Lecture 12 : Continuous Time Markov Chains

Lecture 12 : Continuous Time Markov Chains

Lecture 12 : Continuous Time Markov Chains
Parimal Parag
1
Continuous Time Markov Chains
Consider a continuous time stochastic process {X(t), t ≥ 0} taking on values
on the set of non-negative integers. The process {X(t), t ≥ 0} is continuous
time Markov chain if for all s, t ≥ 0 and i, j ∈ N0 , P (X(t + s)|X(u), u ∈
[0, s]) = P (X(t+s)|X(s)). If the probability is independent of s, the continuous
time Markov chain (CTMC) has homogeneous transitions and we denote the
probability by Pij (t). Suppose X(0) = i, X(u) = i, ∀u ∈ [0, s]. We are
interested in knowing probabilities of the form P (X(v) = i, v ∈ [s, s+t]||X(u) =
i, u ∈ [0, s]). To that end, τi , {t ≥ 0 : X(t) 6= i|X(0) = i}. We could sample
the process at these instants and construct a DTMC out of it and study the
same. Observe that,
P (τi ≥ s + t|τi > s) = P (X(v) = i, v ∈ [s, s + t]|X(0) = i)
= P (τi ≥ t|X(0) = i)
τi is memoryless and exponentially distributed.
1.1
Alternative way of constructing CTMC
A CTMC is a stochastic process that each time it enters state i,
1. τi ∼ exp(νi ).
2. P(Entering state j— state i ) ≡ Pij is such that
P
i6=j
Pij = 1.
Note that Pi,j and τi are not dependent. νi is called rate of state i. νi < ∞. Else
we call the state to be instantaneous. A CTMC is a DTMC with exponential
sojourn time in each state.
Definition: A CTMC is called ”regular” if P (#transitionsin[0, t]isfinite) =
1, ∀t < ∞. Consider the following example of a non-regular CTMC.
Example: Pi,i+1 = 1, νi = i2 . Show that it is regular.
Let Q be a matrix such that
1. qij = νi Pij , ∀i 6= j.
2. qii = −νi .
1
Properties of Q :
1. 0 ≤ −qii < ∞, ∀i.
2. qij ≥ 0, ∀i 6= j.
3.
P
j
qij = 0, ∀i.
From the Q matrix, we can construct the whole CTMC. In DTMC, we
had the result P (n) (i, j) = (P n )i,j . We can generalize this notion in the case of
P
k
CTMC as follows: P = eQ , k∈N0 Qk! . Observe that eQ1 +Q2 = eQ1 eQ2 , enQ =
(eQ )n = P n .
Theorem 1.1. Let Q be a finite sized matrix. Let P (t) = etQ . Then {P (t), t ≥
0} has the following properties:
1. P (s + t) = P (s)P (t), ∀s, t (semi group property).
2. P (t), t ≥ 0 is the unique solution to the forward equation,
P (t)Q, P (0) = I.
3. And the backward equation
4. For all k ∈ N,
dk P (t)
|
dk (t) t=0
dP (t)
dt
dP (t)
dt
=
= QP (t), P (0) = I.
= Qk .
−tQ
Proof. dM (t)e
= 0, M (t)e−tQ is constant. M (t) is any matrix satisfying the
dt
forward equation.
Theorem 1.2. A finite matrix Q is Q matrix if and only if P (t) = etQ is a
stochastic matrix for all t ≥ 0.
Proof. P (t) = I + tQ + O(t2 ) (f (t) = O(t) ⇒ f (t)
≤ c, for small t, c < ∞
t
). qij ≥ 0 if and only if Pij (t) ≥ 0, ∀i 6= j and t ≥ 0 sufficiently small.
P (t) = P ( nt )n . Note that if Q has zero row sums, Qn also has zero row sums.
X
XX
XX
[Qn−1 ]ik Qkj =
Qkj [Qn−1 ]ik = 0.
[Qn ]ij =
j
j
X
j
Conversely
1.2
j
k
k
X tn X
Pij (t) = 1 +
[Qn ]ij = 1 + 0 = 1.
n! j
n∈N
P
j
Pij (t) = 1, ∀t ≥ 0, then
P
j
Qij =
dPij (t)
dt
= 0.
Kolmogorov Differential Equations
Lemma 1.3.
2. limt→0
1. limt→0
Pij (t)
t
1−Pii (t)
t
= νi .
= qij .
Lemma 1.4. For all s, t ≥ 0, Pij (t + s) =
2
P
k∈N0 Pik (t)Pkj (s)
1.3
Chapman Kolmogorov Equation for CTMC
Theorem 1.5. Kolmogorov Backward equation: For all i, j, t ≥ 0, Pij0 (t) =
P
dP (t)
k6=i qik Pkj (t) − νi Pij (t) ,
dt = QP (t)
P
Proof. Pij (t + h) = k Pik (h)Pkj (t).
Pij (t + h) − Pij (t) =
X
Pik (h)Pkj (t) − (1 − Pii (h))Pij (t),
k6=1
P
dPij (t)
divide by h, h → 0, we get dt
= limh→0 Pij0 (t) = k6=i qik Pkj (t) − νi Pij (t).
Now the exchange of limit and summation has to be justified.
lim inf
h→0
=
X
X Pik (h)
h
k6=1
Pkj (t) ≥ lim inf
h→0
X Pik (h)
k6=1
h
Pkj (t), k < N
qjk Pkj (t), k < N.
k6=1
This is true for any finite N . Take supremum over all N . We get
P
P
P
lim inf h→0 k6=1 Pikh(h) Pkj (t) ≥ k6=1 qjk Pkj (t). Suffices to show that lim suph→0 k6=1
P
k6=1 qjk Pkj (t). To that end,
X
LHS ≤ lim sup[
h→0
k6=1,k<N
Pik (h)
Pkj (t) +
h
X
k6=1,k≥N
Pik (h)
]
h
Pik (h)
1 − Pii (h) X Pik (h)
Pkj (t) +
]
h
h
h
h→0
k6=1,k<N
k6=1,k≥N
X
X
=[
qik Pkj (t) + νi −
qik ]
= lim sup[
X
k6=1,k<N
=
X
k6=1,k≥N
qik Pkj (t).
k6=1
Theorem 1.6. Kolmogorov Forward Equation: Under suitablle reqularity
P
conditions, Pij0 (t) = k6=i Pik (t)qkj − Pij (t)νi , i.e. dPdt(t) = P (t)Q.
3
Pik (h)
h Pkj (t)
≤