Stability of Stochastic Delay Hybrid Systems with Jumps

The jump diffusion process has come to play an important role in many branches of science and industry. In their book [25], Øksendal and Sulem have studied the optimal control, optimal stopping and impulse control for jump diffusion processes. In mathematical finance theory, many researchers have developed option pricing theory, for example, Merton [24] was the first to use the jump process to describe the stock dynamics, Bardhan andChao [2] were amongst the first authors to consider market completeness in a discontinuous model. The jump diffusion models have been discussed by Chan [6], Föllmer and Schweizer [9], EI Karoui and Quenez [13], Henderson and Hobson [12], and Merculio and Runggaldier [23], to name a few. In this study, we shall investigate the delay hybrid jump diffusion process. First, the importance of delay equations derives from the fact that many phenomena witnessed