Springer Finance

110 3 Brownian Motion

Let m be a real number, and define the first passage time to level m

Tm = min{t ;;:: 0; W(t) = m}. (3.6.3)

This is the first time the Brownian motion W reaches the level m. If the

Brownian motion never reaches the level m, we set Tm = oo. A martingale

that is stopped ("frozen" would be a more apt description) at a stopping

time is still a martingale and thus must have constant expectation. (The text

following Theorem 4.3.2 of Volume I discusses this in more detail.) Because

of this fact,

1 = Z(O) = IEZ(t 1\ Tm ) =IE [ exp { uW(t 1\ Tm ) -�u^2 (t 1\ Tm )}] , (3.6.4)

where the notation t 1\ Tm denotes the minimum oft and Tm.

For the next step, we assume that u > 0 and m > 0. In this case, the

Brownian motion is always at or below level m for t � T m and so

(3.6.5)

If Tm < oo, the term exp { -�u^2 (t 1\ Tm )} is equal to exp { -�u^2 rm} for large

enough t. On the other hand, if Tm = oo, then the term exp { -�u^2 (t 1\ Tm )}

is equal to exp { -�u^2 t}, and as t ---+ oo, this converges to zero. We capture

these two cases by writing

where the notation n{r,.<oo} is used to indicate the random variable that takes
the value 1 if Tm < oo and otherwise takes the value zero. If Tm < oo, then
exp{uW(t 1\ Tm )} = exp{uW(rm)} = e""m when t becomes large enough. If
Tm = oo, then we do not know what happens to exp{uW(t 1\ Tm )} as t---+ oo,
but we at least know that this term is bounded because of (3.6.5). That is
enough to ensure that the product of exp{uW(t 1\ Tm )} and exp { -�u^2 rm}
has limit zero in this case. In conclusion, we have

t��

exp { uW(t 1\ Tm) -�u^2 (t 1\ Tm )} = H{r,.<oo} exp { um-�u^2 Tm}.

We can now take the limit in (3.6.4)^3 to obtain

1 =IE [n{r,.<oo} exp { um-�u^2 rm}]

or, equivalently,

(^3) The interchange of limit and expectation implicit in this step is justified by the
Dominated Convergence Theorem, Theorem 1.4.9.