On a Parabolic Free Boundary Equation Modeling Price Formation

Abstract

We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results.

On a price formation free boundary model by Lasry and Lions

“…We remark that due to the boundedness of f these sums converge in D (R × [0, ∞)). It is very easy to check that F satisfies, in the sense of distributions, the heat equation with initial datum F (x, t = 0) =: F I (x), given by (4). Clearly, the free boundary p = p(t) is now the zero level set of F .…”

“…This model has been studied in a number of papers, cf. [1,2] and [4]. Here we shall present the first global existence result of a smooth solution on the whole real line.…”

On a Boltzmann-type price formation model

“…The Dirac deltas correspond to trading events that take place at the agreed price p = p(t), shifted by the transaction cost a. The analysis of system (1.3) was studied in a number of papers; see Markowich et al [3], Chayes et al [4] and Caffarelli et al [5,6]. The main difference between system (1.1) and system (1.3) is that the agreed price p = p(t) enters as a free boundary in (1.3).…”

Instability and Bifurcation in a Trend Depending Price Formation Model

“…Both the dynamics of the problem on the whole line and that of the Neumann problem are actually reasonably well understood by the work done in the last few years by several authors, like L.A. Caffarelli [4,5,7,[9][10][11]16]. Essentially, it has been shown that every solution approaches a single equilibrium when time tends to infinity.…”