ABSTRACT
The second term in (8.2.3) goes to zero as T →+∞. For the first term in (8.2.3) we note that function Pρ− ρˆ satisfies the balance condition with respect to the measure p(dx), as
∫ X p(dx)[Pρ− ρˆ] = 0. This is why we can apply the algorithms of diffusion
approximation (see [10]) to the first term in (8.2.3), where instead of function Pρ it needs the function Pρ− ρˆ. Taking limit as T →∞ we obtain from (8.2.3) :
wT (t) = σˆ wˆ(t), (8.2.4)
where wˆ(t) is a standard Wiener process with diffusion coefficient
σˆ := ∫
X p(dx)(Pρ− ρˆ)R0(Pρ− ρˆ)+2−1(Pρ− ρˆ)2]/m, (8.2.5)
which follows from the algorithms of normal deviations [4]. In (8.2.5) R0 is a potential [8] of (xn)n∈Z+ . Thus, we obtain the double approximation for αT (t) in an ergodic normal deviations scheme:
αT (t)' ρˆt+T−1/2σˆ wˆ(t), (8.2.6)
where σˆ is defined in (8.2.5). Hence,
ln ST (t)
S0 ' ρˆt+T−1/2σˆ wˆ(t). (8.2.7)
Corollary 8.1 Ergodic normal deviated GMRP has the form:
ST (t)' S0eρˆt+T−1/2σˆ wˆ(t), (8.2.8)
or, in stochastic differential equation (SDE) form
dST (t) ST (t)
' (ρˆ+ 1 2 +T−1σˆ2)dt+T−1/2σˆdwˆ(t). (8.2.9)
8.2.2 Reducible (merged) normal deviations
Let us suppose that the balance condition is not fulfilled :
ρˆ(k) = ∫
P(x,dy)ρ(y)/m(k) 6= 0, (8.2.10)
for all k = 1,2, . . . ,r where (xn)n∈Z+ is the supporting embedded Markov chain, pk is the stationary density for the ergodic component Xk, m(k) :=
pk(dx)m(x),
0 x of X are fulfilled. consider the normal deviated process
w˜T (t) := √
T (αT (t)− ρ˜(t)), (8.2.11)
where αT (t) := T−1∑ ν(tT ) k=1 ρ(xk), and ρ˜(t) =
∫ t 0 ρˆ(xˆ(s))ds, and ρˆ(k) is defined in
(8.2.10). In this case the construction of the normal deviated process for αT (t) in reducible case consists of the fact see [3, 4, 7, 8], that w˜T (t) is a stochastic Ito integral under large T :
0 σ˜(xˆ(s))dw(s), (8.2.12)
where
σ˜(k) := ∫
Xk pk(dx)[(Pρ− ρˆ(k))Ro(Pρ− ρˆ(k))+2−1(Pρ− ρˆ(k))2]/m(k), (8.2.13)
for all k = 1,2, . . . ,r. Thus, double approximation of GMRP has the following form:
0 σ˜(xˆ(s))dw(s). (8.2.14)
From (8.2.11) and (8.2.14) it follows that
ln ST (t)
S0 ' ρ˜(t)+T−1/2
or, ST (t)' eρ˜(t)+T−1/2
Corollary 8.2 The reducible normal deviated GMRP has the form:
ST (t)' S0eρ˜(t)+T−1/2 ∫ t
or, in the form of SDE
dST (t) ST (t)
' (ρˆ(xˆ(t))+ 1 2
T−1σ˜2(xˆ(t)))dt+T−1/2σ˜(xˆ(t))dw(t), (8.2.17)
where ρˆ(k) and σ˜(k) are defined in (8.2.10) and (8.2.13), respectively.