A Definitions and results from stochastic calculus

Abstract

In the discussion above, we have made extensive use of some definitions and results from the theory of stochastic calculus. To disecumber our discussion from these technicalities, we have confined those which are less known in this section. For a more complete treatment see for instance Protter [3], whose notation is followed here. Let (Ω, ℱ, P, {ℱ _t }) be a filtered complete probability space satisfying the usual conditions (see [3]). Given a stochastic process X on (Ω, ℱ, P) we write X _t instead of X(t, ω) and X _{t −} for lim _s↑t X(s, ω). Moreover, we define ΔX _t = X _t − X _{t −} to be the jump at t. Finally, we set X₀₋ = 0 by convention; remark however that we do not require X₀ = 0.