ABSTRACT
A fundamental concept in contemporary stochastic analysis is the stochastic basis. This is a complete probability space ( Ω , F , P ) $ (\iOmega ,{ \mathcal{F}},\text{ }P) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_1.tif"/> endowed with a filtration F = ( F t ) t ≥ 0 $ F = ({\mathcal{F}}_{t} )_{t \ge 0} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_2.tif"/> , which is a nondecreasing right-continuous family of σ - algebras. F t ⊆ F $ {\mathcal{F}}_{t} \subseteq {\mathcal{F}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_3.tif"/> completed by P-null sets from F $ {\mathcal{F}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_4.tif"/> . As usual, we assume that all stochastic processes X = (X t (ω)) t≥0 under consideration are defined on the stochastic basis ( Ω , F , F , P ) $ (\iOmega ,{ \mathcal{F}},\text{ }F, P) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_5.tif"/> and are F $ {\mathcal{F}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_6.tif"/> -adapted (for each fixed t, the random variable X t is measurable with respect to F t $ {\mathcal{F}}_{t} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_7.tif"/> ). Moreover, for almost all ω ∊ Ω, the trajectories X. (ω) belong to the space D ( R ) $ {\mathbb{D}}({\mathbb{R}}) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_8.tif"/> of functions x(t), t ≥ 0, right continuous with finite left limits, taking the values in a space R . $ {\mathbb{R}}. $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_9.tif"/> As a rule, this space will be the d-dimensional Euclidean space R d , $ {\mathbb{R}}^{d} , $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315118505/6befe1c9-a2f5-4407-a7d2-aff70ef0dad0/content/inline-math1_10.tif"/> d ≥ 1. The measurability of X is understood as the measurability of the mapping:
