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3 Covariation and Multidimensional Itˆ o-Formula 29 Given F ∈ C 2 (IRd ), we use the following notations: ∇F (x) = ∂F ∂F (x) = Fx1 (x), . . e. ∆ = i=1 i=1 dF (x) = ( ∇F (x), dx ) = Fxi (x) dxi i scalar product classical diﬀerential. 4. (d-dimensional Itˆo-formula): For F ∈ C 2 (IRd ) one has t F (Xt ) = F (X0 ) + 1 ∇F (Xs ) dXs + 2 0 d t Fxk ,xl (Xs ) d X k , X l s , k,l=1 0 Itˆ o integral t ∇F (Xti ), (Xti+1 − Xti ) =: and the limit lim n ti ∈ τn ti ≤ t ∇F (Xs ) dXs 0 exists. Proof. The proof is analogous to that of Prop.
5. For a right-continuous ﬁltration the condition (19) is equivalent to [T < t] ∈ Ft (t ≥ 0). Proof. 6. Every stopping time is a decreasing limit of discrete stopping times. Proof. Consider the sequence Dn = K 2−n K = 0, 1, 2, . . n=1,2,... of dyadic partitions of the interval [0, ∞). Deﬁne, for any n, Tn (ω) = K 2−n if T (ω) ∈ [(K − 1) 2−n , K 2−n ) +∞ if T (ω) = ∞ Clearly, [Tn ≤ d] = [Tn < d] ∈ Fd for d = K 2−n ∈ Dn and [Tn ≤ t] = [Tn = d] ∈ Ft . t≥d∈Dn Hence (Tn ) are stopping times and Tn (ω) ↓ T (ω) ∀ ω ∈ Ω.
On the other hand E[XSˆ ] = E[XS ; A] + E[XT ; Ac ], which together with the above equation implies E[XS ; A] = E[XT ; A]. 15. , for (XT ∧n ) uniformly integrable. 6 Stopping Times and Local Martingales 43 As an application of the optional stopping theorem we consider the hitting times of a Brownian motion for an interval a ≤ 0 < b deﬁned by / [a, b]}. 10. P [BTa,b = b] = and |a| b , P [BTa,b = a] = , b−a b−a E[Ta,b ] = |a| · b. Proof. 1) 0 = E[B0 ] = E[BT ] = b · P [BT = b] + a (1 − P [BT = b]) =⇒ P [BT = b] = −a .