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Researcher profile

Huyên Pham

Huyên Pham contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.OCmath.PRq-fin.CPMachine Learningmath.APmath.NAmath.STNeural and Evolutionary ComputingNumerical Analysisq-fin.TR

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Published work

11 published item(s)

preprint2026arXiv

Direct Estimation of Schrödinger Bridge Time-Series Drifts: Finite-Sample, Asymptotic, and Adaptive Guarantees

We study nonparametric estimation of Schrödinger bridge (SB) drifts from i.i.d.\ data observed on a single time interval. Starting from the conditional-ratio form of the Schrödinger bridge time-series (SBTS) drift formula, we analyze a direct Nadaraya--Watson plug-in estimator built from kernelized numerator and denominator terms. Unlike recent SB analyses based on entropic-OT potentials, Sinkhorn iterations, or iterative bridge solvers, our approach works directly at the drift level and isolates \emph{statistical error} from optimization, approximation, and discretization error. Under Hölder regularity, a marginal-density floor, and bounded support, we prove a uniform non-asymptotic bound for admissible bandwidth pairs, a pointwise CLT under genuine undersmoothing, and an adaptive bandwidth selector satisfying an oracle inequality. We also prove a pivot-local minimax lower bound which, through an explicit uniform pivot, yields a global minimax lower bound under transparent compatibility conditions; hence the adaptive selector is minimax-rate optimal up to logarithmic factors. Synthetic experiments provide theorem-targeted diagnostics for finite-sample scaling, Gaussian approximation, and adaptive behavior.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Learning Generative Dynamics with Soft Law Constraints: A McKean-Vlasov FBSDE Approach

We propose a generative framework for learning stochastic dynamics from endpoint and intermediate distributional observations. The method formulates generation as a McKean-Vlasov control problem in which terminal and time-marginal laws are enforced through soft energy constraints. The associated optimality system is a forward-backward stochastic differential equation (FBSDE) whose backward component receives a continuous drift induced by the marginal law penalties. This provides a principled alternative to hard interpolation or optimal transport maps between observed distributions: the model learns a stochastic path law whose dynamics remain globally coupled through the mean-field objective. We derive the reduced FBSDE system for quadratic control cost and constant diffusion, connecting terminal and marginal law flat derivatives to score-like training signals. The resulting neural solver is evaluated on low-dimensional distributional benchmarks, where it recovers smooth stochastic paths matching prescribed marginal laws. In a higher-dimensional ALAE latent space, endpoint supervision is used as a qualitative stress test for transporting non-smiling faces toward smiling ones in a pretrained representation. We then use articulated human motion as a structured high-dimensional case study on a curated AMASS low-to-high position dataset, using SMPL-H pose sequences and reduced pose representations. The experiments show that soft marginal law constraints can produce coherent stochastic trajectories whose intermediate distributions follow the observed evolution of human motion. The code is available at https://github.com/murex/deep-mkv-gen/tree/main.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Optimal control of McKean-Vlasov systems under partial observation and hidden Markov switching

We study a class of mean-field control problems under partial observation. The controlled dynamics are of McKean-Vlasov type and are subject to regime switching driven by a hidden Markov chain. The observation process depends on the control and on the joint distribution of the state and control, which prevents the direct application of standard filtering techniques. The main contribution of this paper is to show how this distribution dependence can be handled within a change-of-probability framework, leading to a well-posed separated control problem. We derive a Zakai equation with a specific structure for the unnormalized filter, and show that the corresponding value function satisfies a dynamic programming principle. This yields a Bellman equation posed on a convex subset of a Wasserstein space, characterizing the optimal control problem under partial observation.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

DeepSets and their derivative networks for solving symmetric PDEs

Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we introduce a class of PDEs that are invariant to permutations, and called symmetric PDEs. Such problems are widespread, ranging from cosmology to quantum mechanics, and option pricing/hedging in multi-asset market with exchangeable payoff. Our main application comes actually from the particles approximation of mean-field control problems. We design deep learning algorithms based on certain types of neural networks, named PointNet and DeepSet (and their associated derivative networks), for computing simultaneously an approximation of the solution and its gradient to symmetric PDEs. We illustrate the performance and accuracy of the PointNet/DeepSet networks compared to classical feedforward ones, and provide several numerical results of our algorithm for the examples of a mean-field systemic risk, mean-variance problem and a min/max linear quadratic McKean-Vlasov control problem.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Differential learning methods for solving fully nonlinear PDEs

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control representation form, and the corresponding optimal feedback control is estimated using a neural network. Next, three different methods are presented to approximate the associated value function, i.e., the solution of the initial PDE, on the entire space-time domain of interest. The proposed deep learning algorithms rely on various loss functions obtained either from regression or pathwise versions of the martingale representation and its differential relation, and compute simultaneously the solution and its derivatives. Compared to existing methods, the addition of a differential loss function associated to the gradient, and augmented training sets with Malliavin derivatives of the forward process, yields a better estimation of the PDE's solution derivatives, in particular of the second derivative, which is usually difficult to approximate. Furthermore, we leverage our methods to design algorithms for solving families of PDEs when varying terminal condition (e.g. option payoff in the context of mathematical finance) by means of the class of DeepOnet neural networks aiming to approximate functional operators. Numerical tests illustrate the accuracy of our methods on the resolution of a fully nonlinear PDE associated to the pricing of options with linear market impact, and on the Merton portfolio selection problem.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes.Using the standard notion of viscosity solution à la Crandall-Lions extended to our Wasserstein setting,we prove a comparison result under general conditions on the drift and reward coefficients, whichcoupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation.This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative $n$-player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharp estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland's variational principle on the Wasserstein space.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Quantitative propagation of chaos for mean field Markov decision process with common noise

We investigate propagation of chaos for mean field Markov Decision Process with common noise (CMKV-MDP), and when the optimization is performed over randomized open-loop controls on infinite horizon. We first state a rate of convergence of order $M_N^γ$, where $M_N$ is the mean rate of convergence in Wasserstein distance of the empirical measure, and $γ\in (0,1]$ is an explicit constant, in the limit of the value functions of $N$-agent control problem with asymmetric open-loop controls, towards the value function of CMKV-MDP. Furthermore, we show how to explicitly construct $(ε+\mathcal{O}(M_N^γ))$-optimal policies for the $N$-agent model from $ε$-optimal policies for the CMKV-MDP. Our approach relies on sharp comparison between the Bellman operators in the $N$-agent problem and the CMKV-MDP, and fine coupling of empirical measures.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Markowitz portfolio selection for multivariate affine and quadratic Volterra models

This paper concerns portfolio selection with multiple assets under rough covariance matrix. We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coefficients, the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equations. This framework includes multivariate rough Heston models and extends the results of \cite{han2019mean}. In the quadratic case, we obtain new analytic formulae for the the Riccati BSDE and we establish their link with infinite dimensional Riccati equations. This covers rough Stein-Stein and Wishart type covariance models. Numerical results on a two dimensional rough Stein-Stein model illustrate the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategy. In particular for positively correlated assets, we find that the optimal strategy in our model is a `buy rough sell smooth' one.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Algorithmic trading in a microstructural limit order book model

We propose a microstructural modeling framework for studying optimal market making policies in a FIFO (first in first out) limit order book (LOB). In this context, the limit orders, market orders, and cancel orders arrivals in the LOB are modeled as Cox point processes with intensities that only depend on the state of the LOB. These are high-dimensional models which are realistic from a micro-structure point of view and have been recently developed in the literature. In this context, we consider a market maker who stands ready to buy and sell stock on a regular and continuous basis at a publicly quoted price, and identifies the strategies that maximize her P\&L penalized by her inventory. We apply the theory of Markov Decision Processes and dynamic programming method to characterize analytically the solutions to our optimal market making problem. The second part of the paper deals with the numerical aspect of the high-dimensional trading problem. We use a control randomization method combined with quantization method to compute the optimal strategies. Several computational tests are performed on simulated data to illustrate the efficiency of the computed optimal strategy. In particular, we simulated an order book with constant/ symmet-ric/ asymmetrical/ state dependent intensities, and compared the computed optimal strategy with naive strategies. Some codes are available on https://github.com/comeh.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Deep backward schemes for high-dimensional nonlinear PDEs

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in \cite{weinan2017deep} when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minimaas it can be the case with the algorithm designed in \cite{weinan2017deep}, and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Mean-variance portfolio selection with tracking error penalization

This paper studies a variation of the continuous-time mean-variance portfolio selection where a tracking-error penalization is added to the mean-variance criterion. The tracking error term penalizes the distance between the allocation controls and a reference portfolio with same wealth and fixed weights. Such consideration is motivated as follows: (i) On the one hand, it is a way to robustify the mean-variance allocation in case of misspecified parameters, by "fitting" it to a reference portfolio that can be agnostic to market parameters; (ii) On the other hand, it is a procedure to track a benchmark and improve the Sharpe ratio of the resulting portfolio by considering a mean-variance criterion in the objective function. This problem is formulated as a McKean-Vlasov control problem. We provide explicit solutions for the optimal portfolio strategy and asymptotic expansions of the portfolio strategy and efficient frontier for small values of the tracking error parameter. Finally, we compare the Sharpe ratios obtained by the standard mean-variance allocation and the penalized one for four different reference portfolios: equal-weights, minimum-variance, equal risk contributions and shrinking portfolio. This comparison is done on a simulated misspecified model, and on a backtest performed with historical data. Our results show that in most cases, the penalized portfolio outperforms in terms of Sharpe ratio both the standard mean-variance and the reference portfolio.

0 citations0 reviewsNo code linkedOpen access