arXiv subjects

Christa Cuchiero

Publications and source records attributed to Christa Cuchiero.

At least 19 recordsLinked to original sources

arXiv2026

Dynamic Universal Approximation via Signature Controlled Differential Equations

We study signature controlled differential equations (Sig-CDEs), that is, path-dependent controlled differential equations (CDEs) whose vector fields factor through the signature map. Working on spaces of stopped Hölder paths, we develop an existence, uniqueness, and stability theory for general path-dependent CDEs, and translate these pathwise well-posedness criteria into conditions on the corresponding signature functionals. We then prove dynamic universality: simply parametrized Sig-CDEs approximate the solution path of any well-posed path-dependent CDE arbitrarily well, uniformly over bounded sets of controls and initial histories, with global variants obtained using weighted spaces. Within this framework, entire maps of group-like elements provide a specific class of Sig-CDEs. Using a new class of limiting tensor spaces, we recast Sig-CDEs as infinite-dimensional classical CDEs and prove their well-posedness via a gauge-type scaling argument, thereby establishing a principled way to lift generic path-dependent dynamics. Lastly, we study truncated Sig-CDEs as finite-dimensional differential equations on step$-N$ Lie groups under intrinsic conditions, that is, with well-posedness formulated in terms of the underlying group metric.

math.CA↗

arXiv2026

Signature SDEs from an affine and polynomial perspective

Signature stochastic differential equations (SDEs) constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are linear maps of their own signature, i.e. of iterated integrals of the process with itself, and therefore allow for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in the set of group-like elements of the extended tensor algebra. By relying on duality theory for affine or polynomial processes, we obtain explicit formulas in terms of converging power series for the Fourier-Laplace transform and the expected value of entire functions of the signature process' marginals. The coefficients of these power series are solutions of Riccati and linear ordinary differential equations (ODEs) with values in the extended tensor algebra, respectively, whose vector fields can be expressed in terms of the characteristics of the corresponding SDEs. We thus construct a class of stochastic processes that is universal (in a sense specified in the introduction) within Itô-diffusions with path-dependent characteristics and allows for an explicit characterization of the Fourier-Laplace transform and hence the full law on path space. The practical applicability of this affine and polynomial approach is illustrated by several numerical examples.

math.PR↗

arXiv2026

Quantitative Halmos-Savage theorems and robust large financial markets

We establish a quantitative version of the classical Halmos-Savage Theorem for convex, potentially non-dominated sets of probability measures and its dual counterpart, generalizing previous quantitative versions. These results are then used to derive robust versions of the fundamental theorem of asset pricing (FTAP) in large financial markets in a one-period setting, characterizing the absence of arbitrage under Knightian uncertainty. To this end, we consider robust formulations of no asymptotic arbitrage of first kind (NAA1), which is the large market analogue of ``No unbounded profit with bounded risk'' (NUPBR), as well as no asymptotic arbitrage of second kind (NAA2). Finally, we characterize asymptotic arbitrage of first and second kind in the robust one-period binomial model in terms of the model parameters.

math.PR↗

arXiv2026

Robust financial calibration: a Bayesian approach for neural SDEs

The paper presents a Bayesian framework for the calibration of financial models using neural stochastic differential equations (neural SDEs), for which we also formulate a global universal approximation theorem based on Barron-type estimates. The method is based on the specification of a prior distribution on the neural network weights and an adequately chosen likelihood function. The resulting posterior distribution can be seen as a mixture of different classical neural SDE models yielding robust bounds on the implied volatility surface. Both, historical financial time series data and option price data are taken into consideration, which necessitates a methodology to learn the change of measure between the risk-neutral and the historical measure. The key ingredient for a robust numerical optimization of the neural networks is to apply a Langevin-type algorithm, commonly used in the Bayesian approaches to draw posterior samples.

q-fin.CP↗

arXiv2025

Global universal approximation of functional input maps on weighted spaces

We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family to map the input weighted space to the hidden layer, on which a non-linear scalar activation function is applied to each neuron, and finally return the output via some linear readouts. Relying on Stone-Weierstrass theorems on weighted spaces, we can prove a global universal approximation result on weighted spaces for continuous functions going beyond the usual approximation on compact sets. This then applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks. As a further application of the weighted Stone-Weierstrass theorem we prove a global universal approximation result for linear functions of the signature. We also introduce the viewpoint of Gaussian process regression in this setting and emphasize that the reproducing kernel Hilbert space of the signature kernels are Cameron-Martin spaces of certain Gaussian processes. This paves a way towards uncertainty quantification for signature kernel regression.

stat.ML↗

arXiv2025

Functional Itô-formula and Taylor expansions for non-anticipative maps of càdlàg rough paths

We derive a functional Itô-formula for non-anticipative maps of rough paths, based on the approximation properties of the signature of càdlàg rough paths. This result is a functional extension of the Itô-formula for càdlàg rough paths (by Friz and Zhang (2018)), which coincides with the change of variable formula formulated by Dupire (2009) whenever the functionals' representations, the notions of regularity, and the integration concepts can be matched. Unlike these previous works, we treat the vertical (jump) pertubation via the Marcus transformation, which allows for incorporating path functionals where the second order vertical derivatives do not commute, as is the case for typical signature functionals. As a byproduct, we show that sufficiently regular non-anticipative maps admit a functional Taylor expansion in terms of the path's signature, leading to an important generalization of the recent results by Dupire and Tissot-Daguette (2022).

math.PR↗

arXiv2025

Polynomial McKean-Vlasov SDEs

We study a new class of McKean-Vlasov stochastic differential equations (SDEs), possibly with common noise, applying the theory of time-inhomogeneous polynomial processes. The drift and volatility coefficients of these SDEs depend on the state variables themselves as well as their conditional moments in a way that mimics the standard polynomial structure. Our approach leads to new results on the existence and uniqueness of solutions to such conditional McKean-Vlasov SDEs which are, to the best of our knowledge, not obtainable using standard methods. Moreover, we show in the case without common noise that the moments of these McKean-Vlasov SDEs can be computed by non-linear ODEs. As a by-product, this also yields new results on the existence and uniqueness of global solutions to certain ODEs.

math.PR↗

arXiv2024

Holomorphic jump-diffusions

We introduce a class of jump-diffusions, called holomorphic, of which the well-known classes of affine and polynomial processes are particular instances. The defining property concerns the extended generator, which is required to map a (subset of) holomorphic functions to themselves. This leads to a representation of the expectation of power series of the process' marginals via a potentially infinite dimensional linear ODE. We apply the same procedure by considering exponentials of holomorphic functions, leading to a class of processes named affine-holomorphic for which a representation for quantities as the characteristic function of power series is provided. Relying on powerful results from complex analysis, we obtain sufficient conditions on the process' characteristics which guarantee the holomorphic and affine-holomorphic properties and provide applications to several classes of jump-diffusions.

math.PR↗

arXiv2024

Signature Methods in Stochastic Portfolio Theory

In the context of stochastic portfolio theory we introduce a novel class of portfolios which we call linear path-functional portfolios. These are portfolios which are determined by certain transformations of linear functions of a collections of feature maps that are non-anticipative path functionals of an underlying semimartingale. As main example for such feature maps we consider the signature of the (ranked) market weights. We prove that these portfolios are universal in the sense that every continuous, possibly path-dependent, portfolio function of the market weights can be uniformly approximated by signature portfolios. We also show that signature portfolios can approximate the growth-optimal portfolio in several classes of non-Markovian market models arbitrarily well and illustrate numerically that the trained signature portfolios are remarkably close to the theoretical growth-optimal portfolios. Besides these universality features, the main numerical advantage lies in the fact that several optimization tasks like maximizing (expected) logarithmic wealth or mean-variance optimization within the class of linear path-functional portfolios reduce to a convex quadratic optimization problem, thus making it computationally highly tractable. We apply our method also to real market data based on several indices. Our results point towards out-performance on the considered out-of-sample data, also in the presence of transaction costs.

q-fin.MF↗

arXiv2024

Joint calibration to SPX and VIX options with signature-based models

We consider a stochastic volatility model where the dynamics of the volatility are described by a linear function of the (time extended) signature of a primary process which is supposed to be a polynomial diffusion. We obtain closed form expressions for the VIX squared, exploiting the fact that the truncated signature of a polynomial diffusion is again a polynomial diffusion. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log-price and the VIX squared as linear functions of the signature of the corresponding augmented process. This feature can then be efficiently used for pricing and calibration purposes. Indeed, as the signature samples can be easily precomputed, the calibration task can be split into an offline sampling and a standard optimization. We also propose a Fourier pricing approach for both VIX and SPX options exploiting that the signature of the augmented primary process is an infinite dimensional affine process. For both the SPX and VIX options we obtain highly accurate calibration results, showing that this model class allows to solve the joint calibration problem without adding jumps or rough volatility.

q-fin.MF↗

arXiv2024

Polynomial interacting particle systems and non-linear SPDEs for market capitalization curves

Motivated by the robustness of the capital distribution curves, we study the behavior of a certain polynomial equity market model as the number of companies goes to infinity. More precisely, we extend volatility-stabilized market models introduced by Fernholz et al. by allowing for a common noise term such that the models remain polynomial. As the number of companies approaches infinity, we show that the limit of the empirical measure of the $N$-company system converges to the unique solution of a degenerate, non-linear SPDE. The obtained limit also has a representation as the conditional probability of the solution to a certain McKean-Vlasov SDE. Together with its conditional, this is again a polynomial process for which we can prove pathwise uniqueness as well as regularity properties for the marginal densities. We also provide conditional propagation of chaos results and numerical implementations of the particle system as well as its limiting equations.

math.PR↗

arXiv2024

Polynomial Volterra processes

We study the class of continuous polynomial Volterra processes, which we define as solutions to stochastic Volterra equations driven by a continuous semimartingale with affine drift and quadratic diffusion matrix in the state of the Volterra process. To demonstrate the versatility of possible state spaces within our framework, we construct polynomial Volterra processes on the unit ball. This construction is based on a stochastic invariance principle for stochastic Volterra equations with possibly singular kernels. Similarly to classical polynomial processes, polynomial Volterra processes allow for tractable expressions of the moments in terms of the unique solution to a system of deterministic integral equations, which reduce to a system of ODEs in the classical case. By applying this observation to the moments of the finite-dimensional distributions we derive a uniqueness result for polynomial Volterra processes. Moreover, we prove that the moments are polynomials with respect to the initial condition, another crucial property shared by classical polynomial processes. The corresponding coefficients can be interpreted as a deterministic dual process and solve integral equations dual to those verified by the moments themselves. Additionally, we obtain a representation of the moments in terms of a pure jump process with killing, which corresponds to another non-deterministic dual process.

math.PR↗

arXiv2023

A new perspective on the fundamental theorem of asset pricing for large financial markets

In the context of large financial markets we formulate the notion of \emph{no asymptotic free lunch with vanishing risk} (NAFLVR), under which we can prove a version of the fundamental theorem of asset pricing (FTAP) in markets with an (even uncountably) infinite number of assets, as it is for instance the case in bond markets. We work in the general setting of admissible portfolio wealth processes as laid down by Y. Kabanov \cite{kab:97} under a substantially relaxed concatenation property and adapt the FTAP proof variant obtained in \cite{CT:14} for the classical small market situation to large financial markets. In the case of countably many assets, our setting includes the large financial market model considered by M. De Donno et al. \cite{DGP:05} and its abstract integration theory. The notion of (NAFLVR) turns out to be an economically meaningful "no arbitrage" condition (in particular not involving weak-$*$-closures), and, (NAFLVR) is equivalent to the existence of a separating measure. Furthermore we show -- by means of a counterexample -- that the existence of an equivalent separating measure does not lead to an equivalent $σ$-martingale measure, even in a countable large financial market situation.

q-fin.MF↗

arXiv2023

Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models

We prove a universal approximation theorem that allows to approximate continuous functionals of càdlàg (rough) paths uniformly in time and on compact sets of paths via linear functionals of their time-extended signature. Our main motivation to treat this question comes from signature-based models for finance that allow for the inclusion of jumps. Indeed, as an important application, we define a new class of universal signature models based on an augmented Lévy process, which we call Lévy-type signature models. They extend continuous signature models for asset prices as proposed e.g. by Arribas et al.(2020) in several directions, while still preserving universality and tractability properties. To analyze this, we first show that the signature process of a generic multivariate Lévy process is a polynomial process on the extended tensor algebra and then use this for pricing and hedging approaches within Lévy-type signature models.

math.PR↗

arXiv2023

Ramifications of generalized Feller theory

Generalized Feller theory provides an important analog to Feller theory beyond locally compact state spaces. This is very useful for solutions of certain stochastic partial differential equations, Markovian lifts of fractional processes, or infinite dimensional affine and polynomial processes which appear prominently in the theory of signature stochastic differential equations. We extend several folklore results related to generalized Feller processes, in particular on their construction and path properties, and provide the often quite sophisticated proofs in full detail. We also introduce the new concept of extended Feller processes and compare them with standard and generalized ones. A key example relates generalized Feller semigroups of algebra homomorphisms via the method of characteristics to transport equations and continuous semiflows on weighted spaces, i.e. a remarkably generic way to treat differential equations on weighted spaces. We also provide a counterexample, which shows that no condition of the basic definition of generalized Feller semigroups can be dropped.

math.PR↗

arXiv2023

Infinite-dimensional Wishart-processes

We introduce and analyse infinite dimensional Wishart processes taking values in the cone $S^+_1(H)$ of positive self-adjoint trace class operators on a separable real Hilbert space $H$. Our main result gives necessary and sufficient conditions for their existence, showing that these processes are necessarily of fixed finite rank almost surely, but they are not confined to a finite-dimensional subspace of $S^+_1(H)$. By providing explicit solutions to operator valued Riccati equations, we prove that their Fourier-Laplace transform is exponentially affine in the initial value. As a corollary, we obtain uniqueness in law as well as the Markov property. We actually show the explicit form of the Fourier-Laplace transform for a wide parameter regime, thereby also extending what is known in the finite-dimensional setting. Finally, under minor conditions on the parameters we prove the Feller property with respect to a slight refinement of the weak-$*$-topology on $S_1^+(H)$. Applications of our results range from tractable infinite-dimensional covariance modelling to the analysis of the limit spectrum of large random matrices.

math.PR↗

arXiv2022

Model-free Portfolio Theory: A Rough Path Approach

Based on a rough path foundation, we develop a model-free approach to stochastic portfolio theory (SPT). Our approach allows to handle significantly more general portfolios compared to previous model-free approaches based on F{ö}llmer integration. Without the assumption of any underlying probabilistic model, we prove a pathwise formula for the relative wealth process which reduces in the special case of functionally generated portfolios to a pathwise version of the so-called master formula of classical SPT. We show that the appropriately scaled asymptotic growth rate of a far reaching generalization of Cover's universal portfolio based on controlled paths coincides with that of the best retrospectively chosen portfolio within this class. We provide several novel results concerning rough integration, and highlight the advantages of the rough path approach by showing that (non-functionally generated) log-optimal portfolios in an ergodic It{ô} diffusion setting have the same asymptotic growth rate as Cover's universal portfolio and the best retrospectively chosen one.

math.PR↗

arXiv2022

Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem

We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash in order to limit defaults to a given proportion of entities. We prove that the value of the central agent's control problem converges as the number of defaultable institutions goes to infinity, and that it satisfies a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a regularized version of the corresponding mean field control problem using a policy gradient method. Our simulations show that the central agent's optimal strategy is to subsidise banks whose equity values lie in a non-trivial time-dependent region.

math.OC↗