We present an example where the representation of a bounded r.v. considered as the terminal variable of a martingale can be explicitly computed. et $B$ be a Brownian motion and $T_a=inf\{t\ge 0:B_t=a\}$ where $a>0$.

1. Using the Doléans-Dade exponential of $\lambda B$, prove that, for $\lambda >0$

$$1_{\{T_a<T\}}=P(T_a<T)+2{\int }_0^{T_a\wedge T}\phi (T-s,B_s-a)d{B}_s.$$

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E(e-/2 Ft) = c +\ 0 Ta^t e-a-Bu)-u/2dBu (1.6.3) and that Ta e-x T/ = e - + S e-a-Bu)-u/2dBu 0 Check that E((e-x(a-Bu)-Xu/2) du) < . Prove that (1.6.3) is not true for <0, i.e., that, in the case := -A 0 the quantities