Explosive and non - explosive solutions of stochastic functional differential equations

The paper covers questions relating to the existence, explosion, and prevention of explosion, of solutions of stochastic functional differential equations (SFDEs) of Itô type. It is well-known that solutions of a deterministic ordinary differential equation x(t) = f(x(t)) can explode in finite time if the function f does not obey a global linear bound; however, uniqueness up to the explosion time is guaranteed if f obeys a local Lipschitz condition. Traditionally, proofs of the global existence and uniqueness of solutions of SFDEs rely upon the imposition of global linear bounds on the coefficients. Alternatively, global existence has been proved under the assumption that there is a sufficiently strong negative feedback in the instantaneous part of the drift of the equation. Here, we adapt ideas from the theory of stochastic differential equations to prove the global existence of solutions when the drift term is of pure delay type, without requiring growth assumptions on the coefficients, while also weakening the standard regularity requirements for existence. We also consider the effect that a noise perturbation can have on the explosion of solutions of a deterministic functional differential equation. In particular, we show that a sufficiently strong noise perturbation can suppress explosions exhibited by the solutions of the deterministic equation. However, this suppression is sensitive to the structure of the noise perturbation: examples can also be given which show that a strong noise perturbation of the appropriate from can cause the almost sure explosion of solutions of a SFDE, while the associated deterministic problem does not have explosive solutions.