| This dissertation paper is primarily focused on the investigation of both the first passage time and the pre-exit time with three or more active and arbitrarily many passive components, on a delayed renewal process marked by a multivariate random walk. Throughout the research, the joint transformation of the first passage index, first passage time, active and passive excess values and pre-exit time and pre-exit values remain the center of attention. The results obtained are in closed and compact form expressions for three active and four active components, generalized from recent results with both the first passage time and the pre-exit time for one and two active components in [12, 13] and three active components in [14], which is done without the pre-exit time.; Firstly, we introduce and analyze a delayed renewal process T = {lcub}tau0,tau1,...{rcub} marked by a multivariate random walk ( S,T ) and its behavior about fixed levels to be crossed by one of the components of ( S,T ). We derive the joint distribution of first passage time tau rho, pre-exit time taurho-1 (i.e. the instant one phase prior to the first passage time), and the respective values of ( S,T ) at taurho and taurho-1 in a closed form for three active components and arbitrarily many passive components.; Then, we continue our studies of multivariate marked recurrent processes (initiated in the nineties) which we refer to as a multivariate random walk with four active components and arbitrarily many passive components. It is formed with a delayed renewal process T = {lcub}tau0,tau1,...{rcub}, along with marks representing a multidimensional recurrent process S .; Finally, we deal with a class of real-valued random processes observed over random epochs of time that forms a delayed renewal process. The model does not restrict the class to merely monotone processes. The objective is to find the first passage of the process exiting a rectangular set and registering the value of the process at this time. The joint transformation of the named random characteristics of the process are derived in a closed form. We conclude with examples, including numerical examples, demonstrating the use of the results as well as practical applications to finance. |
