Parabolic free boundary price formation models under market size   fluctuations
  In this paper we propose an extension of the Lasry-Lions price formation model which includes fluctuations of the numbers of buyers and vendors. We analyze the model in the case of deterministic and stochastic market size fluctuations and present results on the long time asymptotic behavior and numerical evidence and conjectures on periodic, almost periodic and stochastic fluctuations. The numerical simulations extend the theoretical statements and give further insights into price formation dynamics. 
Peter A. Markowich and Josef Teichmann and Marie-Therese Wolfram
math.AP