Estimation in Financial Models - RiskLab

Example 2
The solutions of the stochastic dierential equation
dX t =( + X t ) dt + (X t ) dW t ; (3.54)
where X 0 = x 0 and the function takes positive values in IR, are called
mean-reverting processes (see also model (1.5)). The unknown parameters
are and . Our aim is to be able to calculate the martingale estimating
functions ~G n and G n.
Lemma 1 The function
f(t) E ; (X t jX 0 )
solves
f 0 (t) = + f(t): (3.55)
Proof: Write (3.54) in integral form
X t = X 0 +
Conditioning on X 0 we have
and equivalently
Z t
0
( + X s )ds +
Z t
0
(X s )dW s :
Z t
E ; (X t jX 0 ) = E ; (X 0 jX 0 )+E ; ( + X s )dsjX 0
0
(X s )dW s jX 0 ;
0
| {z }
=0
+E ;
Z t
E ; (X t jX 0 )=X 0 + t +
Z t
0
E ; (X s jX 0 )ds:
We conclude
dE ; (X t jX 0 )
= + E ; (X t jX 0 );
dt
and the claim follows. Note that the function f r (t) =E ; (X t jX r ), 0 r t,
also solves (3.55), for the proof stays the same apart from
E ;
Z t
0
(X s )dW s jX r
=
Z r
0
(X s )dW s ;
which is independent of t and thus plays no role in the derivative. 2
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