By Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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**Example text**

G. the Stratonovich integral). Applying Itˆ o’s formula to the value process V yields 1 dV = Vx dX + V˙ dt + Vxx d X 2 1 = φ dX + V˙ dt + Vxx d X 2 Hence φ is self-ﬁnancing if, and only if, V satisﬁes the diﬀerential equation 1 V˙ dt + Vxx d X = 0 (15) 2 for all t > 0, where V˙ = ∂ V (x, t). g. a ”call” option H = (XT − K)+ ). s. 11. 3 (Black-Scholes model), one has d X t = σt2 Xt2 dt and (15) is equivalent to the (PDE) 1 V˙ + σ 2 X 2 Vxx = 0 2 This is the classical approach to option pricing, as pioneered by BlackScholes (1973) and Merton (1973), which leads to the solution of PDE’s under boundary conditions.

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. 7 by applying the d-dimensional Taylor-formula to the discrete increments of F. In diﬀerential form the Itˆ o-formula can be written as dF (Xt ) = ∇F (Xt ), dXt + 1 2 k,l ∂2F (Xt ) d X k , X l ∂xk ∂xl t which is the chain rule for stochastic diﬀerentials.

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. 24 2 Introduction to Itˆ o-Calculus Examples: 1) F (x) = xn implies t Xtn = X0n Xsn−1 +n t n(n − 1) dXs + 2 Xsn−2 d X s , 0 0 or in short notation d(X n ) = n X n−1 dX + n(n − 1) n−2 dX . X 2 In particular, for n = 2 and Xt = Bt standard Brownian motion, it follows t Bt2 =2 t Bs . dBs + 0 dB s. e. dF = F dX no longer holds for F (Xt ) = eXt with X ≡ 0) .

### Analysis, Manifolds and Physics. Basics by Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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