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Researcher profile

Christoph Reisinger

Christoph Reisinger contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.NANumerical AnalysisMachine Learningmath.PRmath.OCq-fin.CPq-fin.RMq-fin.STComputational Engineering, Finance, and Sciencemath-phmath.APmath.MPq-fin.MF

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Published work

15 published item(s)

preprint2026arXiv

A higher order sparse grid combination technique

We show that a generalised sparse grid combination technique which combines multi-variate extrapolation of finite difference solutions with the standard combination formula lifts a second order accurate scheme on regular meshes to a fourth order combined sparse grid solution. In the analysis, working in a general dimension, we characterise all terms in a multivariate error expansion of the scheme as solutions of a sequence of semi-discrete problems. This is first carried out formally under suitable assumptions on the truncation error of the scheme, stability and regularity of solutions. We then verify the assumptions on the example of the Poisson problem with smooth data, and illustrate the practical convergence in up to seven dimensions.

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preprint2026arXiv

Uniform Scaling Limits in AdamW-Trained Transformers

We study the large-depth limit of transformers trained with AdamW, by modelling the hidden-state dynamics as an interacting particle system (IPS) coupled through the attention mechanism. Under appropriate scaling of the attention heads, we prove that the joint dynamics of the hidden states and backpropagated variables converge in $L^2$, uniformly over the initial condition, to the solution of a forward--backward system of ODEs at rate $\mathcal O(L^{-1}+L^{-1/3}H^{-1/2})$. Here, $L$ and $H$ denote the depth and number of heads of the transformer, respectively. The limiting system of ODEs can be identified with a McKean--Vlasov ODE (MVODE) when the attention heads do not incorporate causal masking. By using the flow maps associated with this MVODE and applying concentration of measure techniques, we obtain bounds on the difference between the discrete and continuous models that are uniform over compact sets of initial conditions. As this is achieved without resorting to a covering argument, the constants in our bounds are independent of the number of tokens. Furthermore, under a suitable adaptation to AdamW, the bounds become independent of the token embedding dimension.

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preprint2026arXiv

Weighted Conditional Flow Matching

Conditional flow matching (CFM) has emerged as a powerful framework for training continuous normalizing flows due to its computational efficiency and effectiveness. However, standard CFM often produces paths that deviate significantly from straight-line interpolations between prior and target distributions, making generation slower and less accurate due to the need for fine discretization at inference. Recent methods enhance CFM performance by inducing shorter and straighter trajectories but typically rely on computationally expensive mini-batch optimal transport (OT). Drawing insights from entropic optimal transport (EOT), we propose Weighted Conditional Flow Matching (W-CFM), a novel approach that modifies the classical CFM loss by weighting each training pair $(x, y)$ with a Gibbs kernel. We show that this weighting recovers the entropic OT coupling up to some bias in the marginals, and we provide the conditions under which the marginals remain nearly unchanged. Moreover, we establish an equivalence between W-CFM and the minibatch OT method in the large-batch limit, showing how our method overcomes computational and performance bottlenecks linked to batch size. Empirically, we test our method on unconditional generation on various synthetic and real datasets, confirming that W-CFM achieves comparable or superior sample quality, fidelity, and diversity to other alternative baselines while maintaining the computational efficiency of vanilla CFM.

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preprint2022arXiv

Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem

The supercooled Stefan problem and its variants describe the freezing of a supercooled liquid in physics, as well as the large system limits of systemic risk models in finance and of integrate-and-fire models in neuroscience. Adopting the physics terminology, the supercooled Stefan problem is known to feature a finite-time blow-up of the freezing rate for a wide range of initial temperature distributions in the liquid. Such a blow-up can result in a discontinuity of the liquid-solid boundary. In this paper, we prove that the natural Euler time-stepping scheme applied to a probabilistic formulation of the supercooled Stefan problem converges to the liquid-solid boundary of its physical solution globally in time, in the Skorokhod M1 topology. In the course of the proof, we give an explicit bound on the rate of local convergence for the time-stepping scheme. We also run numerical tests to compare our theoretical results to the practically observed convergence behavior.

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preprint2022arXiv

Estimating risks of option books using neural-SDE market models

In this paper, we examine the capacity of an arbitrage-free neural-SDE market model to produce realistic scenarios for the joint dynamics of multiple European options on a single underlying. We subsequently demonstrate its use as a risk simulation engine for option portfolios. Through backtesting analysis, we show that our models are more computationally efficient and accurate for evaluating the Value-at-Risk (VaR) of option portfolios, with better coverage performance and less procyclicality than standard filtered historical simulation approaches.

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preprint2022arXiv

Hedging option books using neural-SDE market models

We study the capability of arbitrage-free neural-SDE market models to yield effective strategies for hedging options. In particular, we derive sensitivity-based and minimum-variance-based hedging strategies using these models and examine their performance when applied to various option portfolios using real-world data. Through backtesting analysis over typical and stressed market periods, we show that neural-SDE market models achieve lower hedging errors than Black--Scholes delta and delta-vega hedging consistently over time, and are less sensitive to the tenor choice of hedging instruments. In addition, hedging using market models leads to similar performance to hedging using Heston models, while the former tends to be more robust during stressed market periods.

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preprint2022arXiv

Implicit and fully discrete approximation of the supercooled Stefan problem in the presence of blow-ups

We consider two implicit approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme studied in V. Kaushansky, C. Reisinger, M. Shkolnikov, and Z. Q. Song, arXiv:2010.05281, 2020, but here the flux over the free boundary and its velocity are coupled implicitly. Moreover, we extend the analysis to more general driving processes than Brownian motion. The second scheme is a Donsker-type approximation, also interpretable as an implicit finite difference scheme, for which global convergence is shown under minor technical conditions. With stronger assumptions, which apply in cases without blow-ups, we obtain additionally a convergence rate arbitrarily close to 1/2. Our numerical results suggest that this rate also holds for less regular solutions, in contrast to explicit schemes, and allow a sharper resolution of the discontinuous free boundary in the blow-up regime.

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preprint2020arXiv

A neural network based policy iteration algorithm with global $H^2$-superlinear convergence for stochastic games on domains

In this work, we propose a class of numerical schemes for solving semilinear Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the $H^2$-norm, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to HJBI boundary value problems corresponding to controlled diffusion processes with oblique boundary reflection. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

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preprint2020arXiv

Duality-based a posteriori error estimates for some approximation schemes for optimal investment problems

We consider a Markov chain approximation scheme for utility maximization problems in continuous time, which uses, in turn, a piecewise constant policy approximation, Euler-Maruyama time stepping, and a Gauss-Hermite approximation of the Gaussian increments. The error estimates previously derived in Picarelli and Reisinger (2019) are asymmetric between lower and upper bounds due to the control approximation and improve on known results in the literature in the lower case only. In the present paper, we use duality results to obtain a posteriori upper error bounds which are empirically of the same order as the lower bounds. The theoretical results are confirmed by our numerical tests.

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preprint2020arXiv

Error estimates of penalty schemes for quasi-variational inequalities arising from impulse control problems

This paper proposes penalty schemes for a class of weakly coupled systems of Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) arising from stochastic hybrid control problems of regime-switching models with both continuous and impulse controls. We show that the solutions of the penalized equations converge monotonically to those of the HJBQVIs. We further establish that the schemes are half-order accurate for HJBQVIs with Lipschitz coefficients, and first-order accurate for equations with more regular coefficients. Moreover, we construct the action regions and optimal impulse controls based on the error estimates and the penalized solutions. The penalty schemes and convergence results are then extended to HJBQVIs with possibly negative impulse costs. We also demonstrate the convergence of monotone discretizations of the penalized equations, and establish that policy iteration applied to the discrete equation is monotonically convergent with an arbitrary initial guess in an infinite dimensional setting. Numerical examples for infinite-horizon optimal switching problems are presented to illustrate the effectiveness of the penalty schemes over the conventional direct control scheme.

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preprint2020arXiv

Executive stock option exercise with full and partial information on a drift change point

We analyse the optimal exercise of an executive stock option (ESO) written on a stock whose drift parameter falls to a lower value at a change point, an exponentially distributed random time independent of the Brownian motion driving the stock. Two agents, who do not trade the stock, have differing information on the change point, and seek to optimally exercise the option by maximising its discounted payoff under the physical measure. The first agent has full information, and observes the change point. The second agent has partial information and filters the change point from price observations. This scenario is designed to mimic the positions of two employees of varying seniority, a fully informed executive and a partially informed less senior employee, each of whom receives an ESO. The partial information scenario yields a model under the observation filtration $\widehat{\mathbb{F}}$ in which the stock drift becomes a diffusion driven by the innovations process, an $\widehat{\mathbb{F}}$-Brownian motion also driving the stock under $\widehat{\mathbb{F}}$, and the partial information optimal stopping value function has two spatial dimensions. We rigorously characterise the free boundary PDEs for both agents, establish shape and regularity properties of the associated optimal exercise boundaries, and prove the smooth pasting property in both information scenarios, exploiting some stochastic flow ideas to do so in the partial information case. We develop finite difference algorithms to numerically solve both agents' exercise and valuation problems and illustrate that the additional information of the fully informed agent can result in exercise patterns which exploit the information on the change point, lending credence to empirical studies which suggest that privileged information of bad news is a factor leading to early exercise of ESOs prior to poor stock price performance.

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preprint2020arXiv

Probabilistic error analysis for some approximation schemes to optimal control problems

We introduce a class of numerical schemes for optimal control problems based on a novel Markov chain approximation, which uses, in turn, a piecewise constant policy approximation, Euler-Maruyama time stepping, and a Gauss-Hermite approximation of the Gaussian increments. We provide lower error bounds of order arbitrarily close to 1/2 in time and 1/3 in space for Lipschitz viscosity solutions, coupling probabilistic arguments with regularization techniques as introduced by Krylov. The corresponding order of the upper bounds is 1/4 in time and 1/5 in space. For sufficiently regular solutions, the order is 1 in both time and space for both bounds. Finally, we propose techniques for further improving the accuracy of the individual components of the approximation.

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preprint2020arXiv

Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general Lévy noise.

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preprint2020arXiv

Well-posedness and tamed schemes for McKean-Vlasov Equations with Common Noise

In this paper, we first establish well-posedness of McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs) with common noise, possibly with coefficients having super-linear growth in the state variable. Second, we present stable time-stepping schemes for this class of McKean-Vlasov SDEs. Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for an interacting particle system associated with the McKean-Vlasov equation. We prove stability and strong convergence of order $1/2$ and $1$, respectively. To obtain our main results, we employ techniques from calculus on the Wasserstein space. The proof for the strong convergence of the tamed Milstein scheme only requires the coefficients to be once continuously differentiable in the state and measure component. To demonstrate our theoretical findings, we present several numerical examples, including mean-field versions of the stochastic $3/2$ volatility model and the stochastic double well dynamics with multiplicative noise.

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preprint2019arXiv

A numerical scheme for the quantile hedging problem

We consider the numerical approximation of the quantile hedging price in a non-linear market. In a Markovian framework, we propose a numerical method based on a Piecewise Constant Policy Timestepping (PCPT) scheme coupled with a monotone finite difference approximation. We prove the convergence of our algorithm combining BSDE arguments with the Barles & Jakobsen and Barles & Souganidis approaches for non-linear equations. In a numerical section, we illustrate the efficiency of our scheme by considering a financial example in a market with imperfections.

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