# Edgar Asplund; Lutz Bungart's A first course in integration PDF

By Edgar Asplund; Lutz Bungart

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S. =⇒ (21) Lebesgue 2) The general case can be reduced to 1) by deﬁning the following stopping times Sn (ω) = inf{t : |Xt (ω)| > n} Un (ω) = inf{t : |Ht (ω)| > n}. Consider the stopping time Vn = Sn ∧ Un ∧ Tn where Tn is a localizing sequence of bounded stopping times for (Xt ). By assumption (XTn ∧t ) is a martingale. Since Vn ≤ Tn , the stopping theorem implies that (XVn ∧t ) is a martingale, and hence by 1) also (MVn ∧t ). Clearly Vn ↑ ∞. Hence (Mt ) is a local martingale. 4. If (Xt ) is a local martingale, then Xt2 − X t (t ≥ 0) is a local martingale.

PX ≡ PX distribution of X distribution of X ≡ under P under P For X ∼ N(0, σ 2 ) under P X ∼ N(µ, σ 2 ) under P (↔ X = X − µ ∼ N(0, σ 2 ) under P ) it follows 1 nµ,σ2 (x) dPX (x) = = e σ2 dPX n0,σ2 (x) (µx− 21 µ2 ) . 1 Heuristic Introduction 57 Application to Brownian Motion Let (Bt )0≤t≤1 be a BM on (Ω, (F)t , P ). =⇒ Bt ∼ N(0, t) under P and ∆Bt = Bt+∆t − Bt ∼ N(0, ∆t), independent of Bt Consider now a BM with drift t Bt = Bt − Hs ds 0 for some stochastic process (Hs )0≤s≤1 . Question: Under which measure P is (Bt ) again a BM (without drift) ?

2 Quadratic Variation and 1-dimensional Itˆ o-Formula 23 For n −→ ∞ it follows Rn (ti ) ≤ a) n t≥ti ∈τn · (∆Xti )2 −−−→ 0 . n↑∞ t≥ti ∈τn bounded (F (Xti+1 − F (Xti )) −−−→ F (Xt ) − F (X0 ) b) n↑∞ t≥ti ∈τn c) t 1 1 F (Xti ) (∆Xti )2 −−−→ n↑∞ 2 2 F (Xs ) d Xs . ) Hence also F (Xti ) ∆Xti must converge and there exists t F (Xti ) ∆Xti =: lim n t≥ti ∈τn F (Xs ) dXs . 8. In the classical case ( X ≡ 0 or X ∈ FV) Itˆ o’s formula reduces to t F (Xt ) = F (X0 ) + F (Xs ) dXs 0 or in short notation, for X ∈ C 1 , dF (X) = F (X) dX = F (X) X˙ dt.