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The financial model proposed involves the liquidation process of a portfolio of $n$ assets through sell or (and) buy orders with volatility. We present the rigorous mathematical formulation of this model in a financial setting resulting to…
We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space-time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such…
Motivated by applications in economics and finance, in particular to the modeling of limit order books, we study a class of stochastic second-order PDEs with non-linear Stefan-type boundary interaction. To solve the equation we transform…
We consider the one-dimensional outer stochastic Stefan problem with reflection. The problem admits maximal solutions as long as the velocity of the moving boundary remains bounded, [3,9,10]. We apply Malliavin calculus to the transformed…
We study a nonlinear, pseudomonotone, stochastic diffusion-convection evolution problem on a bounded spatial domain, in any space dimension, with homogeneous boundary conditions and reflection. The additive noise term is given by a…
This work deals with the one-dimensional Stefan problem with a general time-dependent boundary condition at the fixed boundary. Stochastic solutions are obtained using discrete random walks, and the results are compared with analytic…
We derive two weak formulations for the supercooled Stefan problem with transport noise on a half-line: one captures a continuously evolving system, while the other resolves blow-ups by allowing for jump discontinuities in the evolution of…
The dissolution of solids has created spectacular geomorphologies ranging from centimeter-scale cave scallops to the kilometer-scale "stone forests" of China and Madagascar. Mathematically, dissolution processes are modeled by a Stefan…
Most heat transfer models for bulk crystal growth rely on the classical Stefan formulation to evaluate interface motion during phase change. However, when the interface is non-smooth the use of the classical Stefan formulation may lead to…
We study the regularity of the bounded self-similar solution to the one-phase Stefan problem with fractional diffusion posed on the whole line. In terms of the enthalpy $h(x,t)$, the evolution problem reads \[ \begin{cases} \partial_t h +…
We study the stochastic Willmore flow and the stochastic surface diffusion flow for closed or non-closed curves on $\mathbb{R}^2$ in this paper. We equivalently formulate them as a stochastic one-phase Stefan problem (or a stochastic free…
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…
Our Recent advancements in stochastic processes have illuminated a paradox associated with the Einstein model of Brownian motion. The model predicts an infinite propagation speed, conflicting with the second law of thermodynamics. The…
We introduce stochastic volatility models, in which the volatility is described by a time-dependent nonnegative function of a reflecting diffusion. The idea to use reflecting diffusions as building blocks of the volatility came into being…
Moving boundary problems allow to model systems with phase transition at an inner boundary. Driven by problems in economics and finance, in particular modeling of limit order books, we consider a stochastic and non-linear extension of the…
We introduce and analyze a nonlocal version of the one-phase Stefan problem in which, as in the classical model, the rate of growth of the volume of the liquid phase is proportional to the rate at which energy is lost through the…
We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimization approach introduced for the local case, we construct…
Different one-phase Stefan problems for a semi-infinite slab are considered, involving a moving phase change material as well as temperature dependent thermal coefficients. Existence of at least one similarity solution is proved imposing a…
A new comprehensive analysis of Stefan's flow caused by a free growing droplet in vapor-gas atmosphere with several condensing components is presented. This analysis, based on the nonstationary heat and material balance and diffusion…
The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain…