arXiv subjects

Emmanuel Gnabeyeu

Publications and source records attributed to Emmanuel Gnabeyeu.

9 recordsLinked to original sources

arXiv2026

On (fake) Stationarity in Stochastic Volterra Equations with Affine Drift and Regular Kernels

We investigate the fake stationarity properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs) with affine drift and long-memory (regular) kernels, both on finite horizons and in the long-run regime. By either deriving explicit closed-form specifications for the deterministic initial condition $\phi$ and the mean-reversion function $\mu$ appearing in the drift, or by introducing a deterministic stabilizing factor $\varsigma$ in the diffusion coefficient associated with the kernel while keeping $\mu$ fully flexible, we show that it is possible to induce a \textit{fake stationary} regime, in the sense that all marginal distributions share the same mean and variance. Afterwards, using a refined asymptotic analysis, we further establish that, in both frameworks, the time-shifted solutions of these long-memory SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance structure, for suitable classes of diffusion coefficients. These results are applied to a class of exponential-fractional Stochastic Volterra Integral Equations driven by an $\alpha$-gamma fractional integration kernel, in the particular regime $\alpha \geq 1$, which regularizes diffusion paths and invoke textit{ long-term memory}, persistence or long range dependence.

math.PR↗

arXiv2026

On Utility Maximization under Multivariate Fake Stationary Affine Volterra Models

This paper is concerned with Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate fake stationary Volterra--Heston model. Due to the non-Markovianity and non-semimartingality of the underlying processes, the classical stochastic control approach cannot be directly applied in this setting. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). Our approach is inspired by the martingale optimality principle combined with a suitable verification argument. The resulting optimal strategies for Merton's problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations, while the optimal value is expressed using the solution to this original Riccati BSDE. Numerical results on a two dimensional fake stationary rough Heston model illustrate the impact of stationary rough volatilities on the optimal Merton strategies.

math.OC↗

arXiv2026

Optimal Merton's Problem under Multivariate Affine Volterra Models with Jumps

This paper is concerned with portfolio selection for an investor with exponential, power, and logarithmic utility in multi-asset financial markets allowing jumps. We investigate the classical Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate Volterra--Heston model with jumps driven by an independent Poisson random measure. Owing to the non-Markovian and non-semimartingale nature of the model, classical stochastic control techniques are not directly applicable. Instead, the problem is tackled using the martingale optimality principle by constructing a family of supermartingale processes characterized via solutions to an original Riccati backward stochastic differential equation with jumps (Riccati BSDEJ).The resulting optimal strategies for Merton's problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations, while the optimal value is expressed using the solution to this original Riccati BSDEJ. Numerical experiments on a two-dimensional rough Heston model illustrate the impact of both path roughness and jumps components on the value function and optimal strategies in the Merton problem.

math.OC↗

arXiv2026

On Path-dependent Volterra Integral Equations: Strong Well-posedness and Stochastic Numerics

The aim of this paper is to provide a comprehensive analysis of the path-dependent Stochastic Volterra Integral Equations (SVIEs), in which both the drift and the diffusion coefficients are allowed to depend on the whole trajectory of the process up to the current time. We investigate the existence and uniqueness (aka the strong well-posedness) of solutions to such equations in the $L^p$ setting, $p>0$, locally in time and their properties specifically their path regularity and flows. Then, we introduce a numerical approximation method based on an interpolated $K-$integrated Euler-Maruyama scheme to simulate numerically the process, and we prove the convergence, with an explicit rate, of this scheme towards the strong solution in the $L^p$ norm.

math.PR↗

arXiv2026

On explicit solutions to a class of quadratic BSDEJs driven by affine Volterra processes with jumps and applications

In this paper we consider a class of quadratic BSDEs with jumps (quadratic BSDEJs) involving inhomogeneous affine Volterra processes and show that their solution can be reduced to solving a system of generalized inhomogeneous integral Riccati-Volterra ordinary differential equations with L\'evy jump compensators. This yields a rich and flexible class of quadratic BSDEJs that are analytically tractable, in the sense that their solutions are explicit up to the solution of an associated integral Riccati-Volterra ODE with L\'evy jump compensator. As an application, we provide analytically tractable solutions to the continuous-time Markowitz mean-variance portfolio selection problem within a multivariate class of affine Volterra models allowing jumps driven by an independent Poisson random measure. In this non-Markovian and non-semimartingale market framework with unbounded random coefficients, the classical stochastic control approach cannot be directly applied to the associated optimization task. Instead, the problem is tackled using the martingale optimality principle by constructing a family of submartingale processes characterized via solutions to a novel Riccati backward stochastic differential equation with jumps (Riccati BSDEJ), particular subclass of the aforementionned quadratic BSDEJ. Specifically, we obtain analytical closed-form expressions for the optimal feedback control as well as the mean-variance efficient frontier, both of which depend on the solution to the associated multivariate inhomogeneous Riccati-Volterra system, while the optimal value function is expressed using the solution to this original Riccati BSDEJ. Furthermore, numerical experiments on a two-dimensional fake stationary rough Heston model is discussed and used to highlight the impact of stabilized rough volatilities on the Markowitz allocation problem.

math.OC↗

arXiv2026

Fake stationary rough Heston volatility: Microstructure-inspired foundations

This paper investigates the asymptotic behavior of suitably time-modulated Hawkes processes with heavy-tailed kernels in a nearly unstable regime. We show that, under appropriate scaling, both the intensity processes and the rescaled Hawkes processes converge to a mean-reverting, time-inhomogeneous rough fractional square-root process and its integrated counterpart, respectively. In particular, when the original Hawkes process has a stationary first moment (constant marginal mean), the limiting process takes the form of a time-inhomogeneous rough fractional Cox-Ingersoll-Ross (CIR) equation with a constant mean-reversion parameter and a time-dependent diffusion coefficient. This class of equations is particularly appealing from a practical perspective, especially for the so-called $\textit{fake stationary rough Heston}$ model. We further investigate the properties of such limiting scaled time-inhomogeneous Volterra equations, including moment bounds, path regularity and maximal inequality in the $L^p$ setting for every $p>0$.

math.PR↗

arXiv2025

On Inhomogeneous Affine Volterra Processes: Stationarity and Applications to the Volterra Heston Model

True Volterra equations are inherently non stationary and therefore do not admit $\textit{genuine stationary regimes}$ over finite horizons. This motivates the study of the finite-time behavior of the solutions to scaled inhomogeneous affine Stochastic Volterra equations through the lens of a weaker notion of stationarity referred to as $\textit{fake stationary regime}$ in the sense that all marginal distributions share the same expectation and variance. As a first application, we introduce the $\textit{Fake stationary Volterra Heston model}$ and derive a closed-form expression for its characteristic function. Having established this finite-time proxy for stationarity, we then investigate the asymptotic (long-time) behavior to assess whether genuine stationary regimes emerge in the limit. Using an extension of the exponential-affine transformation formula for those processes, we establish in the long run the existence of limiting distributions, which (unlike in the case of classical affine diffusion processes) may depend on the initial state of the process, unless the Volterra kernel coincides with the $α-$ fractional integration kernel, for which the dependence on the initial state vanishes. We then proceed to the construction of stationary processes associated with these limiting distributions. However, the dynamics in this long-term regime are analytically intractable, and the process itself is not guaranteed to be stationary in the classical sense over finite horizons. This highlights the relevance of finite-time analysis through the lens of the aforementioned $\textit{fake stationarity}$, which offers a tractable approximation to stationary behavior in genuinely non-stationary Volterra systems.

math.PR↗

arXiv2025

On a Stationarity Theory for Stochastic Volterra Integral Equations with Affine Drift

This paper investigate the properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs for short) with affine drift, specifically their stationarity, both over a finite horizon and in the long run. We demonstrate that it is possible to induce a $\textit{fake stationary regime}$, in the sense that all marginal distributions share the same expectation and variance. This phenomenon can be achieved either through explicit closed-form specifications of the deterministic initial condition $\phi$ and the mean-reversion function $\mu$ appearing in the drift, or by introducing a deterministic stabilizing factor $\varsigma$ in the diffusion coefficient, associated with the kernel, while keeping the function $\mu$ otherwise fully flexible. We further look at the $L^p$-confluence properties $p>0$ of such processes as time goes to infinity, namely we investigate whether the marginals of solutions associated with different initial values become asymptotically confluent in $L^p$. We finally study the functional weak long-run asymptotics for some classes of diffusion coefficients. More precisely, we establish that, in both settings, the time-shifted solutions of such SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance function. These results are then applied to a class of Exponential-Fractional Stochastic Volterra Integral Equations driven by an $\alpha$-gamma fractional integration kernel,in the particular case $\alpha \in (0,1]$, which corresponds to the $\textit{rough-path}$ regime. Building on these fake stationary Volterra processes, we finally introduce a family of stabilized Rough volatility models.

math.PR↗

arXiv2024

Solving The Dynamic Volatility Fitting Problem: A Deep Reinforcement Learning Approach

The volatility fitting is one of the core problems in the equity derivatives business. Through a set of deterministic rules, the degrees of freedom in the implied volatility surface encoding (parametrization, density, diffusion) are defined. Whilst very effective, this approach widespread in the industry is not natively tailored to learn from shifts in market regimes and discover unsuspected optimal behaviors. In this paper, we change the classical paradigm and apply the latest advances in Deep Reinforcement Learning(DRL) to solve the fitting problem. In particular, we show that variants of Deep Deterministic Policy Gradient (DDPG) and Soft Actor Critic (SAC) can achieve at least as good as standard fitting algorithms. Furthermore, we explain why the reinforcement learning framework is appropriate to handle complex objective functions and is natively adapted for online learning.

q-fin.CP↗