The Geometric Markov Renewal Processes with Application to Finance

Discrete-Time Semi-Markov Random Evolutions and their Applications

“…[5], [18], [19], and [24]), (ii) to geometric Markov renewal processes, which also have many applications, including finance (see, e.g. [43], [44], [45], and [46]), and (iii) to dynamical systems (see, e.g. [10]).…”

Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas

“…In the diffusion approximation scheme for GMRP in scale of time tT 2 from 4.27 we obtain that characteristic m T tT 2 does not go to zero as T → ∞ since …”

“…If m 0 t is the limiting martingale for m T tT 2 in 4.22 as T → ∞, then from 4.31 and 3.10 we obtain 2 and m 0 t are zero-mean martingales then from 4.32 we obtain:…”

“…we obtain, replacing tT 2 by v tT 2 , that process G T t converges weakly to the process G 0 t as T → ∞, which is the solution of such martingale problem:…”

“…In fact, we consider the counting process v t in 2.5 in the new accelerated scale of time tT 2 , that is, v ≡ v tT 2 . Due to more rapid changes of states of the system under the balance condition, the fluctuations are described by a diffusion processes.…”