Here, [L.sub.0] is the

infinitesimal generator of the SFMR process Y.

By this approach, A(t) is an

infinitesimal generator of a [C.sub.0]-quasisemigroup on X.

In general, CTMC process [M(t);t [greater than or equal to] 0} is characterized by its

infinitesimal generator. The

infinitesimal generator is a square matrix whose dimension is same as the dimension of state space.

The

infinitesimal generator matrix Q of the QBD process X is then given by

where A : D(A) [subset or equal to] X [right arrow] X is the

infinitesimal generator of strongly continuous semigroup of bounded linear operators T(t), t > 0.

It turns out that 1/2[D.sub.xx], that is, one-half times the second-order vertical derivative, is not the appropriate

infinitesimal generator, because of path dependence.

With the help of the

infinitesimal generator, we obtain an explicit formula for the value function and the stopping time.

Compared with the model in [23], we will see that the

infinitesimal generator [??] is different, and in our model, [[pi].sub.k0] = 0, k [greater than or equal to] N + 1.

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Caputo fractional derivative of order 0 < [alpha] < 1,T > 0, A : D(A) [subset] E [right arrow] E is the

infinitesimal generator of an [alpha]-resolvent family [([S.sub.[alpha]](t)).sub.t[greater than or equal to]0], the solution operator [([T.sub.[alpha]](t)).sub.t[greater than or equal to]0] is defined on a complex Banach space E, f : [0, T] x B x E [right arrow] E is a given function.

1) Generation of system states and transition matrix (an

infinitesimal generator matrix).

The flow [[PHI].sub.[tau]] is called the exponential map, corresponding to the vector field F; the vector field F is called the

infinitesimal generator of [[PHI].sub.[tau]], or, according to (2), the time derivative operator.

The

infinitesimal generator matrix Q for this model is given below