arXiv subjects

Yumiharu Nakano

Publications and source records attributed to Yumiharu Nakano.

At least 19 recordsLinked to original sources

arXiv2026

Convergence of kernel and neural-network methods for Hamilton--Jacobi--Bellman equations on unbounded domains

We derive error bounds from empirical squared residuals for two non-monotone, mesh-free approximations of Hamilton--Jacobi--Bellman equations on unbounded spatial domains: collocation with Wendland kernels under a native space norm constraint, and physics-informed neural networks with smooth activations under a Sobolev-norm constraint. The main step is to recover the pointwise residual control needed for viscosity-solution error estimates from the empirical loss, while quantifying the discretization or sampling error. For the kernel method, we prove uniform convergence on compact sets under classical and native space regularity and derive an a posteriori quantitative error bound for the unique bounded continuous viscosity solution. For the neural-network method, we prove uniform convergence on compact sets in probability and give a quantitative high-probability error bound under classical Sobolev regularity. The bounds are governed by the fill distance for kernels and by the network and sample sizes for neural networks. The proofs combine empirical-to-population estimates, an interpolation inequality that turns an $L^2$ residual into a sup norm bound, and a controlled diffusion estimate for the error caused by truncating the domain. The regularity assumptions used to obtain convergence from approximation properties are additional to viscosity wellposedness. Numerical experiments in one and two space dimensions illustrate the methods and the estimates.

math.NA↗

arXiv2026

An inverse-stochastic-control Galerkin estimator for drift potentials

We consider the ill-posed inverse problem of recovering the drift potential of an overdamped Langevin diffusion from discrete, possibly irregular position observations. Specializing the suboptimality gap for inverse stochastic control to drift-potential recovery, we parametrize the associated value function in a finite basis. Under a basis nondegeneracy condition, the resulting Galerkin estimator is obtained from one positive-definite linear system whose entries are smooth functionals of the empirical time marginals; no transition density is required. The gap equals one half of the squared energy distance to the true potential, yielding a best-approximation identity and energy-seminorm consistency as the basis becomes complete. At a fixed basis, we separate the sample fluctuation from the time-quadrature error, thereby avoiding the Euler--Maruyama $O(Δt)$ drift-discretization bias. We also analyze the finite basis as regularization, give a dimension-dependent convergence rate for a concrete basis family, and bound observation-noise bias. One- and two-dimensional experiments verify the predicted sampling-interval behavior. An application to irregular animal telemetry illustrates the method on a single long trajectory.

math.OC↗

arXiv2026

Kernel-based potential mean-field games with unbiased random Fourier $U$-statistics

We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For population near-minimizers we prove convergence to the terminal-constrained problem as the penalty diverges, and show that the same limit is recovered almost surely when the learned controls are evaluated on independent finite samples under explicit coupling conditions on the penalty, random-feature count and sample size. The framework includes the kernel-MMD-penalty Schr{ö}dinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schr{ö}dinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.

math.OC↗

arXiv2026

Continuum-marginal optimal transport: a mesh-free kernel method

In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou--Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency. The same computational framework also applies to the Nelson problem.

math.OC↗

arXiv2026

Convergence of differentiable non-monotone schemes for fully nonlinear parabolic equations

We develop a convergence theory for non-monotone approximation schemes for fully nonlinear parabolic partial differential equations. Modern computational methods such as kernel-based collocation, spectral methods, physics-informed neural networks, and deep Galerkin methods are typically non-monotone, since they produce smooth approximate solutions and compute spatial derivatives directly from gradients of the chosen ansatz. Such schemes therefore lie outside the scope of the classical Barles and Souganidis convergence theory. We introduce an abstract framework that replaces strict monotonicity by two pointwise consistency conditions, on the PDE residual and on the terminal mismatch, both directly verifiable for a smooth approximating sequence. The technical key is a max-min representation of the nonlinearity that converts a vanishing classical residual into the viscosity subsolution and supersolution inequalities, and so dispenses with monotonicity in the abstract argument. The framework yields qualitative convergence under standard hypotheses, together with a quantitative error bound for Hamilton-Jacobi-Bellman equations on an unbounded spatial domain, in which the residual on an expanding truncation cylinder is balanced against an exponentially decaying tail term coming from the controlled stochastic differential equation underlying the value function. As a concrete realization, we analyze kernel-based collocation with Wendland radial basis functions, and present numerical experiments on a benchmark Hamilton-Jacobi-Bellman problem in one and two spatial dimensions that confirm the predicted convergence behaviour.

math.NA↗

arXiv2026

Inverse Stochastic Control via Generalized Schrödinger Problems

We propose a variational formulation of an inverse problem in continuous-time stochastic control, aimed at identifying control costs consistent with a given distribution over trajectories. The formulation is based on minimizing the suboptimality gap of observed behavior. We establish a connection between the inverse problem and a generalized dynamic Schrödinger problem, showing that their optimal values coincide. This result links inverse stochastic control with stochastic optimal transport, offering a new conceptual viewpoint on inverse inference in controlled diffusions.

math.OC↗

arXiv2026

Solving Monge problem by Hilbert space embeddings of probability measures

We propose deep learning methods for classical Monge's optimal mass transportation problems, where where the distribution constraint is treated as penalty terms defined by the maximum mean discrepancy in the theory of Hilbert space embeddings of probability measures. We prove that the transport maps given by the proposed methods converge to optimal transport maps in the problem with $L^2$ cost. Several numerical experiments validate our methods. In particular, we show that our methods are applicable to large-scale Monge problems. This is a corrected version of the ICORES 2025 proceedings paper.

math.OC↗

arXiv2026

Structural and Convergence Analysis of Discrete-Time Denoising Diffusion Probabilistic Models

This paper studies the original discrete-time denoising diffusion probabilistic model (DDPM) from a probabilistic point of view. We present three main theoretical results. First, we show that the time-dependent score function associated with the forward diffusion process admits a characterization as the backward component of a forward--backward stochastic differential equation (FBSDE). This result provides a structural description of the score function and clarifies how score estimation errors propagate along the reverse-time dynamics. As a by-product, we also obtain a system of semilinear parabolic PDEs for the score function. Second, we use tools from Schrödinger's problem to relate distributional errors arising in reverse time to corresponding errors in forward time. This approach allows us to control the reverse-time sampling error in a systematic way. Third, combining these results, we derive an explicit upper bound for the total variation distance between the sampling distribution of the discrete-time DDPM algorithm and the target data distribution under general finite noise schedules. The resulting bound separates the contributions of the learning error and the time discretization error. Our analysis highlights the intrinsic probabilistic structure underlying discrete-time DDPMs and provides a clearer understanding of the sources of error in their sampling procedure.

math.PR↗

arXiv2025

A kernel-based method for Schrödinger bridges

We characterize the Schrödinger bridge problems by a family of Mckean-Vlasov stochastic control problems with no terminal time distribution constraint. In doing so, we use the theory of Hilbert space embeddings of probability measures and then describe the constraint as penalty terms defined by the maximum mean discrepancy in the control problems. A sequence of the probability laws of the state processes resulting from $ε$-optimal controls converges to a unique solution of the Schrödinger's problem under mild conditions on given initial and terminal time distributions and an underlying diffusion process. We propose a neural SDE based deep learning algorithm for the Mckean-Vlasov stochastic control problems. Several numerical experiments validate our methods.

math.OC↗

arXiv2025

Convergence of meshfree collocation methods for fully nonlinear parabolic equations

We prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions, provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step. A numerical experiment with a radial basis function demonstrates the convergence property.

math.NA↗

arXiv2025

A deep learning approach to multi-marginal optimal transport via Hilbert space embeddings of probability measures

We propose a numerical method for solving the multi-marginal Monge problem, which extends the classical Monge formulation to settings involving multiple target distributions. Our approach is based on the Hilbert space embedding of probability measures and employs a penalization technique using the maximum mean discrepancy to enforce marginal constraints. The method is designed to be computationally efficient, enabling GPU-based implementation suitable for large-scale problems. We confirm the effectiveness of the proposed method through numerical experiments using synthetic data.

math.OC↗

arXiv2024

Weak approximation of Schrödinger-Föllmer diffusion

We consider the weak convergence of the Euler-Maruyama approximation for Schrödinger-Föllmer diffusions, which are solutions of Schrödinger bridge problems and can be used for sampling from given distributions. We show that the distribution of the terminal random variable of the time-discretized process weakly converges to the target one under mild regularity conditions.

math.PR↗

arXiv2022

Inverse stochastic optimal controls

We study an inverse problem of the stochastic optimal control of general diffusions with performance index having the quadratic penalty term of the control process. Under mild conditions on the system dynamics, the cost functions, and the optimal control process, we show that our inverse problem is well-posed using a stochastic maximum principle. Then, with the well-posedness, we reduce the inverse problem to some root finding problem of the expectation of a random variable involved with the value function, which has a unique solution. Based on this result, we propose a numerical method for our inverse problem by replacing the expectation above with arithmetic mean of observed optimal control processes and the corresponding state processes. The recent progress of numerical analyses of Hamilton-Jacobi-Bellman equations enables the proposed method to be implementable for multi-dimensional cases. In particular, with the help of the kernel-based collocation method for Hamilton-Jacobi-Bellman equations, our method for the inverse problems still works well even when an explicit form of the value function is unavailable. Several numerical experiments show that the numerical method recovers the unknown penalty parameter with high accuracy.

math.OC↗

arXiv2020

Kernel-based collocation methods for Heath-Jarrow-Morton models with Musiela parametrization

We propose kernel-based collocation methods for numerical solutions to Heath-Jarrow-Morton models with Musiela parametrization. The methods can be seen as the Euler-Maruyama approximation of some finite dimensional stochastic differential equations, and allow us to compute the derivative prices by the usual Monte Carlo methods. We derive a bound on the rate of convergence under some decay condition on the inverse of the interpolation matrix and some regularity conditions on the volatility functionals.

q-fin.CP↗

arXiv2019

Kernel-based collocation methods for Zakai equations

We examine an application of the kernel-based interpolation to numerical solutions for Zakai equations in nonlinear filtering, and aim to prove its rigorous convergence. To this end, we find the class of kernels and the structure of collocation points explicitly under which the process of iterative interpolation is stable. This result together with standard argument in error estimation shows that the approximation error is bounded by the order of the square root of the time step and the error that comes from a single step interpolation. Our theorem is well consistent with the results of numerical experiments.

math.NA↗

arXiv2016

On a law of large numbers for insurance risks

This note presents a kind of the strong law of large numbers for an insurance risk caused by a single catastrophic event rather than by an accumulation of independent and identically distributed risks. We derive this result by a large diversification effect resulting from optimal allocation of the risk to many reinsurers or investors.

q-fin.RM↗

arXiv2015

Quasi-Monte Carlo methods for Choquet integrals

We propose numerical integration methods for Choquet integrals where the capacities are given by distortion functions of an underlying probability measure. It relies on the explicit representation of the integrals for step functions and can be seen as quasi-Monte Carlo methods in this framework. We give bounds on the approximation errors in terms of the modulus of continuity of the integrand and the star discrepancy.

math.NA↗

arXiv2013

A liability tracking approach to long term management of pension funds

We propose a long term portfolio management method which takes into account a liability. Our approach is based on the LQG (Linear, Quadratic cost, Gaussian) control problem framework and then the optimal portfolio strategy hedges the liability by directly tracking a benchmark process which represents the liability. Two numerical results using empirical data published by Japanese organizations are served: simulations tracking an artificial liability and an estimated liability of Japanese organization. The latter one demonstrates that our optimal portfolio strategy can hedge his or her liability.

q-fin.PM↗