arXiv subjects

H. Mete Soner

Publications and source records attributed to H. Mete Soner.

At least 19 recordsLinked to original sources

arXiv2026

Learning algorithms for mean field optimal control

We analyze an algorithm to numerically solve the mean-field optimal control problems by approximating the optimal feedback controls using neural networks with problem specific architectures. We approximate the model by an $N$-particle system and leverage the exchangeability of the particles to obtain substantial computational efficiency. In addition to several numerical examples, a convergence analysis is provided. We also developed a universal approximation theorem on Wasserstein spaces.

math.OC↗

arXiv2025

Relative arbitrage problem under eigenvalue lower bounds

We give a new formulation of the relative arbitrage problem from stochastic portfolio theory that asks for a time horizon beyond which arbitrage relative to the market exists in all ``sufficiently volatile'' markets. In our formulation, ``sufficiently volatile'' is interpreted as a lower bound on an ordered eigenvalue of the instantaneous covariation matrix, a quantity that has been studied extensively in the empirical finance literature. Upon framing the problem in the language of stochastic optimal control, we characterize the time horizon in question through the unique upper semicontinuous viscosity solution of a fully nonlinear elliptic partial differential equation (PDE). In a special case, this PDE amounts to the arrival time formulation of the Ambrosio-Soner co-dimension mean curvature flow. Beyond the setting of stochastic portfolio theory, the stochastic optimal control problem is analyzed for arbitrary compact, possibly non-convex, domains, thanks to a boundedness assumption on the instantaneous covariation matrix.

q-fin.MF↗

arXiv2025

Markov Perfect Equilibria in Discrete Finite-Player and Mean-Field Games

We study dynamic finite-player and mean-field stochastic games within the framework of Markov perfect equilibria (MPE). Our focus is on discrete time and space structures without monotonicity. Unlike their continuous-time analogues, discrete-time finite-player games generally do not admit unique MPE. However, we show that uniqueness is remarkably recovered when the time steps are sufficiently small, and we provide examples demonstrating the necessity of this assumption. This result, established without relying on any monotonicity conditions, underscores the importance of inertia in dynamic games. In both the finite-player and mean-field settings, we show that MPE correspond to solutions of the Nash-Lasry-Lions equation, which is known as the master equation in the mean-field case. We exploit this connection to establish the convergence of discrete-time finite-player games to their mean-field counterpart in short time. Finally, we prove the convergence of finite-player games to their continuous-time version on every time horizon.

math.OC↗

arXiv2025

Iterative Schemes for Markov Perfect Equilibria

We study Markov perfect equilibria in continuous-time dynamic games with finitely many symmetric players. The corresponding Nash system reduces to the Nash-Lasry-Lions equation for the common value function, also known as the master equation in the mean-field setting. In the finite-state space problems we consider, this equation becomes a nonlinear ordinary differential equation admitting a unique classical solution. Leveraging this uniqueness, we prove the convergence of both Picard and weighted Picard iterations, yielding efficient computational methods. Numerical experiments confirm the effectiveness of algorithms based on this approach.

math.OC↗

arXiv2025

Optimal Control and Potential Games in the Mean Field

We study a mean field optimal control problem with general non-Markovian dynamics, including both common noise and jumps. We show that its minimizers are Nash equilibria of an associated mean field game of controls. These types of games are necessarily potential, and the Nash equilibria derived as the minimizers of the control problem are closely connected to McKean-Vlasov equations of Langevin type. To illustrate the general theory, we present several examples, including a mean field game of controls with interactions through a price variable, and mean field Cucker-Smale Flocking and Kuramoto models. We also establish the invariance property of the value function, a key ingredient used in our proofs.

math.OC↗

arXiv2025

Mean Field Games of Control and Cryptocurrency Mining

This paper studies Mean Field Games (MFGs) in which agent dynamics are given by jump processes of controlled intensity, with mean-field interaction via the controls and affecting the jump intensities. We establish the existence of MFG equilibria in a general discrete-time setting, and prove a limit theorem as the time discretization goes to zero, establishing equilibria in the continuous-time setting for a class of MFGs of intensity control. This motivates numerical schemes that involve directly solving discrete-time games as opposed to coupled Hamilton-Jacobi-Bellman and Kolmogorov equations. As an example of the general theory, we consider cryptocurrency mining competition, modeled as an MFG both in continuous and discrete time, and illustrate the effectiveness of the discrete-time algorithm to solve it.

math.OC↗

arXiv2025

Stopping Times of Boundaries: Relaxation and Continuity

We study the properties of the free boundaries and the corresponding hitting times in the context of optimal stopping in discrete time. We first prove the continuity of the map from the boundaries to the expected value of the corresponding stopping policy both in the supremum norm and also in a weaker, novel topology induced by the relaxed $L^\infty$ metric that we introduce. The latter is particularly useful when the optimal stopping boundary is only proved to be semicontinuous. Secondly, we study the connection between the hitting times, and their relaxations as widely employed in recent numerical methods. All these results together with the universal approximation capability of neural networks and the notion of inf/sup convolution are then used to provide a convergence analysis for the algorithm in [Reppen, Soner, and Tissot-Daguette, Neural Optimal Stopping Boundary, 2025] for the numerical resolution of the exercise regions arising in the analysis of Bermudan type option.

math.PR↗

arXiv2024

Controlled Occupied Processes and Viscosity Solutions

We consider the optimal control of occupied processes which record all positions of the state process. Dynamic programming yields nonlinear equations on the space of positive measures. We develop the viscosity theory for this infinite dimensional parabolic $occupied$ PDE by proving a comparison result between sub and supersolutions, and thus provide a characterization of the value function as the unique viscosity solution. Toward this proof, an extension of the celebrated Crandall-Ishii-Lions (second order) Lemma to this setting, as well as finite-dimensional approximations, is established. Examples including the occupied heat equation, and pricing PDEs of financial derivatives contingent on the occupation measure are also discussed.

math.OC↗

arXiv2024

Synchronization Games

We propose a new mean-field game model with two states to study synchronization phenomena, and we provide a comprehensive characterization of stationary and dynamic equilibria along with their stability properties. The game undergoes a phase transition with increasing interaction strength. In the subcritical regime, the uniform distribution, representing incoherence, is the unique and stable stationary equilibrium. Above the critical interaction threshold, the uniform equilibrium becomes unstable and there is a multiplicity of stationary equilibria that are self-organizing. Under a discounted cost, dynamic equilibria spiral around the uniform distribution before converging to the self-organizing equilibria. With an ergodic cost, however, unexpected periodic equilibria around the uniform distribution emerge.

math.OC↗

arXiv2024

Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

Dynamic programming equations for mean field control problems with a separable structure are Eikonal equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to obtain a comparison result among sub and super solutions that are Lipschitz continuous in the Wasserstein-one norm which is directly verified for the value function. A key technical result provides a point-wise upper bound for the derivative of the linear derivative of a function in terms of its Lipschitz norm.

math.OC↗

arXiv2023

Deep Level-set Method for Stefan Problems

We propose a level-set approach to characterize the region occupied by the solid in Stefan problems with and without surface tension, based on their recent probabilistic reformulation. The level-set function is parameterized by a feed-forward neural network, whose parameters are trained using the probabilistic formulation of the Stefan growth condition. The algorithm can handle Stefan problems where the liquid is supercooled and can capture surface tension effects through the simulation of particles along the moving boundary together with an efficient approximation of the mean curvature. We demonstrate the effectiveness of the method on a variety of examples with and without radial symmetry.

math.NA↗

arXiv2023

Neural Optimal Stopping Boundary

A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.

q-fin.PR↗

arXiv2022

Viscosity Solutions for McKean-Vlasov Control on a torus

An optimal control problem in the space of probability measures, and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a novel smooth Fourier Wasserstein metric. A comparison result between the Lipschitz viscosity sub and super solutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution.

math.OC↗

arXiv2022

Synchronization in a Kuramoto Mean Field Game

The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions become stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes.

math.OC↗

arXiv2022

Deep Empirical Risk Minimization in finance: looking into the future

Many modern computational approaches to classical problems in quantitative finance are formulated as empirical loss minimization (ERM), allowing direct applications of classical results from statistical machine learning. These methods, designed to directly construct the optimal feedback representation of hedging or investment decisions, are analyzed in this framework demonstrating their effectiveness as well as their susceptibility to generalization error. Use of classical techniques shows that over-training renders trained investment decisions to become anticipative, and proves overlearning for large hypothesis spaces. On the other hand, non-asymptotic estimates based on Rademacher complexity show the convergence for sufficiently large training sets. These results emphasize the importance of synthetic data generation and the appropriate calibration of complex models to market data. A numerically studied stylized example illustrates these possibilities, including the importance of problem dimension in the degree of overlearning, and the effectiveness of this approach.

stat.ML↗

arXiv2022

Deep Stochastic Optimization in Finance

This paper outlines, and through stylized examples evaluates a novel and highly effective computational technique in quantitative finance. Empirical Risk Minimization (ERM) and neural networks are key to this approach. Powerful open source optimization libraries allow for efficient implementations of this algorithm making it viable in high-dimensional structures. The free-boundary problems related to American and Bermudan options showcase both the power and the potential difficulties that specific applications may face. The impact of the size of the training data is studied in a simplified Merton type problem. The classical option hedging problem exemplifies the need of market generators or large number of simulations.

q-fin.CP↗

arXiv2021

Viability and Arbitrage under Knightian Uncertainty

We reconsider the microeconomic foundations of financial economics. Motivated by the importance of Knightian Uncertainty in markets, we present a model that does not carry any probabilistic structure ex ante, yet is based on a common order. We derive the fundamental equivalence of economic viability of asset prices and absence of arbitrage. We also obtain a modified version of the Fundamental Theorem of Asset Pricing using the notion of sublinear pricing measures. Different versions of the Efficient Market Hypothesis are related to the assumptions one is willing to impose on the common order.

econ.GN↗

arXiv2020

Martingale optimal transport duality

We obtain a dual representation of the Kantorovich functional defined for functions on the Skorokhod space using quotient sets. Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. These sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given. Another important ingredient of our analysis is a regularized version of Jakubowski's $S$-topology on the Skorokhod space.

math.FA↗