  In this paper we introduce a completely continuous and time-variate model of
the evolution of market limit orders based on the existence, uniqueness, and
regularity of the solutions to a type of stochastic partial differential
equations obtained in Zheng and Sowers (2012). In contrary to several models
proposed and researched in literature, this model provides complete continuity
in both time and price inherited from the stochastic PDE, and thus is
particularly suitable for the cases where transactions happen in an extremely
fast pace, such as those delivered by high frequency traders (HFT's).
  We first elaborate the precise definition of the model with its associated
parameters, and show its existence and uniqueness from the related mathematical
results given a fixed set of parameters. Then we statistically derive parameter
estimation schemes of the model using maximum likelihood and least
mean-square-errors estimation methods under certain criteria such as AIC to
accommodate to variant number of parameters . Finally as a typical economics
and finance use case of the model we settle the investment optimization problem
in both static and dynamic sense by analysing the stochastic (It\^{o})
evolution of the utility function of an investor or trader who takes the model
and its parameters as exogenous. Two theorems are proved which provide criteria
for determining the best (limit) price and time point to make the transaction.
