Applications of state space models in finance

48 4 Markov regime switching
While γij(l) describes the conditional probability of being in state j at time t + l,
with the Markov chain starting from state i at time t, it does not provide the marginal
probability of being in state i at a given time t. With the probability distribution of the
initial state, π(1) := (π1(1), . . . , πm(1)) = (P (S1 = 1), . . . , P (S1 = m)), the probability
function of the state at time t is given by
π(t) := (P (St = 1), . . . , P (St = m)) = π(t)Γ l−1 . (4.10)
For a homogeneous and irreducible Markov chain, π(t) can be shown to converge to a
fixed vector πs := (π1, . . . , πm) for large t. This unique vector of dimension m satisfies
πs = πsΓ, (4.11)
and is called the vector of stationary transition probabilities. For existing πs, a Markov
chain is referred to as being stationary if πs describes the marginal distribution of the
states for all t = 1, . . . , T .
For more details on the well developed theory of Markov chains and further references,
see, for example, Hamilton (1994b, ¢ 22).
4.2 The basic hidden Markov model
The sequence of observations and hidden states in an independent mixture model are
independent by definition. Any potential correlation between the states cannot be captured
by an independent mixture as it does not take into account the respective information.
One method of modeling serially correlated time series is to use an unobserved
Markov chain to select the parameters. This yields the hidden Markov model as a special
dependent mixture model.
With {Xt} = {Xt, t = 1, . . . , T } denoting a sequence of observations and {St} =
{St, t = 1, . . . , T } denoting a Markov chain in the state space {1, . . . , m}, their respective
histories up to time t can be written as
X (t) := {X1, . . . , Xt}, (4.12)
S (t) := {S1, . . . , St}. (4.13)
Consider a stochastic process that consists of two elements: (i) an underlying and unobserved
parameter process {St} for which the Markov property (4.7) holds, and (ii) a
state-dependent observation process {Xt}, which fulfills the conditional independence
property
P (Xt = xt|X (t−1) = x (t−1) , S (t) = s (t) ) = P (Xt = xt | St = st). (4.14)
This means that with known St, Xt only depends on St and not on the history of states
or observations. The pair of stochastic processes {Xt} and {St} is referred to as an
m-state hidden Markov model whose basic structure is illustrated in Figure 4.3, which
is taken from Bulla (2006, ¢ 2).
Generally, different distributions are imposed for the various states. In this thesis, the
Markov chain with transition probability matrix Γ will be assumed to be homogeneous