arXiv subjects

Emmet Lawless

Publications and source records attributed to Emmet Lawless.

7 recordsLinked to original sources

arXiv2026

Infinite Horizon Optimal Consumption: Intertemporal Hedging under Epstein-Zin Preferences

We study an infinite-horizon optimal consumption-investment problem for an investor with Epstein-Zin stochastic differential utility in an incomplete market with stochastic investment opportunities. Risk aversion and intertemporal substitution are separated, and we work in the regime $\theta\in(0,1)$, where there exists a unique generalised utility process for arbitrary non-negative progressively measurable consumption streams. Our main contribution is a variational characterisation of the value function. We show that the value function is the unique minimiser of a functional whose Euler-Lagrange equation coincides with the Hamilton-Jacobi-Bellman equation. Although the functional may be non-convex, the direct method yields existence, and we prove that every minimiser is a strictly positive, bounded classical solution. A verification theorem identifies any minimiser with the value function and gives feedback representations for optimal consumption and investment policies. The proof combines a change of measure to the myopic probability with uniqueness results for Epstein-Zin BSDEs and a perturbation argument for optimality. Examples with stochastic volatility, Gaussian excess returns, and fat-tailed excess returns illustrate the scope of the framework and its implications for intertemporal hedging.

q-fin.MF↗

arXiv2026

Solution space characterisation of perturbed linear discrete and continuous stochastic Volterra convolution equations: the $\ell^p$ and $L^p$ cases

In this article, we are concerned with characterising when solutions of perturbed linear stochastic Volterra summation equations are almost surely $p$-summable and when their continuous time counterparts, perturbed linear stochastic Volterra integro-differential equations, are almost surely $p$-integrable. In the discrete case, we find it necessary and sufficient that perturbing functions are $p$-summable in order to ensure paths of the discrete equation are almost surely $p$-summable, while in the continuous case, it transpires one can have almost surely $p$-integrable sample paths with non-integrable perturbation functions. For the continuous equation, the main converse is clinched by considering an appropriate discretisation and applying results from the discrete case. We also conduct a thorough study of the asymptotic behaviour of the trajectories of solutions to the continuous equation in the regime of $p$-integrable paths and provide a characterisation of almost sure convergence to zero in the case of diagonal noise. Additionally, we highlight how all proof methods can be applied to obtain stronger results for stochastic functional differential equations.

math.DS↗

arXiv2025

Characterisation of asymptotic behaviour of perturbed deterministic and stochastic pantograph equations

This paper considers the asymptotic behaviour of deterministically and stochastically forced linear pantograph equations. The asymptotic behaviour is studied in the case when all solutions of the pantograph equation without forcing tend to a trivial equilibrium. In all cases, we give necessary and sufficient conditions on the forcing terms which enable all solutions to converge to the equilibrium of the unforced equation, and which enable solutions to remain bounded. In the deterministic case, we give sharp conditions on forcing terms which enable the solutions of the forced equations to inherit the power law behaviour of the unforced equation, as well as slower rates of decay or growth not present in the unforced equation. Extensions to equations with general unbounded delay, and to finite--dimensional equations are also presented.

math.CA↗

arXiv2024

Solution space characterisation of perturbed linear functional and integrodifferential Volterra convolution equations: Cesàro limits

In this article we discuss the requirements needed in order to characterise the solution space of perturbed linear integro-differential Volterra convolution equations. We highlight in general how the pointwise behaviour of perturbation functions does not necessarily propagate through to the solution which the classical literature seems to suggest. To illustrate this general idea we show the Cesàro mean of the solution can converge even in cases when the Cesàro mean of the perturbation function diverges. Furthermore we provide a characterisation of when such convergence takes place and explicitly identify the limit in terms of the problem data. Additionally we prove how all results can also be applied to perturbed linear functional differential equations.

math.CA↗

arXiv2024

Weighted $L_\infty$ Asymptotic Characterisation of Perturbed Autonomous Linear Ordinary and Stochastic Differential Equations: Part I -- ODEs

This is the first of a two-part paper which determines necessary and sufficient conditions on the asymptotic behaviour of forcing functions so that the solutions of additively pertubed linear differential equations obey certain growth or decay estimates. Part I considers deterministic equations, and part II Itô-type stochastic differential equations. Results from part I are used to deal with deterministically and stochastically forced equations in the second part. Results apply to both scalar and multi-dimensional equations, and connect the asymptotic behaviour of time averages of the forcing terms on finite intervals with the growth or decay rate of the solution. Mainly, results deal with large perturbations, but some indications of how results extend to tackle subdominant perturbations are also sketched.

math.CA↗

arXiv2023

Mean square asymptotic stability characterisation of perturbed linear stochastic functional differential equations

In this paper we investigate the mean square asymptotic stability of a perturbed scalar linear stochastic functional differential equation. Specifically, we are able to give necessary and sufficient conditions on the forcing terms for convergence of the mean square, exponential convergence of the mean square, and integrability of the mean square of solutions. It is also essential that the underlying unperturbed SFDE is mean square asymptotically stable for these results to hold.

math.PR↗

arXiv2023

Solution Space Characterisation of Perturbed Linear Volterra Integrodifferential Convolution Equations: the Lp case

In this paper we characterise the Lp stability of perturbed linear Volterra integrodifferential convolution equations. Additionally we provide a framework which points to necessary and sufficient conditions on the forcing function that ensures the solution lies in a particular function space. We highlight how such a result is of interest when studying perturbed Stochastic Functional Differential Equations (SFDEs).

math.CA↗