Price formation model with zero-Neumann boundary condition

ABSTRACT

This paper concerns the mathematical analysis of a mathematical model for price formation. We take a large number of rational buyers and vendors in the market who are trading the same good into consideration. Each buyer or vendor will choose his optimal strategy to buy or sell goods. Since markets seldom stabilize, our model mimics the real market behavior. We introduce three models. All of them are modifications of the original J.-M. Lasry and P. L. Lions evolution model. In the first modified model, a random term is added to mimic the randomness of trading in the real market. This reflects markets with low volatility, where it might be difficulty to buy or sell goods at specific price. In the second model, we use cumulative density function instead of density function. We give numerical simulations on these two models in order to have a general picture on the solution. In the third model, we add a term associated with the parameter R to destabilize the original Larsy–Lions model and study oscillations and wave solutions depending on different values of R. We also study existence and uniqueness of the solution. Moreover, Several plots are given to demonstrate these results corresponding to the theoretical prediction.

COMMUNICATED BY:

• Yongzhi Xu

KEYWORDS:

• Price formation model
• bitcoin model
• free boundary problem
• instability analysis

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35K61 Nonlinear initial-boundary valueproblems for nonlinear parabolic equations
• 35K05 heat equation

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Chubo Deng http://orcid.org/0000-0003-3469-5624