The multi-dimensional Stochastic Stefan Financial Model for a portfolio of assets

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 an \(n\)-dimensional outer parabolic Stefan problem with noise. In particular, our aim is to estimate for a short time period the areas of zero trading, and their diameter which approximates the minimum of the \(n\) spreads of the portfolio assets for orders from the \(n\) limit order books of each asset respectively. In dimensions \(n=3\), and for zero volatility, this problem stands as a mean field model for Ostwald ripening, and has been proposed and analyzed by Niethammer. Therein, when the initial moving boundary consists of well separated spheres, a first order approximation system of odes had been rigorously derived for the dynamics of the interfaces and the asymptotic profile of the solution. In our financial case, we propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices, while each sphere may correspond to a different market; the relevant radii represent the half of the minimum spread. We apply Itô calculus and provide second order formal asymptotics for the stochastic version dynamics, written as a system of stochastic differential equations for the radii evolution in time. A second order approximation seems to disconnect the financial model from the large diffusion assumption for the trading density. Moreover, we solve the approximating systems numerically.