent control problems with finite state and action spaces where the dynamics and cost structures are symmetric and homogeneous, and are affected by the distribution of the agents. A standard solution method for these problems is to consider the infinite population limit as an approximation and use symmetric solutions of the limit problem to achieve near optimality. The control policies, and in particular the dynamics, depend on the population distribution in the finite population setting, or the marginal distribution of the state variable of a representative agent for the infinite population setting. Hence, learning and planning for these control problems generally require estimating the reaction of the system to all possible state distributions of the agents. To overcome this issue, we consider linear function approximation for the control problem and provide coordinated and independent learning methods. We rigorously establish error upper bounds for the performance of learned solutions. The performance gap stems from (i) the mismatch due to estimating the true model with a linear one, and (ii) using the infinite population solution in the finite population problem as an approximate control. The provided upper bounds quantify the impact of these error sources on the overall performance.

math.OC↗
arXiv2025

Walsh Spider Diffusions as Time Changed Multi-parameter Processes

Inspired by allocation strategies in multi-armed bandit model, we propose a pathwise construction of Walsh spider diffusions. For any infinitesimal generator on a star shaped graph, there exists a unique time change associated with a multi-parameter process such that the time change of this multi-parameter process is the desired diffusion. The time change has an interpretation of time allocation of the process on each edge, and it can be derived explicitly from a family of equations.

math.PR↗
arXiv2025

Probabilistic cellular automata with local transition matrices: synchronization, ergodicity, and inference

We introduce a new class of probabilistic cellular automata that are capable of exhibiting rich dynamics such as synchronization and ergodicity and can be easily inferred from data. The system is a finite-state locally interacting Markov chain on a circular graph. Each site's subsequent state is random, with a distribution determined by its neighborhood's empirical distribution multiplied by a local transition matrix. We establish sufficient and necessary conditions on the local transition matrix for synchronization and ergodicity. Also, we introduce novel least squares estimators for inferring the local transition matrix from various types of data, which may consist of either multiple trajectories, a long trajectory, or ensemble sequences without trajectory information. Under suitable identifiability conditions, we show the asymptotic normality of these estimators and provide non-asymptotic bounds for their accuracy.

math.PR↗
arXiv2025

Equilibrium transport with time-inconsistent costs

Given two probability measures on sequential data, we investigate the transport problem with time-inconsistent preferences in a discrete-time setting. Motivating examples are nonlinear objectives, state-dependent costs, and regularized optimal transport with general $f$-divergence. Under the bicausal constraint, we introduce the concept of equilibrium transport. Existence is proved in the semi-discrete Markovian case and the continuous non-Markovian case with strict quasiconvexity, while uniqueness also holds in the second case. We apply our framework to study mean-variance dynamic matching, nonlinear or state-dependent objectives with Gaussian data, and mismatches in job markets. Numerical results indicate a positive relationship between mismatches and state dependence.

math.OC↗
arXiv2025

Finite Approximations for Mean Field Type Multi-Agent Control and Their Near Optimality

We study a multi-agent mean field type control problem in discrete time where the agents aim to find a socially optimal strategy and where the state and action spaces for the agents are assumed to be continuous. The agents are only weakly coupled through the distribution of their state variables. The problem in its original form can be formulated as a classical Markov decision process (MDP), however, this formulation suffers from several practical difficulties. In this work, we attempt to overcome the curse of dimensionality, coordination complexity between the agents, and the necessity of perfect feedback collection from all the agents (which might be hard to do for large populations.) We provide several approximations: we establish the near optimality of the action and state space discretization of the agents under standard regularity assumptions for the considered formulation by constructing and studying the measure valued MDP counterpart for finite and infinite population settings. It is a well known approach to consider the infinite population problem for mean-field type models, since it provides symmetric policies for the agents which simplifies the coordination between the agents. However, the optimality analysis is harder as the state space of the measure valued infinite population MDP is continuous (even after space discretization of the agents). Therefore, as a final step, we provide further approximations for the infinite population problem by focusing on smaller sized sub-population distributions.

math.OC↗
arXiv2025

Uniform-in-time weak propagation of chaos for consensus-based optimization

We study the uniform-in-time weak propagation of chaos for the consensus-based optimization (CBO) method on a bounded searching domain. We apply the methodology for studying long-time behaviors of interacting particle systems developed in the work of Delarue and Tse (ArXiv:2104.14973). Our work shows that the weak error has order $O(N^{-1})$ uniformly in time, where $N$ denotes the number of particles. The main strategy behind the proofs are the decomposition of the weak errors using the linearized Fokker-Planck equations and the exponential decay of their Sobolev norms. Consequently, our result leads to the joint convergence of the empirical distribution of the CBO particle system to the Dirac-delta distribution at the global minimizer in population size and running time in Wasserstein-type metrics.

math.OC↗