Stochastic Hybrid Processes with Hybrid Jumps

Abstract Switching Diffusion processes can be represented as pathwise unique solutions of SDE's in a hybrid state space that are driven by Brownian motion and Poisson random measure. This paper extends these SDE's to switching jump-diffusions, the jumps of which i) happen simultaneously with mode switching, and ii) depend on the mode after the switching. Jumps satisfying both i) and ii) are referred to as hybrid jumps. Because of ii) there is an anticipation effect in the SDE, which makes the hybrid jump extension challenging.

[1]  Mark H. A. Davis.  Piecewise‐Deterministic Markov Processes: A General Class of Non‐Diffusion Stochastic Models , 1984 .

[2]  A. Bensoussan,et al.  Stochastic Hybrid Control , 2000 .

[3]  Henk A. P. Blom,et al.  Petri-Nets and Hybrid-State Markov Processes in a Power-Hierarchy of Dependability Models , 2003, ADHS.

[4]  B. Marchal,et al.  Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel , 1976 .

[5]  Juan Ye.  Optimal control of piecewise deterministic Markov processes. , 1990 .

[6]  Philip Protter,et al.  Point Process Differentials with Evolving Intensities , 1983 .

[7]  E. Renshaw,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS , 1974 .

[8]  M. K. Ghosh,et al.  Ergodic Control of Switching Diffusions , 1997 .

[9]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[10]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[11]  M. K. Ghosh,et al.  Optimal Control of a Stochastic Hybrid System with Discounted Cost , 1999 .

[12]  S. Shankar Sastry,et al.  Conflict resolution for air traffic management: a study in multiagent hybrid systems , 1998, IEEE Trans. Autom. Control..

[13]  J. Jacod,et al.  Quelques remarques sur un nouveau type d'équations différentielles stochastiques , 1982 .

[14]  A. Skorokhod.  Asymptotic Methods in the Theory of Stochastic Differential Equations , 2008 .

[15]  Pierre-Etienne Labeau,et al.  Dynamic reliability: towards an integrated platform for probabilistic risk assessment , 2000, Reliab. Eng. Syst. Saf..

[16]  David D. Sworder,et al.  Estimation Problems in Hybrid Systems , 1999 .

[17]  H.A.P. Blom.  Bayesian estimation for decision-directed stochastic control , 1990 .

[18]  Stanley B. Gershwin,et al.  Hierarchical flow control: a framework for scheduling and planning discrete events in manufacturing systems , 1989, Proc. IEEE.

[19]  Henk A. P. Blom,et al.  Accident Risk Assessment for Advanced Air Traffic Management , 2001 .

[20]  D. Verms.  Optimal control of piecewise deterministic markov process , 1985 .

[21]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[22]  John Lygeros,et al.  Towars a Theory of Stochastic Hybrid Systems , 2000, HSCC.

[23]  M. K. Ghosh,et al.  Optimal control of switching diffusions with application to flexible manufacturing systems , 1993 .