arXiv subjects

Liviu Ignat

Publications and source records attributed to Liviu Ignat.

6 recordsLinked to original sources

arXiv2026

Optimal Investment with Switching Preferences

Major life events can significantly increase individuals' risk aversion over a sustained period of time, as empirical studies reveal. How such an event-triggered shift of risk preferences impacts optimal investment is the focus of this paper. On a finite time horizon where a major life event may occur independently of the financial market, an investor aims to maximize her expected utility from terminal wealth while foreseeing a potential change in her risk aversion. We find that the associated Hamilton--Jacobi--Bellman (HJB) equation involves the post-event optimal value function (under elevated but fixed risk aversion after the event's occurrence), and the Fenchel--Legendre transform fails to linearize this HJB equation: it yields a parabolic equation with a fully nonlinear term, induced precisely by the post-event optimal value function. Through a combination of fixed-point, compactness, and verification arguments, we establish the existence of a positive convex classical solution with suitable growth to the fully-nonlinear parabolic equation. The convex conjugate of this solution is shown to satisfy the HJB equation and coincides with the pre-event optimal value function. The optimal trading strategy is obtained by concatenating the optimal pre-event and post-event strategies -- the former is expressed in terms of the solution to the HJB equation and the latter is traditional Merton's ratio.

math.OC↗

arXiv2022

Asymptotic behaviour of solutions to fractional diffusion-convection equations

We consider a convection-diffusion model with linear fractional diffusion in the sub-critical range. We prove that the large time asymptotic behavior of the solution is given by the unique entropy solution of the convective part of the equation. The proof is based on suitable a-priori estimates, among which proving an Oleinik type inequality plays a key role.

math.AP↗

arXiv2021

A non-local coupling model involving three fractional laplacians

In this article we study a non-local diffusion problem that involves three different fractional Laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator serves as a coupling mechanism between the two of them. The model proposed is the gradient flow of a non-local energy functional. In the first part of the article we provide results about existence of solutions and the conservation of mass. The second part encompasses results about the Lp decay of the solutions. The third part is devoted to study the asymptotic behavior of the solutions of the problem when the two domains are a ball and its complementary. Exterior fractional Sobolev and Nash inequalities of independent interest are also provided in an appendix.

math.AP↗

arXiv2016

Long time behavior for a nonlocal convection diffusion equation

In this paper we consider a nonlocal viscous Burgers equation and study the well-posedness and asymptotic behaviour of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.

math.AP↗

arXiv2011

Convergence rates for dispersive approximation schemes to nonlinear Schr\"odinger equations

This article is devoted to the analysis of the convergence rates of several nu- merical approximation schemes for linear and nonlinear Schr\"odinger equations on the real line. Recently, the authors have introduced viscous and two-grid numerical approximation schemes that mimic at the discrete level the so-called Strichartz dispersive estimates of the continuous Schr\"odinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L2(R), a fact that can not be proved in the nonlinear setting for standard conservative schemes unless more regularity of the initial data is assumed. In the present article we obtain explicit convergence rates and prove that dispersive schemes fulfilling the Strichartz estimates are better behaved for Hs(R) data if 0 < s < 1/2. Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic decay rates.

math.NA↗

arXiv2011

Dispersion for the Schrödinger Equation on Networks

In this paper we consider the Schrödinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. We give an explicit description of the solution of the linear Schrödinger equation with constant coefficients. This allows us to prove dispersive estimates, which in turn are useful for solving the nonlinear Schrödinger equation. The proof extends also to the laminar case of positive step-function coefficients having a finite number of discontinuities.

math.AP↗