arXiv subjects

Roger J. A. Laeven

Publications and source records attributed to Roger J. A. Laeven.

At least 19 recordsLinked to original sources

arXiv2026

Graphon Hawkes Processes: Sparse Estimation and Network Recovery

Learning interaction structures from spatiotemporal event data is a key statistical problem in fields such as neuroscience, epidemiology, and seismology. We study this problem for Hawkes processes on a continuous spatial domain, where events may trigger future events at different locations. Following the framework developed in ``Spatiotemporal Hawkes processes with a graphon-induced connectivity structure'' by Baars, Laeven, and Mandjes (2026), we model the spatial interaction mechanism using an underlying graphon that characterizes how the intensities' excitation strength varies across locations. By expanding the baseline intensity, temporal excitation kernel and graphon in orthogonal bases, we represent the problem as a high-dimensional parametric inference problem. We propose a sparse penalized least-squares estimator that recovers the effective interaction structure while controlling model complexity. Our analysis establishes non-asymptotic error bounds in a high-dimensional regime where the number of parameters may grow with the observation horizon, without requiring the conditional intensity to be almost surely bounded. We further establish sparsistency results, showing that the latent interaction network can be consistently recovered. Simulation studies demonstrate the good finite sample performance of the proposed method and an application to large-scale neuronal spike data, comprising 1.81 million events, showcases its practical effectiveness.

math.ST↗

arXiv2026

Asymptotically distribution-free goodness-of-fit testing for point processes

Consider an observed path from a multivariate temporal point process $N$ with law $\mathbb P$ on the time interval $[0,T]$. To test the null hypothesis that $\mathbb P$ belongs to a given parametric family, we construct a compensated empirical process to which we apply a suitable innovation martingale transformation. We prove that the resulting test process converges weakly to a standard Wiener process. Consequently, taking a testing functional of this process yields an asymptotically distribution-free goodness-of-fit test for parametric point processes. Furthermore, for standard test statistics based on the increments of this test process, we establish consistency under alternative hypotheses. We assess the performance of the proposed testing procedure through a Monte Carlo simulation study and illustrate its practical utility with two real-data examples.

math.ST↗

arXiv2026

Optimal Stopping with Randomly Arriving Stopping Opportunities

We develop simulation-based methods to solve general optimal stopping problems with opportunities to stop that arrive randomly. Such problems occur naturally in a wide variety of applications with market frictions, such as limited liquidity, incomplete information and transaction costs. Our approach employs random time scales to map the original problem to an integer-time stopping problem with (possibly) infinite horizon. We introduce an infinite-horizon policy iteration algorithm to generate lower-biased estimates of the value function and establish a martingale dual representation to generate upper-biased estimates. Furthermore, we extend a family of backward dynamic programming methods that includes least-squares Monte Carlo. We illustrate the performance of our methods, their general applicability and the managerial implications of randomly arriving opportunities to stop in three examples: a stopped jump-diffusion that can be analyzed in closed form, a random-exercise version of a multi-dimensional max-call contract, and a real options problem in an illiquid market.

math.PR↗

arXiv2026

Geometric BSDEs

We introduce Geometric Backward Stochastic Differential Equations (GBSDEs) and two-driver BSDEs, which arise naturally in the geometric dynamics of dynamic return risk measures and of recursive portfolio choice. Through a reduction to auxiliary ordinary BSDEs with logarithmic and singular quadratic (LN-Q) growth rate $y|\ln (y)|+|z|^2/y$, we establish existence, regularity, uniqueness and stability of solutions, under both bounded and unbounded driver coefficients and terminal conditions, and we transfer these results to the original two-driver equations. We then deploy the theory in two applications. We solve a portfolio optimization problem under stochastic differential utility, in which the opportunity process satisfies an endogenously derived two-driver BSDE and optimality is established via our two-driver comparison theorem. We further apply GBSDEs to dynamic return and star-shaped risk measures, including (robust) $L^p$-norms, and characterize their positive homogeneity, star-shapedness and multiplicative convexity.

math.PR↗

arXiv2026

Generalized Orlicz premia

We introduce a generalized class of Orlicz premia based on possibly nonconvex loss functions, extending the classical framework of Haezendonck and Goovaerts (1982). Without the usual convexity requirement on the loss function $Φ$, the Orlicz framework naturally encompasses quantiles, expectiles and $L^p$-quantiles while preserving the fundamental properties of Orlicz premia. We show that within this framework cash-additivity axiomatizes $L^p$-quantiles, generalizing the classical `collapse-to-the-mean' result for cash-additive convex Orlicz premia into a `collapse-to-$L^p$-quantiles' result, with expectiles as a special case. We focus on two natural classes of nonconvex loss functions: concave-convex Orlicz functions, which mimic the idea of S-shaped value functions in prospect theory, and GA-convex Orlicz functions, which can be described in terms of comparative convexity with respect to a logarithmic reference, and for which the corresponding Orlicz premium is geometrically convex. Finally, we show that a suitable subclass of generalized Orlicz premia coincides with the class of law-invariant, monotone, positive, positively homogeneous, normalized functionals that are weakly lower semicontinuous, continuous from above, and whose level sets are convex with respect to mixtures (the so-called CxLS property).

q-fin.RM↗

arXiv2026

Asymptotic Analysis of Risk Premia Under Linear Risk Sharing with Law-Invariant Risk Measures

We investigate the asymptotic behavior of the risk premium associated with a linear risk sharing contract in an infinitely expanding risk pool. We consider general preferences represented by law-invariant robust utility functionals. These preferences encompass the rank-dependent utility model as a special case. We also examine Pareto optimality of general and, in particular, linear risk sharing rules with these preferences. Our analysis is not limited to the classical i.i.d. setting, but allows for heterogeneous risks. Two case studies on actuarial pricing for independent but heterogeneous risks illustrate our results.

q-fin.RM↗

arXiv2026

Financial Resilience Evaluation: From Conditional Expectations to Dynamic Convex Risk Measures

Financial resilience concerns the rate at which a position recovers, or further deteriorates, in response to adverse conditions. As a first step, Laeven, Ferrari, Rosazza Gianin, and Zullino (arXiv:2505.07502) introduced the resilience rate, defined as the expected instantaneous rate of (favorable) change of a price or risk-assessment process. Since this quantity captures only the conditional mean of future increments, it cannot distinguish between positions having the same expected recovery but different conditional risk profiles. We obtain a richer characterization by evaluating such increments through a genuine, possibly nonlinear, dynamic risk measure. More precisely, for an Itô process $π$ and a normalized, cash-additive dynamic risk measure $ρ$, we define the resilience evaluation by \[\mathcal D_s^ρπ_t := L^1\text{-}\lim_{\varepsilon\to0^+} \frac{1}{\varepsilon}ρ_s(π_{t+\varepsilon}-π_t), \qquad 0\leq s\leq t<T,\] whenever the limit exists. When $ρ$ is a convex dynamic risk measure induced by a BSDE with a Lipschitz or quadratic driver, we prove that this limit is well-posed and admits an explicit dual representation. It is given by the worst-case conditional expectation, over a zero-penalty class of measure changes, of an effective drift combining the drift of $π$ with the risk adjustment assigned by $ρ$ to its volatility. We further establish attainment of the optimal scenario and illustrate the scope of the construction, as well as the role of the assumptions, through examples and counterexamples.

q-fin.MF↗

arXiv2026

Goodness-of-Fit Testing for Point Processes in Large Populations

Suppose we have an observed path from a point process counting event occurrences in a large population. Based on the observed path, we would like to test the null hypothesis that the conditional intensity of the point process belongs to a particular parametric family. We propose a novel approach to conducting such goodness-of-fit tests. The idea is to construct a unitary transformation of a natural parametric testing process such that it converges weakly to a ``standard'' target process, independent of the particular parametric form assumed under the null hypothesis. This transformation therefore paves the way for asymptotically distribution-free goodness-of-fit testing of parametric point processes. We demonstrate the good finite-sample performance of our approach through Monte Carlo simulations of Aalen-type survival processes, without and with censoring, mixture cure models, and software reliability models, and we illustrate its applicability with observed human lifetimes as well as real software failures.

math.ST↗

arXiv2026

A Rank-Dependent Theory for Decision under Risk and Ambiguity

This paper axiomatizes, in a two-stage setup, a new theory for decision under risk and ambiguity. The axiomatized preference relation $\succeq$ on the space $\tilde{V}$ of random variables induces an ambiguity index $c$ on the space $Δ$ of probabilities, a probability weighting function $ψ$, generating the measure $ν_ψ$ by transforming an objective probability measure, and a utility function $ϕ$, such that, for all $\tilde{v},\tilde{u}\in\tilde{V}$, \begin{align*} \tilde{v}\succeq\tilde{u} \Leftrightarrow \min_{Q \in Δ} \left\{\mathbb{E}_Q\left[\intϕ\left(\tilde{v}^{\centerdot}\right)\,\mathrm{d}ν_ψ\right]+c(Q)\right\} \geq \min_{Q \in Δ} \left\{\mathbb{E}_Q\left[\intϕ\left(\tilde{u}^{\centerdot}\right)\,\mathrm{d}ν_ψ\right]+c(Q)\right\}. \end{align*} Our theory extends the rank-dependent utility model of Quiggin (1982) for decision under risk to risk and ambiguity, reduces to the variational preferences model when $ψ$ is the identity, and is dual to variational preferences when $ϕ$ is affine in the same way as the theory of Yaari (1987) is dual to expected utility. As a special case, we obtain a preference axiomatization of a decision theory that is a rank-dependent generalization of the popular maxmin expected utility theory. We characterize ambiguity aversion in our theory.

math.OC↗

arXiv2026

Measuring Financial Resilience Using Backward Stochastic Differential Equations

We introduce the resilience rate as a measure of financial resilience. It captures the expected rate at which a dynamic risk measure recovers, i.e., bounces back, when the risk-acceptance set is breached. We develop the corresponding stochastic calculus by establishing representation theorems for expected time-derivatives of solutions to backward stochastic differential equations (BSDEs) with jumps, evaluated at stopping times. These results reveal that the resilience rate can be represented as a suitable expectation of the generator of a BSDE. We analyze the main properties of the resilience rate and the formal connection of these properties to the BSDE generator. We also introduce resilience-acceptance sets and study their properties in relation to both the resilience rate and the dynamic risk measure. We illustrate our results in several canonical financial examples and highlight their implications via the notion of resilience neutrality.

q-fin.MF↗

arXiv2025

Delayed Hawkes birth-death processes

We introduce, and formally establish, a variant of the Hawkes-fed birth-death process -- the delayed Hawkes birth-death process -- in which the conditional intensity does not increase at arrivals but at departures from the system. In a scaling limit where sojourn times are stretched out by a factor $\sqrt T$, after which time gets contracted by a factor $T$, the delayed Hawkes process behaves markedly differently from its classical counterpart. We design a family of models admitting a cluster representation and containing the Hawkes and delayed Hawkes processes as special cases. The cluster representation allows for transform characterizations by a fixed-point equation and for analysis of heavy-tailed asymptotics. We compare the delayed Hawkes process to the classical Hawkes process using stochastic ordering, which enables us to describe stationary distributions and heavy-traffic behavior. In the Markovian network case, a recursive procedure is presented to calculate the $d$th-order moments analytically.

math.PR↗

arXiv2025

Sample Path Large Deviations for Multivariate Heavy-Tailed Hawkes Processes and Related Lévy Processes

In this paper, we develop sample path large deviations for multivariate Hawkes processes with heavy-tailed mutual excitation rates. Our results address a broad class of rare events in Hawkes processes at the sample path level and, via the cluster representation of Hawkes processes and a recent result on the tail asymptotics of the cluster sizes, unravel the most likely configurations of (multiple) large clusters that could trigger the target events. Our proof hinges on establishing the asymptotic equivalence, in terms of M-convergence, between a suitably scaled multivariate Hawkes process and a coupled Lévy process with multivariate hidden regular variation. Hence, along the way, we derive a sample path large deviations principle for a class of Lévy processes with multivariate hidden regular variation, which not only plays an auxiliary role in our analysis but is also of independent interest.

math.PR↗

arXiv2025

Robust Optimization of Rank-Dependent Models with Uncertain Probabilities

This paper studies distributionally robust optimization for a rich class of risk measures with ambiguity sets defined by $ϕ$-divergences. The risk measures are allowed to be non-linear in probabilities, are represented by Choquet integrals possibly induced by a probability weighting function, and encompass many well-known examples. Optimization for this class of risk measures is challenging due to their rank-dependent nature. We show that for various shapes of probability weighting functions, including concave, convex and inverse $S$-shaped, the robust optimization problem can be reformulated into a rank-independent problem. In the case of a concave probability weighting function, the problem can be reformulated further into a convex optimization problem that admits explicit conic representability for a collection of canonical examples. While the number of constraints in general scales exponentially with the dimension of the state space, we circumvent this dimensionality curse and develop two types of algorithms. They yield tight upper and lower bounds on the exact optimal value and are formally shown to converge asymptotically. This is illustrated numerically in a robust newsvendor problem and a robust portfolio choice problem.

math.OC↗

arXiv2025

Tail Asymptotics of Cluster Sizes in Multivariate Heavy-Tailed Hawkes Processes

We examine a distributional fixed-point equation related to a multi-type branching process that is key in the cluster sizes analysis of multivariate heavy-tailed Hawkes processes. Specifically, we explore the tail behavior of its solution and demonstrate the emergence of a form of multivariate hidden regular variation. Large values of the cluster size vector result from one or several significant jumps. A discrete optimization problem involving any given rare event set of interest determines the exact configuration of these large jumps and the degree of hidden regular variation. Our proofs rely on a detailed probabilistic analysis of the spatiotemporal structure of multiple large jumps in multi-type branching processes.

math.PR↗

arXiv2025

Higher-Order Ambiguity Attitudes

We introduce a model-free preference under ambiguity, as a primitive trait of behavior, which we apply once as well as repeatedly. Its single and double application yield simple, easily interpretable definitions of ambiguity aversion and ambiguity prudence. We derive their implications within canonical models for decision under risk and ambiguity. We establish in particular that our new definition of ambiguity prudence is equivalent to a positive third derivative of: (i) the capacity in the Choquet expected utility model, (ii) the dual conjugate of the divergence function under variational divergence preferences and (iii) the ambiguity attitude function in the smooth ambiguity model. We show that our definition of ambiguity prudent behavior may be naturally linked to an optimal insurance problem under ambiguity.

q-fin.RM↗

arXiv2024

Constructing Uncertainty Sets for Robust Risk Measures: A Composition of $ϕ$-Divergences Approach to Combat Tail Uncertainty

Risk measures, which typically evaluate the impact of extreme losses, are highly sensitive to misspecification in the tails. This paper studies a robust optimization approach to combat tail uncertainty by proposing a unifying framework to construct uncertainty sets for a broad class of risk measures, given a specified nominal model. Our framework is based on a parametrization of robust risk measures using two (or multiple) $ϕ$-divergence functions, which enables us to provide uncertainty sets that are tailored to both the sensitivity of each risk measure to tail losses and the tail behavior of the nominal distribution. In addition, our formulation allows for a tractable computation of robust risk measures, and elicitation of $ϕ$-divergences that describe a decision maker's risk and ambiguity preferences.

math.OC↗

arXiv2024

Spatiotemporal Hawkes processes with a graphon-induced connectivity structure

We introduce a spatiotemporal self-exciting point process $(N_t(x))$, boundedly finite both over time $[0,\infty)$ and space $\mathscr X$, with excitation structure determined by a graphon $W$ on $\mathscr{X}^2$. This graphon Hawkes process generalizes both the multivariate Hawkes process and the Hawkes process on a countable network, and despite being infinite-dimensional, it is surprisingly tractable. After proving existence, uniqueness and stability results, we show, both in the annealed and in the quenched case, that for compact, Euclidean $\mathscr X\subset\mathbb R^m$, any graphon Hawkes process can be obtained as the suitable limit of $d$-dimensional Hawkes processes $\tilde N^d$, as $d\to\infty$. Furthermore, in the stable regime, we establish an FLLN and an FCLT for our infinite-dimensional process on compact $\mathscr X\subset\mathbb R^m$, while in the unstable regime we prove divergence of $N_T(\mathscr X)/T$, as $T\to\infty$. Finally, we exploit a cluster representation to derive fixed-point equations for the Laplace functional of $N$, for which we set up a recursive approximation procedure. We apply these results to show that, starting with multivariate Hawkes processes $\tilde N^d_t$ converging to stable graphon Hawkes processes, the limits $d\to\infty$ and $t\to\infty$ commute.

math.PR↗

arXiv2024

On Geometrically Convex Risk Measures

Geometrically convex functions constitute an interesting class of functions obtained by replacing the arithmetic mean with the geometric mean in the definition of convexity. As recently suggested, geometric convexity may be a sensible property for financial risk measures ([7,13,4]). We introduce a notion of GG-convex conjugate, parallel to the classical notion of convex conjugate introduced by Fenchel, and we discuss its properties. We show how GG-convex conjugation can be axiomatized in the spirit of the notion of general duality transforms introduced in [2,3]. We then move to the study of GG-convex risk measures, which are defined as GG-convex functionals defined on suitable spaces of random variables. We derive a general dual representation that extends analogous expressions presented in [4] under the additional assumptions of monotonicity and positive homogeneity. As a prominent example, we study the family of Orlicz risk measures. Finally, we introduce multiplicative versions of the convex and of the increasing convex order and discuss related consistency properties of law-invariant GG-convex risk measures.

q-fin.RM↗