Existence and uniqueness of stochastic equations of optional semimartingales under monotonicity condition

ABSTRACT

This paper is devoted to the question of existence and uniqueness of strong solutions of stochastic differential equations with respect to optional semimartingales. Optional semimartingales have right and left limits (ladlag) but are not necessarily continuous, therefore, defined on un usual probability spaces. Integration theory with respect to optional semimartingales is well developed. However, not much attention is given to stochastic equations of optional semimartingales. An existence and uniqueness result for strong solutions of a very general class of optional stochastic equations with non-Lipschitz coefficients is given. Moreover, potential applications to physics and finance are given.

KEYWORDS:

• Stochastic integral equations
• stochastic differential equations
• non-Lipschitz conditions
• optional semimartingales
• Ladlag processes
• comparison theorem
• Gronwall lemma
• Picard approximation
• stochastic particles collision
• backward stochastic differential equations

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is supported by the NSERC Discovery [grant number 5901].