arXiv subjects

Scander Mustapha

Publications and source records attributed to Scander Mustapha.

3 recordsLinked to original sources

arXiv2024

Strong existence and uniqueness of a calibrated local stochastic volatility model

We study a two-dimensional McKean-Vlasov stochastic differential equation, whose volatility coefficient depends on the conditional distribution of the second component with respect to the first component. We prove the strong existence and uniqueness of the solution, establishing the well-posedness of a two-factor local stochastic volatility (LSV) model calibrated to the market prices of European call options. In the spirit of [Jourdain and Zhou, 2020, Existence of a calibrated regime switching local volatility model.], we assume that the factor driving the volatility of the log-price takes finitely many values. Additionally, the propagation of chaos of the particle system is established, giving theoretical justification for the algorithm [Julien Guyon and Henry-Labord\`ere, 2012, Being particular about calibration.].

math.PR↗

arXiv2023

Well-posedness of the supercooled Stefan problem with oscillatory initial conditions

We study the one-phase one-dimensional supercooled Stefan problem with oscillatory initial conditions. In this context, the global existence of so-called physical solutions has been shown recently in [CRSF20], despite the presence of blow-ups in the freezing rate. On the other hand, for regular initial conditions, the uniqueness of physical solutions has been established in [DNS22]. Here, we prove the uniqueness of physical solutions for oscillatory initial conditions by a new contraction argument that replaces the local monotonicity condition of [DNS22] with an averaging condition. We verify this weaker condition for fairly general oscillating probability densities, such as the ones given by an almost sure trajectory of $(1+W_x-\sqrt{2x|\log{|\log{x}|}|})_{+}\wedge 1$ near the origin, where $W$ is a standard Brownian motion. We also permit typical deterministically constructed oscillating densities, including those of the form $(1+\sin{1/x})/2$ near the origin. Finally, we provide an example of oscillating densities for which it is possible to go beyond our main assumption via further complementary arguments.

math.PR↗

arXiv2020

Trend to equilibrium for granular media equations under non-convex potential and application to log gases

We derive new HWI inequalities for the granular media equation, which external potential $V$ and interaction potential $W$ are only strictly convex on complementary parts of the space. Particularly, potentials are not assumed convex. After solving technicalities related to the singularity of a logarithmic $W$, we apply our result to obtain stability rates of log gases under non-strictly convex or quartic external potentials. We prove that the distribution of a log gas converges towards an equilibrium with respect to the Wasserstein distance at a square root rate. Finally, we establish exponential stability of log gases under the double-well potential $V(x) = \frac{x^4}{4} + c\frac{x^2}{2}$, $c < 0$ and the non-confining potential $V(x) = g\frac{x^4}{4} + \frac{x^2}{2}$, $g<0$ for $|c|$ and $|g|$ small enough.

math.AP↗