  In this paper we introduce a new methodology to determine an optimal
coefficient of penalized functional regression. We assume the dependent,
independent variables and the regression coefficients are functions of time and
error dynamics follow a stochastic differential equation. First we construct
our objective function as a time dependent residual sum of square and then
minimize it with respect to regression coefficients subject to different error
dynamics such as LASSO, group LASSO, fused LASSO and cubic smoothing spline.
Then we use Feynman-type path integral approach to determine a
Schr\"odinger-type equation which have the entire information of the system.
Using first order conditions with respect to these coefficients give us a
closed form solution of them.
