Abstract
We consider a simple transition model where agent of type changes to type
with the probability
at each step of time. Transition probabilities are specified as non-linear functions of ,
(1) We compare expectation of type at the stationary state, in the sense where detailed-balanced conditions hold in (1), and the solution of a resemble deterministic differential equation,
(2) Our conclusion is that the expectation of in (1), denoted by , is always smaller than the solution of differential equation (2), denoted by , i.e.,
for
(Theorem 1).
１. Introduction
We consider a simple transition model where agent of type changes to type
with the probability
at each step of time. Transition probabilities are specified as non-linear function of ,
(1) We compare expectation of type at the stationary state, in the sense where detailed-balanced conditions hold in (1) and the solution of a resemble deterministic differential equation,
(2) They have different solutions (Aoki, 2002, p.40, see also Kendall, 1949) . But Aoki (2002) does not show inequality in general but only specific evaluation.
We note the expectation of at detailed balanced situation of (1) as , i.e.,
where means
detailed balanced distribution of . On the other hand, the solution of the differential
equation (2) is .
We will prove the inequality in general case,
.
２. Theorem and Lemma Proposition 1
Detailed-balanced distribution of (1) is described as
99
非線形確率モデルと決定論的モデルの比較
A Deterministic Analysis of Nonlinear Effects Will Mislead.
石 垣 建 志
where
Proof of Proposition 1
Equation (1) has a detailed balanced distribution , whose solution is
, where
and is a constant (see Aoki (2002) for the proof).
The summation of is unit.
Therefore is a rational function of ,
Theorem 1
The expectation of k in (1) satisfying detailed balanced conditions, , is described as a function of ,
or more concisely,
, where
, , and
Proof of Theorem 1 (i)
The expectation is a function of and . We will omit integral interval (0, ) for simplicity of notation hereafter.
(*)
Especially, we have
()
Partially integrating denominator of the expectation (*),
茨城大学人文学部紀要 社会科学論集 100
We have recursive equations of with
where
Above equation is rewritten as follows.
By denoting as ,
, the function is thus defined recursively,
.
()
Coefficients of are given as,
Let and ,
then
＝ ＝
,
Let , then
where
Lemma 1
The expectation of in (1) satisfying the detailed-balanced conditions, , is always smaller than , the solution of deterministic differential equation (2), i.e.,
Proof of Lemma 1 (i)
If, then .
()
We prove for
by induction of n and s.
Firstly
…, since
Secondly, by assumption of induction, i.e.,, 石垣：非線形確率モデルと決定論的モデルの比較 101
.
For ,
. We have shown that
if .
This completes the induction.
３. Conclusion
Showing specific example, Aoki ( 2002 ) wrote (a) deterministic analysis of nonlinear effects may mislead! ( p.40).
We proved above that the analysis is misleading in general.
References
Aoki, M. ( 2002),
：
Cambridge University Press
Kendall, D.G. (1949), "Stochastic Process and Population Growth", Series B, 11, p.230-264.
茨城大学人文学部紀要 社会科学論集 102
