Mou-Hsiung Chang  Tao Pang   Moustapha  Pemy
Abstract：
We consider a finite time horizon optimal stopping problem for a
system of stochastic functional differential equations with a bounded
memory. Under some sufficiently smooth conditions, a Hamilton-Jacobi-
Bellman (HJB) variational inequality for the value function is derived
via dynamical programming principle. It is shown that the value function
is the unique viscosity solution of the HJB variational inequality.
As an application of the results obtained, a pricing problem is considered
for American options in a financial market with one riskless
bank account that grows according to a deterministic linear functional
differential equation and one stock whose price dynamics follows a nonlinear
stochastic functional differential equation. It is shown that the
option pricing can be formulated into an optimal stopping problem
considered in this paper and therefore all results obtained are applicable
under very realistic assumptions.