arXiv subjects

Max Nendel

Publications and source records attributed to Max Nendel.

At least 19 recordsLinked to original sources

arXiv2026

Extreme points of sets of probability measures and $\varphi$-divergences

In this work, we prove several equivalent characterizations of the extreme points of convex sets of probability measures of the form $\mathcal{M}=\mathcal{P} \cap H$, where $\mathcal{P}$ denotes the set of all probability measures on an arbitrary measurable space $(\Omega,\mathcal{F})$ and $H$ is an affine set of signed measures on $(\Omega,\mathcal{F})$ with finite variation. We first give a precise measure-theoretic formulation of the heuristic that extreme measures have minimal support. We then connect this with the notion of minimality with respect to absolute continuity, and prove that points that dominate no other element of $\mathcal{M}$ are the only ones realizing the blow-up of a certain divergence map for any suitable $\phi$-divergence. Finally, considering a different class of $\phi$-divergences, we recover a characterization of the extreme points of $\mathcal{M}$ as strict local maximizers of $\phi$-divergences relative to any suitable dominating measures. We apply our result to recover and complement results from the literature in the context of finite spaces, sets of measures defined by integral constraints, multi-marginal couplings, and dominated sets of probability measures.

math.PR↗

arXiv2026

A Small-Noise Analysis of Controlled Functional Differential Equations with Gaussian Noise

We study small-noise asymptotics for controlled functional differential equations driven by additive Gaussian noise. The Gaussian noise is modeled on an abstract Wiener space, covering both classical Brownian perturbations and non-Markovian perturbations such as fractional Brownian motion. The drift coefficient is assumed to be non-anticipative, Lipschitz continuous in the state path, and of linear growth. For bounded uniformly continuous cost functionals, we prove game-theoretic lower and upper bounds for the small-noise logarithmic value functions and identify their limit whenever the associated deterministic zero-sum game has a value. In the limiting game, one player chooses the drift control, while the other selects a Cameron--Martin shift of the Gaussian noise, penalized by the corresponding quadratic energy cost. We further provide sufficient Fan-type convexity and concavity conditions under which the game has a value, thereby obtaining a full small-noise Laplace principle. The proof combines the Bou\'e--Dupuis variational representation on abstract Wiener spaces with pathwise stability of the controlled solution map and adapted finite-dimensional approximations of Cameron--Martin shifts.

math.PR↗

arXiv2026

Optimal Market Making in Prediction Markets

Prediction markets are attracting growing attention as trading volumes rise and their practical relevance increases. To ensure efficient price discovery, liquidity provision becomes ever more important. Due to the binary settlement structure in prediction markets, optimal market making leads to an optimization problem that is fundamentally different from the ones studied in classical settings. In this paper, we develop a stochastic control framework for prediction markets in which the market price is modeled as a conditional probability of the outcome that is generated by a transformed latent belief diffusion. A market maker selects bid and ask quotes to maximize expected terminal wealth while controlling both mark-to-market inventory risk and the settlement risk of remaining positions at resolution. We derive the associated Hamilton--Jacobi--Bellman equation and characterize the unique optimal bid and ask quotes. By transforming the equation to the latent belief space and using a fixed-point argument, we prove existence and uniqueness of a classical solution and verify the resulting optimal quoting strategy. In addition, we provide a numerical analysis, which reveals how optimal liquidity provision in prediction markets depends on inventory, market beliefs, time to resolution, and risk aversion. Further, we demonstrate that the optimal quoting strategy substantially improves downside protection while preserving most of its expected profit relative to a myopic benchmark that maximizes the instantaneous expected mark-to-market profit.

q-fin.TR↗

arXiv2026

Hidden Dependence and Aggregate Tail Risk

We study risk aggregation problems for arbitrary non-decreasing aggregation functions and tail risk measures under dependence uncertainty in a distributionally robust setting. To this end, we introduce the notion of hidden dependence for random vectors, which is built on the concepts of risk concentration and common tail events developed in Wang and Zitikis (2020). We show that, starting from a tail event $A$ of the aggregate loss for an arbitrary random vector $Y$, one can construct a random vector with hidden dependence that dominates $Y$ on the tail event $A$. We then focus on the case in which model uncertainty is described by small perturbations of the distribution of a random vector with respect to a suitable probability distance without changing the marginals. We show that these perturbations of the reference distribution are compatible with hidden dependence and thus lead to the same worst-case risk bounds as in the unconstrained case for arbitrary $\gamma$-tail risk measures with a suitable level $\gamma$. Finally, we apply our results in a credit risk context and quantify the potential underestimation of portfolio risk arising from uncertainty in the dependence structure. In particular, we show that even small deviations from a reference Gaussian dependence model can, in principle, justify dramatic increases in capital requirements.

q-fin.RM↗

arXiv2026

Absolute Continuity of Monotone Aggregations under Positive Regression Dependence

In this paper, we provide a sufficient condition for the absolute continuity of one-dimensional push-forwards of dependent random vectors. Suppose that $X$ has an absolutely continuous distribution and that the conditional distribution of an $\mathbb{R}^d$-valued random vector $Y$ given $X=x$ is nondecreasing in $x\in \mathbb{R}$ in the usual stochastic order. For Borel maps $g\colon \mathbb{R}\times\mathbb{R}^d\to\mathbb{R}$ satisfying a coordinatewise monotonicity condition in $Y$ and a uniform lower-increment condition in $X$, we prove that $g(X,Y)$ has an absolutely continuous distribution. The result requires neither independence nor a joint density, and allows the marginal law of $Y$ to be completely arbitrary. Moreover, the result remains valid if $\mathbb{R}^d$ is replaced by an arbitrary measurable space endowed with a reflexive binary relation. We discuss consequences for monotone risk aggregation and extensions of the familiar regularization by convolution beyond independent random variables.

math.PR↗

arXiv2026

An optimal transport foundation for a class of dynamically consistent risk measures

We study a class of dynamically consistent risk measures that robustify a time-homogeneous Markovian reference model by allowing for distributional uncertainty in its transition laws. We start from one-step convex risk evaluations in which ambiguity is captured by penalized worst-case expectations over alternative transition laws. Imposing time consistency then yields a convex monotone semigroup on bounded continuous payoff functions, and this semigroup represents the associated dynamic risk measure. The semigroup is uniquely characterized by its risk generator. Under a lower bound on the family of penalties in terms of suitable optimal transport costs relative to the reference laws, we identify the generator on smooth test functions. For optimal transport bounds with linear small-time scaling, this produces a first-order, drift-type correction given by a convex Hamiltonian acting on the gradient. Under martingale transport constraints and a different scaling, however, the leading correction is genuinely of second order and is described by a convex monotone functional acting on the Hessian. We illustrate both regimes for Wasserstein and martingale Wasserstein penalizations and derive explicit formulas via convex conjugates of the underlying transport costs. The associated dynamic risk measures admit stochastic control representations in which the control acts on the drift in the first-order case and on the volatility in the second-order case.

q-fin.MF↗

arXiv2026

Asymptotic Behaviour of Unexpected Losses and Risk Ratios for Co-Monotonic Alternatives

The aggregation of individual risks in large credit and insurance portfolios is guided by diversification and the law of large numbers, which formalizes the convergence of sample averages to their means. At the same time, regulatory capital requirements and insurance premia are designed to provide a capital buffer or risk margin above the mean. The resulting excess, given by the difference between the nonlinear valuation of the aggregate loss and the corresponding mean, reflects the idea of protection against unexpected losses in the sense of banking and insurance regulation. This paper studies the asymptotic behaviour of this excess for large weighted portfolios. The main result shows that, for monotone cash-additive risk measures on Banach-lattice-valued Orlicz spaces, convergence along weighted averages satisfying a weak law of large numbers together with a uniform integrability condition is equivalent to scalar continuity at the origin. If the risk measure is positively homogeneous, this continuity condition is automatically satisfied, and we prove that the unexpected losses of large weighted portfolios are of order $o(n\overline\lambda_n)$, where $\overline\lambda_n$ denotes the average weight assigned to the first $n$ random variables. We establish analogous asymptotic results for Choquet insurance premia. Finally, we derive risk-ratio limits that quantify the potential underestimation arising when diversified portfolios are compared with co-monotonic alternatives.

q-fin.RM↗

arXiv2025

A Strict Comparison Principle for Integro-Differential Hamilton-Jacobi-Bellman Equations on Domains with Boundary

This work provides a comparison principle for viscosity solutions to boundary value problems on (partially) bounded, cylindrical spaces. The comparison principle is based on a test function framework, that allows for the simultaneous treatment of diffusive as well as jump terms. Estimates in the proof of the comparison principle incorporate the use of Lyapunov functions that act as growth bounds for the solutions, effectively yielding a theory for unbounded viscosity solutions. We apply the results to a wide class of parabolic equations and elliptic problems on a space with corners.

math.AP↗

arXiv2025

Scaling limits of multi-period distributionally robust optimization problems

We examine the scaling limit of multi-period distributionally robust optimization (DRO) problems via a semigroup approach. Each period involves a worst-case maximization over distributions in a Wasserstein ball around the transition probability of a reference process with radius proportional to the length of the period, and the multi-period DRO problem arises through its sequential composition. We show that the scaling limit of the multi-period DRO, as the length of each period tends to zero, is a strongly continuous monotone semigroup on $\mathrm{C_b}$. Furthermore, we show that its infinitesimal generator is equal to the generator associated with the non-robust scaling limit plus an additional perturbation term induced by the Wasserstein uncertainty. As an application, we show that when the reference process follows an It\^o process, the viscosity solution of the associated nonlinear PDE coincides with the value of continuous-time robust optimization problems under parametric uncertainty.

math.OC↗

arXiv2025

Chernoff-Mehler Approximation for L\'evy Processes with Drift

In this paper, we study an approximation scheme for L\'evy processes with drift in terms of a representation that is akin to the celebrated Mehler formula for L\'evy-Ornstein-Uhlenbeck processes. The approximation scheme is based on a variant of the Chernoff product formula on the space of bounded continuous functions. In a first step, we provide sufficient and necessary conditions for arbitrary families of probability measures, indexed by positive real numbers, to give rise to a convolution semigroup via a Chernoff approximation on the space of bounded continuous functions, equipped with the mixed topology. In this context, we provide explicit criteria both for the convergence of subsequences and the entire family, and discuss fine properties related to the domain of the associated generator of the L\'evy process and the infinitesimal behavior of the approximating family of measures. In a second step, we enrich the family of measures by a deterministic component and derive explicit conditions that ensure both the convergence of subsequences and the entire family to a L\'evy process with drift under a Chernoff approximation. In a series of examples, we show that our general conditions on the dynamics are satisfied, for example, by flows of Lipschitz ordinary differential equations, Euler schemes, and arbitrary Runge-Kutta methods, and that the Central Limit Theorem can be subsumed under our framework.

math.PR↗

arXiv2025

Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups

In this work, we provide conditions for nonlinear monotone semigroups on locally convex vector lattices to give rise to a generalized notion of viscosity solutions to a related nonlinear partial differential equation. The semigroup needs to satisfy a convexity estimate, so called $K$-convexity, w.r.t. another family of operators, defined on a potentially larger locally convex vector lattice. We then show that, under mild continuity requirements on the bounding family of operators, the semigroup yields viscosity solutions to the abstract Cauchy problem given in terms of its generator in the larger locally convex vector lattice. We apply our results to drift control problems for infinite-dimensional L\'evy processes and robust optimal control problems for infinite-dimensional Ornstein-Uhlenbeck processes.

math.AP↗

arXiv2025

Hopf-Lax approximation for value functions of L\'evy optimal control problems

In this paper, we investigate stochastic versions of the Hopf-Lax formula which are based on compositions of the Hopf-Lax operator with the transition kernel of a L\'evy process taking values in a separable Banach space. We show that, depending on the order of the composition, one obtains upper and lower bounds for the value function of a stochastic optimal control problem associated to the drift controlled L\'evy dynamics. Dynamic consistency is restored by iterating the resulting operators. Moreover, the value function of the control problem is approximated both from above and below as the number of iterations tends to infinity, and we provide explicit convergence rates and guarantees for the approximation procedure.

math.OC↗

arXiv2024

Pasting of Equilibria and Donsker-type Results for Mean Field Games

This paper studies the relation between equilibria in single-period, discrete-time and continuous-time mean field game models. First, for single-period mean field games, we establish the existence of equilibria and then prove the propagation of the Lasry-Lions monotonicity to the optimal equilibrium value, as a function of the realization of the initial condition and its distribution. Secondly, we prove a pasting property for equilibria; that is, we construct equilibria to multi-period discrete-time mean field games by recursively pasting the equilibria of suitably initialized single-period games. Then, we show that any sequence of equilibria of discrete-time mean field games with discretized noise converges (up to a subsequence) to some equilibrium of the continuous-time mean field game as the mesh size of the discretization tends to zero. When the cost functions of the game satisfy the Lasry-Lions monotonicity property, we strengthen this convergence result by providing a sharp convergence rate.

math.OC↗

arXiv2024

A comparison principle based on couplings of partial integro-differential operators

This paper is concerned with a comparison principle for viscosity solutions to Hamilton-Jacobi (HJ), -Bellman (HJB), and -Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with L\'evy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to differential, difference, and integral operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous L\'evy processes.

math.AP↗

arXiv2024

Upper Comonotonicity and Risk Aggregation under Dependence Uncertainty

In this paper, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multi-marginal couplings, as a generalization of well-known correlation statistics such as the Pearson correlation. The first main result states that even an arbitrarily small positive dependence between losses can result in perfectly correlated tails beyond a certain threshold and seemingly complete independence before this threshold. In a second step, we focus on the aggregation of individual risks with known marginal distributions by means of arbitrary nondecreasing left-continuous aggregation functions. In this context, we show that under an arbitrarily small positive dependence, the tail risk of the aggregate loss might coincide with the one of perfectly correlated losses. A similar result is derived for expectiles under mild conditions. In a last step, we discuss our results in the context of credit risk, analyzing the potential effects on the value at risk for weighted sums of Bernoulli distributed losses.

q-fin.RM↗

arXiv2023

A hypothesis test for the long-term calibration in rating systems with overlapping time windows

We present a statistical test that can be used to verify supervisory requirements concerning overlapping time windows for the long-term calibration in rating systems. In a first step, we show that the long-run default rate is approximately normally distributed with respect to random effects in default realization. We then perform a detailed analysis of the correlation effects caused by the overlapping time windows and solve the problem of an unknown distribution of default probabilities for the long-run default rate. In this context, we present several methods for a conservative calibration test that can deal with the unknown variance in the test statistic. We present a test for individual rating grades, and then pass to the portfolio level by suitably adapting the test statistic. We conclude with comparative statics analysing the effect of persisting customers and the number of customers per reference date.

q-fin.RM↗

arXiv2023

Risk measures based on weak optimal transport

In this paper, we study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, uncertainty on path spaces, moment constrains, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting.

q-fin.MF↗

arXiv2023

Convergence of infinitesimal generators and stability of convex monotone semigroups

Based on the convergence of their infinitesimal generators in the mixed topology, we provide a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions. In contrast to previous results, we do not rely on the theory of viscosity solutions but use a recent comparison principle which uniquely determines the semigroup via its $\Gamma$-generator defined on the Lipschitz set and therefore resembles the classical analogue from the linear case. The framework also allows for discretizations both in time and space and covers a variety of applications. This includes Euler schemes and Yosida-type approximations for upper envelopes of families of linear semigroups, stability results and finite-difference schemes for convex HJB equations, Freidlin-Wentzell-type results and Markov chain approximations for a class of stochastic optimal control problems and continuous-time Markov processes with uncertain transition probabilities.

math.AP↗