Estimation in Financial Models - RiskLab

Chapter 3
Parameter Estimation
3.1 Diusion models
Consider the general type of a diusion process X =(X t ) t0 dened as the
solution to the stochastic dierential equation
dX t = b(t; X t ; ) dt + (t; X t ; ) dW t ; X 0 = x 0 ; t 0; (3.1)
where W is an r-dimensional Wiener process, 2 IR p , b(; ; ) :[0; 1)
IR d 7! IR d and (; ; ) : [0; 1) IR d 7! M dr are "nice" 1 functions where
M dr denotes the set of real d r matrices.
The situation will be discussed where a realization of the process X is observed,
but the parameter is unknown to the observer. Hence, we have
to construct suciently good estimators of and examine their properties.
Deriving estimates of there are two dierent kinds of observations to distinguish:
continuous observations of X as considered in section 3.1.1, and
discrete observations that will be considered in section 3.1.2.
3.1.1 Continuous observations
Suppose that X satises
dX t = b(; X t )dt + (; X t )dW t ; X 0 = x 0 ; t 0; (3.2)
where for convenience X and W are one-dimensional processes, b and are
smooth functions, and the parameter 2 IR p is to be estimated by
1 b and are Lipschitz continuous and satisfy a growth condition. Then there exists a
unique strong solution of (3.1), see [65], p.128 and p.136.
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