ABSTRACT
Let ( Ω , ℱ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739891/cc702078-b4a4-43bb-ab6a-a5f697d4ea00/content/inline271_1.tif"/> be a measurable space and {P θ .θεΘ}, Θ⊂R k be a family of probability measures on ( Ω , ℱ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739891/cc702078-b4a4-43bb-ab6a-a5f697d4ea00/content/inline271_2.tif"/> such that, under each P θ , θεΘ, X is a semimartingale with respect to the filtration { ℱ t , t ≥ 0 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739891/cc702078-b4a4-43bb-ab6a-a5f697d4ea00/content/inline271_3.tif"/> generated by X. Observe that the statistical experiment ( Ω , F t , { P θ | F t ; θ ε Θ } ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739891/cc702078-b4a4-43bb-ab6a-a5f697d4ea00/content/inline271_4.tif"/> corresponds to the observation of the semimartraingale X up to time t. Here, P θ | ℱ t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739891/cc702078-b4a4-43bb-ab6a-a5f697d4ea00/content/inline271_5.tif"/> denotes the probability measure P θ restricted to the σ-algebra ℱ t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739891/cc702078-b4a4-43bb-ab6a-a5f697d4ea00/content/commonft.tif"/> contained in ℱ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739891/cc702078-b4a4-43bb-ab6a-a5f697d4ea00/content/commonf.tif"/> .
