Positivity and stabilisation for nonlinear stochastic delay differential equations

Abstract

We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero.

We do not require that any component of the coefficients of the equations satisfy Lipschitz conditions. Instead, we require that the functional part of each coefficient which feeds back the present state of the process admit to bounds imposed by a member of a particular class of concave functions. Lipschitz conditions are included as a special case of these bounds.

We generalise these results to the finite dimensional case, also constructing perturbations that can destabilise the otherwise stable solutions of a deterministic system of equations.

Keywords:

• nonlinear stochastic delay differential equations
• uniqueness
• positivity
• stabilisation
• destabilisation

AMS Subject Classification:

• 60H10
• 93E15
• 37H10
• 34K50

Acknowledgements

John Appleby was partially supported by an Albert College Fellowship awarded by Dublin City University's Research Advisory Panel. Cónall Kelly was partially supported by SFI research programme 04/RP1/L512, “Probability and its applications”. Alexandra Rodkina was partially supported by a London Mathematical Society grant and a Mona Research Fellowship Programme awarded by University of the West Indies, Mona.