A stability criterion for Stochastic Hybrid Systems
A. Abate, L. Shi, S. Simic and S.S. Sastry
Framework
Dynamics: ODE’s, possibly
nonlinear (flows have bounded
Lipschitz constant)
Underlying Markov Chain
Temporal transitions t
(statistically distributed)
Single shared equilibrium q
Reset maps: bounded Lip
constant.
Theorem
LTI systems; Define:
l
p
i
n = Si=1,..,nLip(ji ) i
p
P
i
m = Si,j=1,..,nLip(Rij) ij;
•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
Stability in Probability: q is
are the identity, jumps at fixed
(asymptotically) stable in prob. times.
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 , with flows ji
Reset maps Rij
Steady-state distr. p=[ p1,...,pn]
E[ti] = li
May 10, 2004
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:
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 longterm dynamics of the stocks’ prices
Future work: investigate other kinds of stability.