arXiv subjects

Eduardo Abi Jaber

Publications and source records attributed to Eduardo Abi Jaber.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Investment and Consumption in Financial Markets with Integrated Variance Clocks

We study the infinite-horizon optimal investment and consumption problem in a general class of continuous financial markets, where uncertainty is driven by a continuous non-decreasing stochastic clock representing accumulated variance. This framework encompasses classical Markovian and non-Markovian stochastic volatility models as well as singular realized-variance models in which no spot volatility process exists. We characterize the value process and optimal investment and consumption strategies in terms of a non-linear infinite-horizon backward stochastic differential equation driven jointly by calendar time and the stochastic clock. We develop a general well-posedness theory for this new class of IVC-BSDEs based on the method of sub- and supersolutions, establishing existence, uniqueness, and stability under natural conditions that might be of independent interest beyond the financial application at hand. We are moreover able to identify the sign of the $Z$-component of the solution using Malliavin calculus. We then apply our results to Volterra Heston models with locally integrable kernels, covering both rough and hyper-rough regimes. Exploiting the affine structure of the model, we verify the optimality of the candidate strategies in incomplete markets and obtain an explicit representation of the solution in the complete market case. Owing to the generality of the framework and the weak assumptions imposed on the stochastic clock, our results unify and extend several existing results for optimal investment and consumption, including classical Markovian stochastic volatility models.

q-fin.MF↗

arXiv2026

The Volterra Stein-Stein model with stochastic interest rates

We introduce the Volterra Stein-Stein model with stochastic interest rates, where both volatility and interest rates are driven by correlated Gaussian Volterra processes. This framework unifies various well-known Markovian and non-Markovian models while preserving analytical tractability for pricing and hedging financial derivatives. We derive explicit formulas for pricing zero-coupon bond and interest rate cap or floor, along with a semi-explicit expression for the characteristic function of the log-forward index using Fredholm resolvents and determinants. This allows for fast and efficient derivative pricing and calibration via Fourier methods. We calibrate our model to market data and observe that our framework is flexible enough to capture key empirical features, such as the humped-shaped term structure of ATM implied volatilities for cap options and the concave ATM implied volatility skew term structure (in log-log scale) of the S&P 500 options. Finally, we establish connections between our characteristic function formula and expressions that depend on infinite-dimensional Riccati equations, thereby making the link with conventional linear-quadratic models.

q-fin.MF↗

arXiv2026

From Hyper Roughness to Jumps as $H \to -1/2$

We investigate the weak limit of the hyper-rough square-root process as the Hurst index $H$ goes to $-1/2\,$. This limit corresponds to the fractional kernel $t^{H - 1 / 2}$ losing integrability at zero. We establish the joint convergence of the couple $(X, M)\,$, where $X$ is the hyper-rough process and $M$ the associated martingale, to a fully correlated inverse Gaussian Lévy jump process. This unveils the existence of a continuum between hyper-rough continuous models and jump processes, as a function of the Hurst index. Since we prove a convergence of continuous to discontinuous processes, the usual Skorokhod $J_1$ topology is not suitable for our problem. Instead, we obtain the weak convergence in the Skorokhod $M_1$ topology for $X$ and in the non-Skorokhod $S$ topology for $M$.

math.PR↗

arXiv2026

Stochastic invariance in infinite dimension beyond Lipschitz coefficients

We establish necessary and sufficient conditions for stochastic invariance of closed subsets in Hilbert spaces for solutions to infinite-dimensional stochastic differential equations (SDEs) under mild assumptions on the coefficients. Our first characterization is formulated in terms of certain normal vectors to the invariance set and requires differentiability only of the dispersion operator, but not of the diffusion coefficient itself. The condition involves a suitable corrected drift expressed through the dispersion operator and its Moore-Penrose pseudoinverse, extending the classical Stratonovich correction term to the present low-regularity setting. Our second characterization is given in terms of the positive maximum principle for the infinitesimal generator of the associated diffusion process. We illustrate our characterizations in the case of invariant manifolds.

math.PR↗

arXiv2026

Equilibrium in Functional Stochastic Games with Mean-Field Interaction

We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in $L^2$. We propose a novel approach for deriving the Nash equilibrium of the game semi-explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their solution in semi-explicit form. Furthermore, by proving stability results for the system of stochastic Fredholm equations, we derive the convergence of the equilibrium of the $N$-player game to the corresponding mean-field equilibrium. As a by-product, we also derive an $\varepsilon$-Nash equilibrium for the mean-field game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the mean-field limit are less restrictive than in the finite-player game. Finally, we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.

math.OC↗

arXiv2026

State spaces of multifactor approximations of nonnegative Volterra processes

We show that the state spaces of multifactor Markovian processes, coming from approximations of nonnegative Volterra processes, are given by explicit linear transformation of the nonnegative orthant. We demonstrate the usefulness of this result for applications, including simulation schemes and PDE methods for nonnegative Volterra processes.

math.PR↗

arXiv2026

Affine Structure of the Brownian Signature

We establish an infinite-dimensional affine transform theory for the time-augmented Brownian signature. Our first main result shows that, for a suitable class of linear functions of the signature, the conditional Fourier-Laplace transform admits an entire signature expansion. We prove that the associated coefficients solve an infinite-dimensional linear differential equation on the extended tensor algebra. Our second main result shows that the logarithm admits a local signature expansion whose coefficients satisfy a Riccati equation on the extended tensor algebra, revealing a generalized affine structure of the Brownian signature in a genuinely path-dependent setting. In contrast to conventional affine processes, we show that this representation is intrinsically local: zeros of the Fourier-Laplace transform in the complex plane prevent any global expansion. To recover global representations, we introduce a new class of randomized Riccati equations with path-dependent terminal conditions through a recentering argument. Furthermore, we establish uniqueness of solutions to the linear and Riccati equations within a suitable class of solutions. Our results provide a theoretical framework for transform methods in non-Markovian settings, with applications to the computation of conditional distributions.

math.PR↗

arXiv2026

Stochastic control with signatures via Riccati equations on the tensor algebra

We solve in semi-explicit form a class of non-Markovian stochastic optimal control problems with path-dependent rewards, using path signatures. We reformulate the control problem as the computation of Laplace transforms of signature functionals thanks to the Boué-Dupuis representation. Exploiting recent signature representations of such transforms on tensor algebras, we determine the value process and the optimal control through an infinite-dimensional system of Riccati equations on the extended tensor algebra. We establish an explicit feedback representation of the optimal control and the value process as an infinite linear combination of the time-extended signature of the controlled process, with time-dependent coefficients. The expansions being intrinsically local, we propose a dynamic recentering algorithm to ensure a global representation over the entire time horizon. We illustrate the approach on genuinely path-dependent, non-linear examples that go beyond the tractable linear-quadratic setting, including the tracking of linear functionals of the signature and signature lifts of Volterra control problems.

math.OC↗

arXiv2026

Volterra clocks and their pure-jump limits: hitting times of curved boundaries

We introduce a class of continuous Volterra processes, called Volterra clocks, and study their singular limit as the memory kernel collapses to a Dirac mass at zero. The dynamics are parametrised by a function $f$ acting as a nonlinear time-change, generalising the Volterra square-root process and recovering it when $f$ is affine. In the singular limit, the continuous Volterra clock converges weakly to a pure-jump process given by first passage times of a Brownian motion to curved boundaries, including affine and square-root boundaries when $f$ is, respectively, affine or quadratic. Outside the affine setting, characteristic function methods are no longer available, and we instead identify the limit directly from the dynamics. We do this through a topological framework adapted to the time-change structure which involves Skorokhod's $M_1$ topology and a decorated notion of convergence. Our analysis unifies several regimes of interest for general Volterra clocks, including large-time asymptotics, fast mean reversion, and hyper-roughness. In particular, this subsumes and extends existing results in the affine setting.

math.PR↗

arXiv2026

Exponentially Fading Memory Signature

We introduce the exponentially fading memory (EFM) signature, a time-invariant transformation of an infinite (possibly rough) path that serves as a mean-reverting analogue of the classical path signature. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity. The EFM-signature retains many of the key algebraic and analytical properties of classical signatures, including a suitably modified Chen identity, the linearization property, path-determinacy, and the universal approximation property. From the probabilistic perspective, the EFM-signature provides a "stationarized" representation, making it particularly well-suited for time-series analysis and signal processing overcoming the shortcomings of the standard signature. In particular, the EFM-signature of time-augmented Brownian motion evolves as a group-valued Ornstein-Uhlenbeck process. We establish its stationarity, Markov property, and exponential ergodicity in the Wasserstein distance, and we derive an explicit formula à la Fawcett for its expected value in terms of Magnus expansions. We also study linear combinations of EFM-signature elements and the computation of associated characteristic functions in terms of a mean-reverting infinite dimensional Riccati equation.

math.PR↗

arXiv2026

Weak solutions of Stochastic Volterra Equations in convex domains with general kernels

We establish new weak existence results for $d$-dimensional Stochastic Volterra Equations (SVEs) with continuous coefficients and possibly singular one-dimensional non-convolution kernels. These results are obtained by introducing an approximation scheme and showing its convergence. A particular emphasis is made on the stochastic invariance of the solution in a closed convex set. To do so, we extend the notion of kernels that preserve nonnegativity introduced in \cite{Alfonsi23} to non-convolution kernels and show that, under suitable stochastic invariance property of a closed convex set by the corresponding Stochastic Differential Equation, there exists a weak solution of the SVE that stays in this convex set. We present a family of non-convolution kernels that satisfy our assumptions, including a non-convolution extension of the well-known fractional kernel. We apply our results to SVEs with square-root diffusion coefficients and non-convolution kernels, for which we prove the weak existence and uniqueness of a solution that stays within the nonnegative orthant. We derive a representation of the Laplace transform in terms of a non-convolution Riccati equation, for which we establish an existence result.

math.PR↗

arXiv2026

Heath-Jarrow-Morton meet lifted Heston in energy markets for joint historical and implied calibration

In energy markets, joint historical and implied calibration is of paramount importance for practitioners, yet notoriously challenging due to the need to align historical correlations of futures contracts with implied volatility smiles from the option market. We address this crucial problem with a multiplicative multi-factor Heath-Jarrow-Morton (HJM) model for forward curves, combined with a stochastic volatility factor coming from the lifted Heston model. We develop a sequential fast calibration procedure leveraging the Kemna-Vorst approximation of futures contracts: (i) historical correlations and the Variance Swap (VS) volatility term structure are captured through Level, Slope, and Curvature factors, (ii) the VS volatility term structure can then be corrected for a perfect match via a fixed-point algorithm, (iii) implied volatility smiles are calibrated using Fourier-based techniques. The main advantage of the proposed calibration framework is the decoupling of the calibration steps: each step tackles a simpler calibration subproblem and guaranties that the previously optimized parameters remain unchanged. Our model displays remarkable joint historical and implied calibration fits on the German power market and enables realistic interpolation within the implied volatility hypercube.

q-fin.MF↗

arXiv2026

Malliavin calculus for signatures with applications to finance

Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous Itô processes. As a consequence, we obtain closed-form expressions for the Clark--Ocone representation, the Ornstein--Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.

math.PR↗

arXiv2026

Capturing Smile Dynamics with the Quintic Volatility Model: SPX, Skew-Stickiness Ratio and VIX

We introduce the two-factor Quintic Ornstein-Uhlenbeck (OU) model, where volatility is modelled as a degree-five polynomial of the sum of two Ornstein-Uhlenbeck processes driven by the same Brownian motion, each mean-reverting at a different speed. We demonstrate that the model effectively captures the volatility surfaces of SPX and VIX while aligning with the skew-stickiness ratio (SSR) across maturities ranging from a few days to over two years. Furthermore, it is consistent with key empirical stylized facts, notably reproducing the Zumbach effect.

q-fin.MF↗

arXiv2026

Optimal Portfolio Choice with Cross-Impact Propagators

We consider a class of optimal portfolio choice problems in continuous time where the agent's transactions create both transient cross-impact driven by a matrix-valued Volterra propagator, as well as temporary price impact. We formulate this problem as the maximization of a revenue-risk functional, where the agent also exploits available information on a progressively measurable price predicting signal. We solve the maximization problem explicitly in terms of operator resolvents, by reducing the corresponding first order condition to a coupled system of stochastic Fredholm equations of the second kind and deriving its solution. We then give sufficient conditions on the matrix-valued propagator so that the model does not permit price manipulation. We also provide an implementation of the solutions to the optimal portfolio choice problem and to the associated optimal execution problem. Our solutions yield financial insights on the influence of cross-impact on the optimal strategies and its interplay with alpha decays.

q-fin.PM↗

arXiv2025

Signature approach for pricing and hedging path-dependent options with frictions

We introduce a novel signature approach for pricing and hedging path-dependent options with instantaneous and permanent market impact under a mean-quadratic variation criterion. Leveraging the expressive power of signatures, we recast an inherently nonlinear and non-Markovian stochastic control problem into a tractable form, yielding hedging strategies in (possibly infinite) linear feedback form in the time-augmented signature of the control variables, with coefficients characterized by non-standard infinite-dimensional Riccati equations on the extended tensor algebra. Numerical experiments demonstrate the effectiveness of these signature-based strategies for pricing and hedging general path-dependent payoffs in the presence of frictions. In particular, market impact naturally smooths optimal trading strategies, making low-truncated signature approximations highly accurate and robust in frictional markets, contrary to the frictionless case.

q-fin.PM↗

arXiv2025

Efficient Simulation of Hawkes Processes using their Affine Volterra Structure

We introduce a novel and efficient simulation scheme for Hawkes processes on a fixed time grid, leveraging their affine Volterra structure. The key idea is to first simulate the integrated intensity and the counting process using Inverse Gaussian and Poisson distributions, from which the jump times can then be easily recovered. Unlike conventional exact algorithms based on sampling jump times first, which have random computational complexity and can be prohibitive in the presence of high activity or singular kernels, our scheme has deterministic complexity which enables efficient large-scale Monte Carlo simulations and facilitates vectorization. Our method applies to any nonnegative, locally integrable kernel, including singular and non-monotone ones. By reformulating the scheme as a stochastic Volterra equation with a measure-valued kernel, we establish weak convergence to the target Hawkes process in the Skorokhod $J_1$-topology. Numerical experiments confirm substantial computational gains while preserving high accuracy across a wide range of kernels, with remarkably improved performance for a variant of our scheme based on the resolvent of the kernel.

math.PR↗

arXiv2025

Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model

Fourier-based methods are central to option pricing and hedging when the Fourier-Laplace transform of the log-price and integrated variance is available semi-explicitly. This is the case for the Volterra Stein-Stein stochastic volatility model, where the characteristic function is known analytically. However, naive evaluation of this formula can produce discontinuities due to the complex square root of a Fredholm determinant, particularly when the determinant crosses the negative real axis, leading to severe numerical instabilities. We analyze this phenomenon by characterizing the determinant's crossing behavior for the joint Fourier-Laplace transform of integrated variance and log-price. We then derive an expression for the transform to account for such crossings and develop efficient algorithms to detect and handle them. Applied to Fourier-based pricing in the rough Stein-Stein model, our approach significantly improves accuracy while drastically reducing computational cost relative to existing methods.

q-fin.MF↗