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Fundamentals of Stochastic Filtering with Applications_3

Tham khảo tài liệu 'fundamentals of stochastic filtering with applications_3', kinh tế - quản lý, kinh tế học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Interest Rate Model and Non-Linear Filtering 101 which are measures of the statistical properties of the noise process. The fact that these quantities satisfy the finite-dimensional stochastic dynamical system dZ0 t -I 2 t d t dZ t Zl_1 t -l z t gdt i 1 2.r is the crucial observation which allows US to reduce the original non-Markovian system to Markovian form of the dimension indicated. For the purposes of estimation of HJM models we need to consider the stochastic differential equation for the bond price P t T . However we have considered instead the stochastic differential equation for the log of the bond price since the Markovian system developed in equation 5.13 turns out to be linear in the state variables. This latter result will be most convenient from the point of view of implementing estimation procedures Bharand Chiarella 1995b . Since the volatility vector V t is independent of the state variables and is a function of time only the stochastic dynamic system 5.13 is Gaussian. The quantities t for all i which summarize the history of the noise process as well as the instantaneous spot rate of interest r t are not readily observable. Certainly there exists a one-to-one mapping between the t and yields drawn from the term structure and it would be possible to take n 1 such points to tie downZ iệt . While this would in principle provide an alternative estimation procedure it has the practical drawback that it would require continuous observations of the term structure. Many empirical studies attempt to proxy r t by some short-term rate e.g. 30-day treasury bill rates. Thus in developing estimation techniques we need also to consider the observation vector which in this case reduces to a scalar Y t cs t C 1 O . f. 5.15 The system 5.13 with the observation vector 5.14 is now in a form to which we are able to apply Kalman filter estimation techniques to form the log-likelihood function and hence estimate the parameters specifying the volatility function. 102 .