Real semimartingales and stochastic integrals
The usual setup for the general theory of processes is a complete probability space (Ω,ℱ,ℙ) endowed with a filtration (ℱ t )t ≥0: each ℱtis a sub-σ-field of ℱ, contains all negligible events in ℱand ℱ t =∩ε>0 ℱ t+ε ; this equality says that t↦ℱ t is increasing and right-continuous. Increasingness is the main point; the other conditions are mere technical assumptions. Removing them is possible, but rarely useful, and makes some results heavier.
KeywordsStochastic Differential Equation Variable Formula Quadratic Variation Stochastic Integral Stochastic Calculus
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