Applications of state space models in finance

4.4 Forecasting and decoding 55
models, the Viterbi algorithm is employed for this purpose. The Viterbi algorithm represents
an efficient dynamic programming algorithm. It was originally proposed by Viterbi
(1967) as an error-correction procedure in noisy digital communication applications.
According to Rabiner (1989) the Viterbi algorithm can be summarized as follows. At
first, it is necessary to define the quantity
νt(i) := max P
i1,...,it−1
�
S1 = i1, . . . , St−1 = it−1, St = i, X (t) = x (t)�
, (4.40)
which represents the highest likelihood along a single path at date t accounting for the
first t observations X (t) = x (t) and ending at state St = i. It can be shown that νt(i)
satisfies
νt+1(j) = [max(νt(i)γij)]
· pj(xt+1), (4.41)
i
from which a T × m matrix of values νt(i) for t = 1, . . . , T , and j = 1, . . . , m, can be
derived. The required sequence of hidden states (i∗ 1, . . . , i∗ T ) is computed in four steps:
1. Initialization:
ν1(i) = πi · pi(x1), 1 ≤ i ≤ m, (4.42)
κ1(i) = 0, (4.43)
where κt(i) is another auxiliary variable whose role becomes clear in the second step.
2. Recursion:
νt(j) = max
1≤i≤m [νt−1(i)γij] · pj(xt), 1 ≤ j ≤ m, 2 ≤ t ≤ T, (4.44)
κt(j) = argmax[νt−1(i)γij],
1 ≤ j ≤ m, 2 ≤ t ≤ T, (4.45)
1≤j≤m
where κt(j) yields the state i at time t−1 which is most likely to lead to state j at time t.
3. Termination: The first recursion is terminated by maximizing νT (i) and storing the
likelihood of the most probable path as a vector p ∗ . The most probable state at time T
is stored in i ∗ T :
p ∗ = max
1≤i≤m [νT (i)], (4.46)
i ∗ T = argmax[νT
(i)]. (4.47)
1≤i≤m
4. Path backtracking: Starting with the most probable last state i∗ T , the most likely
path can be computed by tracking back from t = T − 1 to t = 1 in a second recursion
using κt:
i ∗ t = κt+1i ∗ t+1 , t = T − 1, T − 2, . . . , 1. (4.48)
The resulting optimal state sequence is referred to as Viterbi path. In order to avoid
numerical underflow, the Viterbi algorithm can alternatively be stated in terms of logarithms.