arXiv subjects

Nils Detering

Publications and source records attributed to Nils Detering.

At least 19 recordsLinked to original sources

arXiv2025

Monte Carlo on a single sample

In this paper, we consider a Monte Carlo simulation method (MinMC) that approximates prices and risk measures for a range $\Gamma$ of model parameters at once. The simulation method that we study has recently gained popularity [HS20, FPP22, BDG24], and we provide a theoretical framework and convergence rates for it. In particular, we show that sample-based approximations to $\mathbb{E}_{\theta}[X]$, where $\theta$ denotes the model and $\mathbb{E}_{\theta}$ the expectation with respect to the distribution $P_\theta$ of the model $\theta$, can be obtained across all $\theta \in \Gamma$ by minimizing a map $V:H\rightarrow \mathbb{R}$ with $H$ a suitable function space. The minimization can be achieved easily by fitting a standard feedforward neural network with stochastic gradient descent. We show that MinMC, which uses only one sample for each model, significantly outperforms a traditional Monte Carlo method performed for multiple values of $\theta$, which are subsequently interpolated. Our case study suggests that MinMC might serve as a new benchmark for parameter-dependent Monte Carlo simulations, which appear not only in quantitative finance but also in many other areas of scientific computing.

math.ST↗

arXiv2025

Learning from one graph: transductive learning guarantees via the geometry of small random worlds

Since their introduction by Kipf and Welling in $2017$, a primary use of graph convolutional networks is transductive node classification, where missing labels are inferred within a single observed graph and its feature matrix. Despite the widespread use of the network model, the statistical foundations of transductive learning remain limited, as standard inference frameworks typically rely on multiple independent samples rather than a single graph. In this work, we address these gaps by developing new concentration-of-measure tools that leverage the geometric regularities of large graphs via low-dimensional metric embeddings. The emergent regularities are captured using a random graph model; however, the methods remain applicable to deterministic graphs once observed. We establish two principal learning results. The first concerns arbitrary deterministic $k$-vertex graphs, and the second addresses random graphs that share key geometric properties with an Erd\H{o}s-R\'{e}nyi graph $\mathbf{G}=\mathbf{G}(k,p)$ in the regime $p \in \mathcal{O}((\log (k)/k)^{1/2})$. The first result serves as the basis for and illuminates the second. We then extend these results to the graph convolutional network setting, where additional challenges arise. Lastly, our learning guarantees remain informative even with a few labelled nodes $N$ and achieve the optimal nonparametric rate $\mathcal{O}(N^{-1/2})$ as $N$ grows.

stat.ML↗

arXiv2025

A class of locally state-dependent models for forward curves

We present a dynamic model for forward curves within the Heath-Jarrow-Morton framework under the Musiela parametrization. The forward curves take values in a function space H, and their dynamics follows a stochastic partial differential equation with state-dependent coefficients. In particular, the coefficients are defined through point-wise operating maps on H, resulting in a locally state-dependent structure. We first explore conditions under which these point-wise operators are well defined on H. Next, we determine conditions to ensure that the resulting coefficient functions satisfy local growth and Lipschitz properties, so to guarantee the existence and uniqueness of mild solutions. The proposed model captures the behavior of the entire forward curve through a single equation, yet retains remarkable simplicity. Notably, we demonstrate that certain one-dimensional projections of the model are Markovian and satisfy a one-dimensional stochastic differential equation. This connects our Hilbert-space approach to well established models for forward contracts with fixed delivery times, for which existing formulas and numerical techniques can be applied. This link allows us to examine also conditions for maintaining positivity of the solutions. As concrete examples, we analyze Hilbert-space valued variants of an exponential model and of a constant elasticity of variance model.

math.PR↗

arXiv2025

Solving Optimal Execution Problems via In-Context Operator Networks

We propose a novel transformer-based neural network architecture (ICON-OCnet) for solving optimal order execution problems in the presence of unknown price impact. Our architecture facilitates data-driven in-context operator learning for the incurred price impact by merging offline pre-training with online few-shot prompting inference. First, the operator learning component (ICON) learns the prevailing price impact environment from only a few executed trade and price impact trajectories (time series data) provided as context. Second, we employ ICON as a surrogate operator to train a neural network policy (OCnet) for the optimal order execution strategy for the price impact regime inferred from the in-context examples. We study the efficiency of our approach for linear propagator models with path-dependent transient price impact and explicitly known optimal execution strategies. In this model class, price impact persists and decays over time according to some propagator kernel. We illustrate that ICON is capable of accurately inferring the underlying price impact model from the data prompts, even for propagator kernels not seen in the training data. Moreover, we demonstrate that ICON-OCnet correctly retrieves the exact optimal order execution strategy for the model generating the in-context examples. Our introduced methodology is very general, offering a new approach to solving path-dependent optimal stochastic control problems sample-based with unknown state dynamics.

q-fin.TR↗

arXiv2024

Structure-informed operator learning for parabolic Partial Differential Equations

In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of functions. We utilize Fr\'echet space neural networks (Benth et al. (2023)) to address this operator learning problem. Our approach provides an alternative to Deep Operator Networks (DeepONets), using basis functions to span the relevant function spaces rather than relying on finite-dimensional approximations through censoring. With this method, structural information encoded in the basis coefficients is leveraged in the learning process. This results in a neural network designed to learn the mapping between infinite-dimensional function spaces. Our numerical proof-of-concept demonstrates the effectiveness of our method, highlighting some advantages over DeepONets.

math.AP↗

arXiv2024

Optimal Support for Distressed Subsidiaries -- a Systemic Risk Perspective

We consider a network of bank holdings, where every holding has two subsidiaries of different types. A subsidiary can trade with another holding's subsidiary of the same type. Holdings support their subsidiaries up to a certain level when they would otherwise fail to honor their financial obligations. We investigate the spread of contagion in this banking network when the number of bank holdings is large, and find the final number of defaulted subsidiaries under different rules for the holding support. We also consider resilience of this multilayered network to small shocks. Our work sheds light onto the role that holding structures can play in the amplification of financial stress. We find that depending on the capitalization of the network, a holding structure can be beneficial as compared to smaller separated entities. In other instances, it can be harmful and actually increase contagion. We illustrate our results in a numerical case study and also determine the optimal level of holding support from a regulator perspective.

q-fin.RM↗

arXiv2023

Reinforcement Learning Algorithm for Mixed Mean Field Control Games

We present a new combined \textit{mean field control game} (MFCG) problem which can be interpreted as a competitive game between collaborating groups and its solution as a Nash equilibrium between groups. Players coordinate their strategies within each group. An example is a modification of the classical trader's problem. Groups of traders maximize their wealth. They face cost for their transactions, for their own terminal positions, and for the average holding within their group. The asset price is impacted by the trades of all agents. We propose a three-timescale reinforcement learning algorithm to approximate the solution of such MFCG problems. We test the algorithm on benchmark linear-quadratic specifications for which we provide analytic solutions.

math.OC↗

arXiv2022

Reinforcement Learning for Intra-and-Inter-Bank Borrowing and Lending Mean Field Control Game

We propose a mean field control game model for the intra-and-inter-bank borrowing and lending problem. This framework allows to study the competitive game arising between groups of collaborative banks. The solution is provided in terms of an asymptotic Nash equilibrium between the groups in the infinite horizon. A three-timescale reinforcement learning algorithm is applied to learn the optimal borrowing and lending strategy in a data driven way when the model is unknown. An empirical numerical analysis shows the importance of the three-timescale, the impact of the exploration strategy when the model is unknown, and the convergence of the algorithm.

math.OC↗

arXiv2022

Bootstrap Percolation in Random Graphs of Unbounded Rank

Bootstrap percolation in (random) graphs is a contagion dynamic among a set of vertices with certain threshold levels. The process is started by a set of initially infected vertices, and an initially uninfected vertex with threshold $k$ gets infected as soon as the number of its infected neighbors reaches $k$. This process has been studied extensively in \textit{rank one} models. These models can generate random graphs with heavy-tailed degree sequences but they are not capable of generating networks with a flexible stochastic block structure. In this paper, we treat a class of random graphs of unbounded rank that can generate flexible stochastic block structures. Our main result determines the limit in probability of the final fraction of infected vertices from the fixed point of a non-linear operator defined on a suitable function space. We propose a neural network based algorithm to calculate this fixed point efficiently. We further derive criteria based on the Fr\'echet derivative of the operator that allow one to determine whether small infections spread through the entire graph or rather stay local.

math.PR↗

arXiv2022

Neural Networks in Fréchet spaces

We define a neural network in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fréchet space $\X$ into a Banach space $\Y$. The approximation results are generalising the well known universal approximation theorem for continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$, where approximation is done with (multilayer) neural networks [15, 25, 18, 29]. Our infinite dimensional networks are constructed using activation functions being nonlinear operators and affine transforms. Several examples are given of such activation functions. We show furthermore that our neural networks on infinite dimensional spaces can be projected down to finite dimensional subspaces with any desirable accuracy, thus obtaining approximating networks that are easy to implement and allow for fast computation and fitting. The resulting neural network architecture is therefore applicable for prediction tasks based on functional data.

math.FA↗

arXiv2022

Pricing options on flow forwards by neural networks in Hilbert space

We propose a new methodology for pricing options on flow forwards by applying infinite-dimensional neural networks. We recast the pricing problem as an optimization problem in a Hilbert space of real-valued function on the positive real line, which is the state space for the term structure dynamics. This optimization problem is solved by facilitating a novel feedforward neural network architecture designed for approximating continuous functions on the state space. The proposed neural net is built upon the basis of the Hilbert space. We provide an extensive case study that shows excellent numerical efficiency, with superior performance over that of a classical neural net trained on sampling the term structure curves.

q-fin.PR↗

arXiv2021

Suffocating Fire Sales

Fire sales are among the major drivers of market instability in modern financial systems. Due to iterated distressed selling and the associated price impact, initial shocks to some institutions can be amplified dramatically through the network induced by portfolio overlaps. In this paper, we develop a mathematical framework that allows us to investigate central characteristics that drive or hinder the propagation of distress. We investigate single systems as well as ensembles of systems that are alike, where similarity is measured in terms of the empirical distribution of all defining properties of a system. This asymptotic approach ensures a great deal of robustness to statistical uncertainty and temporal fluctuations. A characterization of those systems that are resilient to small shocks emerges, and we provide criteria that regulators might exploit in order to assess the stability of a financial system. We illustrate the application of these criteria for some exemplary configurations in the context of capital requirements and test the applicability of our results for systems of moderate size by Monte Carlo simulations.

q-fin.RM↗

arXiv2021

Accuracy of Deep Learning in Calibrating HJM Forward Curves

We price European-style options written on forward contracts in a commodity market, which we model with an infinite-dimensional Heath-Jarrow-Morton (HJM) approach. For this purpose we introduce a new class of state-dependent volatility operators that map the square integrable noise into the Filipović space of forward curves. For calibration, we specify a fully parametrized version of our model and train a neural network to approximate the true option price as a function of the model parameters. This neural network can then be used to calibrate the HJM parameters based on observed option prices. We conduct a numerical case study based on artificially generated option prices in a deterministic volatility setting. In this setting we derive closed pricing formulas, allowing us to benchmark the neural network based calibration approach. We also study calibration in illiquid markets with a large bid-ask spread. The experiments reveal a high degree of accuracy in recovering the prices after calibration, even if the original meaning of the model parameters is partly lost in the approximation step.

q-fin.MF↗

arXiv2021

Managing Default Contagion in Inhomogeneous Financial Networks

The aim of this paper is to quantify and manage systemic risk caused by default contagion in the interbank market. We model the market as a random directed network, where the vertices represent financial institutions and the weighted edges monetary exposures between them. Our model captures the strong degree of heterogeneity observed in empirical data and the parameters can easily be fitted to real data sets. One of our main results allows us to determine the impact of local shocks, where initially some banks default, to the entire system and the wider economy. Here the impact is measured by some index of total systemic importance of all eventually defaulted institutions. As a central application, we characterize resilient and non-resilient cases. In particular, for the prominent case where the network has a degree sequence without second moment, we show that a small number of initially defaulted banks can trigger a substantial default cascade. Our results complement and extend significantly earlier findings derived in the configuration model where the existence of a second moment of the degree distribution is assumed. As a second main contribution, paralleling regulatory discussions, we determine minimal capital requirements for financial institutions sufficient to make the network resilient to small shocks. An appealing feature of these capital requirements is that they can be determined locally by each institution without knowing the complete network structure as they basically only depend on the institution's exposures to its counterparties.

q-fin.RM↗

arXiv2020

Abstract polynomial processes

We suggest a novel approach to polynomial processes solely based on a polynomial action operator. With this approach, we can analyse such processes on general state spaces, going far beyond Banach spaces. Moreover, we can be very flexible in the definition of what "polynomial" means. We show that "polynomial process" universally means "affine drift". Simple assumptions on the polynomial action operators lead to stronger characterisations on the polynomial class of processes. In our framework we do not need to specify polynomials explicitly but can work with a general sequence of graded vector spaces of functions on the state space. Elements of these graded vector spaces form the monomials by introducing a sequence of vector space complements. The basic tool of our analysis is the polynomial action operator, which is a semigroup of operators mapping conditional expected values of monomials acting on the polynomial process to monomials of the same or lower grade. Unlike the classical Euclidean case, the polynomial action operator may not form a finite-dimensional subspace after a finite iteration, a property we call locally finite. We study abstract polynomial processes under both algebraic and topological assumptions on the polynomial actions, and establish an affine drift structure. Moreover, we characterize the covariance structure under similar but slightly stronger conditions. A crucial part in our analysis is the use of the (algebraic or topological) dual of the monomials of grade one, which serves as a linearization of the state space of the polynomial process. Our general framework covers polynomial processes with values in Banach spaces recently studied by Cuchiero and Svaluto-Ferro.

math.PR↗

arXiv2020

Independent increment processes: A multilinearity preserving property

We observe a multilinearity preserving property of conditional expectation for infinite dimensional independent increment processes defined on some abstract Banach space $B$. It is similar in nature to the polynomial preserving property analysed greatly for finite dimensional stochastic processes and thus offers an infinite dimensional generalisation. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessary a Banach algebra. In the special case of $B$ being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite dimensional polynomial processes in general.

math.PR↗

arXiv2020

Stochastic Volterra integral equations and a class of first order stochastic partial differential equations

We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued \Levy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.

math.PR↗

arXiv2019

Financial Contagion in a Generalized Stochastic Block Model

One of the most defining features of the global financial network is its inherent complex and intertwined structure. From the perspective of systemic risk it is important to understand the influence of this network structure on default contagion. Using sparse random graphs to model the financial network, asymptotic methods turned out powerful to analytically describe the contagion process and to make statements about resilience. So far, however, they have been limited to so-called {\em rank one} models in which informally the only network parameter is the degree sequence (see (Amini et. al. 2016) and (Detering et. al. 2019) for example) and the contagion process can be described by a one dimensional fix-point equation. These networks fail to account for a pronounced block structure such as core/periphery or a network composed of different connected blocks for different countries. We present a much more general model here, where we distinguish vertices (institutions) of different types and let edge probabilities and exposures depend on the types of both, the receiving and the sending vertex plus additional parameters. Our main result allows to compute explicitly the systemic damage caused by some initial local shock event, and we derive a complete characterisation of resilient respectively non-resilient financial systems. This is the first instance that default contagion is rigorously studied in a model outside the class of rank one models and several technical challenges arise. Moreover, in contrast to previous work, in which networks could be classified as resilient or non resilient, independent of the distribution of the shock, information about the shock becomes important in our model and a more refined resilience condition arises. Among other applications of our theory we derive resilience conditions for the global network based on subnetwork conditions only.

q-fin.RM↗