A stability criterion for Stochastic Hybrid Systems

1. A stability criterion for Stochastic Hybrid Systems A. Abate, L. Shi, S. Simic and S.S. Sastry University of California at Berkeley Application:Stocks Pricing Market has fixed number of equities (n), with an equilibrium price; X = # of stockholders willing to buy 1 title; Y = # of operators willing to sell 1 title. 3 regions: Equilibrium, Overpricing, Depreciation: Framework Dynamics: ODE’s, possibly nonlinear (flows have bounded Lipschitz constant) Underlying Markov Chain Temporal transitions t (statistically distributed) Single sharedequilibrium q Reset maps: bounded Lip constant. Stability in Probability: q is (asymptotically) stable in prob. if, for every D, q ε D, there exists a region E, included in D, s.t. the hybrid flow starting in any point in E will end up evolving in D, as time goes to infinity, with Probability 1. {limt P[|x(t)-q|>e]=0, for all e} Additional assumptions: n Domains (scalability: it works with n->∞) Vector fields fi , withflows ji Reset maps Rij Steady-state distr. p=[ p1,...,pn] E[ti] = li • Theorem • LTI systems; Define: • n = Si=1,..,nLip(ji li)pi • m = Si,j=1,..,nLip(Rij)piPij; • If nm <1, then equilibrium q is stable in probability (sufficient condition). • Extension: valid for NL vector fields, with fixed-time switches. • Simulations: HS with 5 nodes, linear vector fields, reset maps are the identity, jumps at fixed times. At every switch, one transaction can be made: 2D birth-death, continuous-time MC. Rationale: Given starting domain (status of the market) and equities’ value, prediction of the long-term dynamics of the stocks’ prices Future work: investigate other kinds of stability. May 10, 2004