By George Kempf

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17) Combining (16) and (17) gives the PDE 1 V˙ + r X Vx + σ 2 X 2 Vxx = r · V 2 (Black-Scholes PDE). (18) This PDE holds for any derivative of the form F (XT ). A simple example is a forward contract on XT ﬁxed at time t = 0 at price K. It is easy to check that F (t) = Xt − K e−r(T −t) = V (Xt , t) is the (unique) solution of (18) under the boundary constraint F (T ) = V (XT , T ) = XT − K. 6 Stopping Times and Local Martingales In the previous sections we have concentrated on real-valued functions and shown how Itˆ o’s calculus comes into play for functions of unbounded variations.

3) dSt = µ(t) dt + σ(t) dBt St The above SDE deﬁnes a diﬀusion or Itˆ o process. It is the standard model used in ﬁnance for the returns of a security price process St with inﬁnitesimal drift µ(t)dt and stochastic noise σ(t)dBt , where σ(t) is called the volatility of St . We show that the SDE has the solution t St = S0 exp t 1 µ(s) − σ 2 (s) ds + 2 0 σ(s) dBs . (13) 0 Proof. We give a proof using Itˆ o’s product formula. The process (13) can be written as t µ(s) ds · E(Mt ) = Yt · Zt St = S0 exp 0 t with Mt = t σ(s) dBs and M t σ 2 (s) ds.

Using only elementary facts of independent normally distributed random variables, it leads to the Dol´eans-Dade exponential as new density under a change of measure for the Brownian motion. Sect. 2 deals with the Girsanov transformation in general form, as can be found in Revuz-Yor (1991). The proofs are straightforward applications of tools developed in Chap. 2. This section, which at ﬁrst sight looks rather abstract, is basic for the applications to ﬁnance in Chapters 4 and 5, where the general Girsanov transformation is repeatedly used.

### Abelian integrals by George Kempf

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