arXiv subjects

Gudmund Pammer

Publications and source records attributed to Gudmund Pammer.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic Knothe-Rosenblatt: Light-speed Calibration of Stochastic Local Volatility Models

European option smiles determine the risk-neutral marginal laws of an asset, but not their intertemporal coupling, which is decisive for many applications. The Bass martingale construction selects, among all calibrated martingales, the one closest to Bachelier dynamics; it permits fast calibration at discrete maturities and recovers the Dupire local-volatility (LV) model as the maturity grid is refined. This article develops a modular calibration overlay for existing stochastic and path-dependent volatility models. We sequentially construct a martingale that matches all prescribed marginals exactly while remaining as close as possible, in an adapted Knothe-Rosenblatt sense, to the reference dynamics. As with the Bass LV model, each calibration step is amenable to an efficient Martingale Sinkhorn algorithm. We develop the theoretical foundations, numerical implementation and consider convergence to the SLV model. We also benchmark the method for Heston and Bergomi finite-factor path-dependent volatility dynamics and develop the multi-asset extension.

q-fin.PR↗

arXiv2026

Dynamic reinsurance via martingale transport

We formulate a dynamic reinsurance problem in which the insurer seeks to satisfy prescribed terminal moment or risk-based constraints while minimizing the $L^2$-norm of the ceded risk. As a tool for this analysis, we first use techniques from martingale optimal transport to study the auxiliary problem in which the insurer matches a given terminal distribution of the surplus process. We show that, under suitable assumptions, this auxiliary problem admits a tractable solution analogous to the Bass martingale. We then relax this condition by only requiring certain moment or risk-based constraints.

q-fin.RM↗

arXiv2026

A weak transport approach to the Schrödinger-Bass bridge

We study the Schrödinger-Bass problem, a one-parameter family of semimartingale optimal transport problems indexed by $β>0$, whose limiting regimes interpolate between the classical Schrödinger bridge, the Brenier-Strassen problem, and, after rescaling, the martingale Benamou-Brenier (Bass) problem. For each $β>0$, we prove that the dynamic Schrödinger-Bass problem is equivalent to a static weak optimal transport (WOT) problem with explicit cost $C_{\mathrm{SB}}^β$. This yields primal and dual attainment, as well as a structural characterization of the optimal semimartingales. The cost $C_{\mathrm{SB}}^β$ is constructed via infimal convolution and deconvolution of the Schrödinger cost with the Wasserstein distance. In a broader setting, we show that such infimal convolutions preserve the WOT structure and inherit continuity, coercivity, and stability of both values and optimizers with respect to the marginals. Building on this formulation, we propose a dual ascent algorithm for numerical computation. We establish monotone improvement of the dual objective and convergence of the iteration to the unique optimizer under suitable integrability assumptions.

math.PR↗

arXiv2026

Tighter Learning Guarantees on Digital Computers via Concentration of Measure on Finite Spaces

Machine learning models with inputs in a Euclidean space $\mathbb{R}^d$, when implemented on digital computers, generalize, and their generalization gap converges to $0$ at a rate of $c/N^{1/2}$ concerning the sample size $N$. However, the constant $c>0$ obtained through classical methods can be large in terms of the ambient dimension $d$ and machine precision, posing a challenge when $N$ is small to realistically large. In this paper, we derive a family of generalization bounds $\{c_m/N^{1/(2\vee m)}\}_{m=1}^{\infty}$ tailored for learning models on digital computers, which adapt to both the sample size $N$ and the so-called geometric representation dimension $m$ of the discrete learning problem. Adjusting the parameter $m$ according to $N$ results in significantly tighter generalization bounds for practical sample sizes $N$, while setting $m$ small maintains the optimal dimension-free worst-case rate of $\mathcal{O}(1/N^{1/2})$. Notably, $c_{m}\in \mathcal{O}(m^{1/2})$ for learning models on discretized Euclidean domains. Furthermore, our adaptive generalization bounds are formulated based on our new non-asymptotic result for concentration of measure in finite metric spaces, established via leveraging metric embedding arguments.

cs.LG↗

arXiv2026

Implicit Regularization of Large Neural Networks via Mean-Field Formulation

We propose a mathematical framework to explain implicit regularization from early stopping during the training of overparametrized neural networks. In the mean-field limit, the parameter distribution evolves according to a gradient flow on the space of probability measures. We show that these dynamics admit an equivalent McKean-Vlasov stochastic control formulation through the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The control viewpoint yields a Dynamic Programming Principle (DPP), which we use to define a new metric on probability measures. This metric can be viewed as a mean-field generalization of the control representation of the Wasserstein-2 distance, and it naturally appears as a regularization term selected by early stopping. We further obtain non-asymptotic bounds describing how the induced regularization depends on the stopping time.

math.OC↗

arXiv2026

The Martingale Sinkhorn Algorithm

We develop a numerical method for the martingale analogue of the Benamou--Brenier optimal transport problem, which seeks a martingale interpolating two prescribed marginals which is closest to the Brownian motion. Recent contributions have established existence of the optimal martingale under finite second moment assumptions on the marginals, but numerical methods exist only in the one-dimensional setting. We introduce an iterative scheme, a martingale analogue of the celebrated Sinkhorn algorithm, and prove that it yields a Bass potential in arbitrary dimension under minimal assumptions. In particular, we show that this holds when the marginals have finite moments of order $p > 1$, thereby extending the known theory beyond the finite-second-moment regime. The proof relies on a strict descent property for the dual value of the martingale Benamou--Brenier problem. While the descent property admits a direct verification in the case of compactly supported marginals, obtaining uniform control on the iterates without assuming compact support is substantially more delicate and constitutes the main technical challenge.

q-fin.CP↗

arXiv2026

The geometry of the adapted Bures--Wasserstein space

The adapted Bures--Wasserstein space consists of Gaussian processes endowed with the adapted Wasserstein distance. It can be viewed as the analogue of the classical Bures--Wasserstein space in optimal transport for the setting of stochastic processes, where the standard Wasserstein distance is inadequate and has to be replaced by its adapted counterpart. We develop a comprehensive geometric theory for the adapted Bures--Wasserstein space, thereby also providing the first results on the fine geometric structure of adapted optimal transport. In particular, we show that the adapted Bures--Wasserstein space is an Alexandrov space with non-negative curvature and provide explicit descriptions of tangent cones and exponential maps. Moreover, we show that Gaussian processes satisfying a natural non-degeneracy condition form a geodesically convex subspace. This subspace is characterized precisely by the property that its tangent cones are linear and hence coincide with the tangent space.

math.PR↗

arXiv2025

On the Inverse Flow Matching Problem in the One-Dimensional and Gaussian Cases

This paper studies the inverse problem of flow matching (FM) between distributions with finite exponential moment, a problem motivated by modern generative AI applications such as the distillation of flow matching models. Uniqueness of the solution is established in two cases - the one-dimensional setting and the Gaussian case. The general multidimensional problem remains open for future studies.

cs.LG↗

arXiv2025

A Brenier Theorem on $(P_2 (...P_2(H)...), W_2 )$ and Applications to Adapted Transport

We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\geq 1$, we construct a full-support probability $Λ$ on $P_2^{N}(H)= P_2(... P_2(H)...)$ that is transport regular: for every $Q$ with finite second moment, transporting $Λ$ to $Q$ with cost $W_2^2$ admits a unique optimizer, and this optimizer is of Monge type. The analysis rests on a characterization of optimal couplings on $P_2(H)$ and, more generally, on $P_2^{N}(H)$ via convex potentials on the Lions lift; in the latter case we employ a new adapted version of the lift tailored to the $N$-step structure. A key idea is a new identification between optimal-transport $c$-conjugation (with $c$ given by maximal covariance) and classical convex conjugation on the lift. A primary motivation comes from the adapted Wasserstein distance $AW_2$: our results yield a first Brenier theorem for $AW_2$ and characterize $AW_2^2$-optimal couplings through convex functionals on the space of $L_2$-processes.

math.PR↗

arXiv2025

Estimating causal distances with non-causal ones

The adapted Wasserstein ($AW$) distance refines the classical Wasserstein ($W$) distance by incorporating the temporal structure of stochastic processes. This makes the $AW$-distance well-suited as a robust distance for many dynamic stochastic optimization problems where the classical $W$-distance fails. However, estimating the $AW$-distance is a notably challenging task, compared to the classical $W$-distance. In the present work, we build a sharp estimate for the $AW$-distance in terms of the $W$-distance, for smooth measures. This reduces estimating the $AW$-distance to estimating the $W$-distance, where many well-established classical results can be leveraged. As an application, we prove a fast convergence rate of the kernel-based empirical estimator under the $AW$-distance, which approaches the Monte-Carlo rate ($n^{-1/2}$) in the regime of highly regular densities. These results are accomplished by deriving a sharp bi-Lipschitz estimate of the adapted total variation distance by the classical total variation distance.

math.PR↗

arXiv2025

Denseness of biadapted Monge mappings

Adapted or causal transport theory aims to extend classical optimal transport from probability measures to stochastic processes. On a technical level, the novelty is to restrict to couplings which are bicausal, i.e. satisfy a property which reflects the temporal evolution of information in stochastic processes. We show that in the case of absolutely continuous marginals, the set of bicausal couplings is obtained precisely as the closure of the set of (bi-) adapted processes. That is, we obtain an analogue of the classical result on denseness of Monge couplings in the set of Kantorovich transport plans: bicausal transport plans represent the relaxation of adapted mappings in the same manner as Kantorovich transport plans are the appropriate relaxation of Monge-maps.

math.PR↗

arXiv2025

Entropic adapted Wasserstein distance on Gaussians

The adapted Wasserstein distance is a metric for quantifying distributional uncertainty and assessing the sensitivity of stochastic optimization problems on time series data. A computationally efficient alternative to it, is provided by the entropically regularized adapted Wasserstein distance. Suffering from similar shortcomings as classical optimal transport, there are only few explicitly known solutions to those distances. Recently, Gunasingam--Wong provided a closed-form representation of the adapted Wasserstein distance between real-valued stochastic processes with Gaussian laws. In this paper, we extend their work in two directions, by considering multidimensional ($\mathbb{R}^d$-valued) stochastic processes with Gaussian laws and including the entropic regularization. In both settings, we provide closed-form solutions.

math.OC↗

arXiv2025

Strassen's theorem for biased convex order

Strassen's theorem asserts that for given marginal probabilities $μ,ν$ there exists a martingale starting in $μ$ and terminating in $ν$ if and only if $μ,ν$ are in convex order. From a financial perspective, it guarantees the existence of market-consistent martingale pricing measures for arbitrage-free prices of European call options and thus plays a fundamental role in robust finance. Arbitrage-free prices of American options demand a stronger version of martingales which are 'biased' in a specific sense. In this paper, we derive an extension of Strassen's theorem that links them to an appropriate strengthening of the convex order. Moreover, we provide a characterization of this order through integrals with respect to compensated Poisson processes.

math.PR↗

arXiv2025

Calibration of the Bass Local Volatility model

The Bass local volatility model introduced by Backhoff-Veraguas, Beiglböck, Huesmann, and Källblad is a Markov model perfectly calibrated to vanilla options at finitely many maturities, that approximates the Dupire local volatility model. Conze and Henry-Labordère show that its calibration can be achieved by solving a fixed-point equation. In this paper we complement the analysis and show existence and uniqueness of the solution to this equation, and that the fixed-point iteration scheme converges at a linear rate.

q-fin.MF↗

arXiv2025

Absolutely Continuous Curves of Stochastic Processes

We study absolutely continuous curves in the adapted Wasserstein space of filtered processes. We provide a probabilistic representation of such curves as flows of adapted processes on a common filtered probability space, extending classical results to the adapted setting. Moreover, we characterize geodesics in this space and derive an adapted Benamou--Brenier-type formula by reformulating adapted optimal transport as an energy minimization problem. As an application, we obtain a Skorokhod-type representation for sequences of filtered processes under the adapted weak topology.

math.PR↗

arXiv2025

Energy-Guided Continuous Entropic Barycenter Estimation for General Costs

Optimal transport (OT) barycenters are a mathematically grounded way of averaging probability distributions while capturing their geometric properties. In short, the barycenter task is to take the average of a collection of probability distributions w.r.t. given OT discrepancies. We propose a novel algorithm for approximating the continuous Entropic OT (EOT) barycenter for arbitrary OT cost functions. Our approach is built upon the dual reformulation of the EOT problem based on weak OT, which has recently gained the attention of the ML community. Beyond its novelty, our method enjoys several advantageous properties: (i) we establish quality bounds for the recovered solution; (ii) this approach seamlessly interconnects with the Energy-Based Models (EBMs) learning procedure enabling the use of well-tuned algorithms for the problem of interest; (iii) it provides an intuitive optimization scheme avoiding min-max, reinforce and other intricate technical tricks. For validation, we consider several low-dimensional scenarios and image-space setups, including non-Euclidean cost functions. Furthermore, we investigate the practical task of learning the barycenter on an image manifold generated by a pretrained generative model, opening up new directions for real-world applications. Our code is available at https://github.com/justkolesov/EnergyGuidedBarycenters.

cs.LG↗

arXiv2025

Multicausal transport: barycenters and dynamic matching

We introduce a multivariate version of causal transport, which we name multicausal transport, involving several filtered processes among which causality constraints are imposed. Subsequently, we consider the barycenter problem for stochastic processes with respect to causal and bicausal optimal transport, and study its connection to specific multicausal transport problems. Attainment and duality of the aforementioned problems are provided. As an application, we study a matching problem in a dynamic setting where agent types evolve over time. We link this to a causal barycenter problem and thereby show existence of equilibria.

math.PR↗

arXiv2025

The Fundamental Theorem of Weak Optimal Transport

The fundamental theorem of classical optimal transport establishes strong duality and characterizes optimizers through a complementary slackness condition. Milestones such as Brenier's theorem and the Kantorovich-Rubinstein formula are direct consequences. In this paper, we generalize this result to non-linear cost functions, thereby establishing a fundamental theorem for the weak optimal transport problem introduced by Gozlan, Roberto, Samson, and Tetali. As applications we provide concise derivations of the Brenier--Strassen theorem, the convex Kantorovich--Rubinstein formula and the structure theorem of entropic optimal transport. We also extend Strassen's theorem in the direction of Gangbo--McCann's transport problem for convex costs. Moreover, we determine the optimizers for a new family of transport problems which contains the Brenier--Strassen, the martingale Benamou--Brenier and the entropic martingale transport problem as extreme cases.

math.PR↗