The turnover of shares in belief h we define to be equal to

if zth>0 then Volth=|zth-#longt(→ h)| +|#shortt(→ h)|;
if zth<0 then Volth=|zth+#shortt(→ h)|+|#longt(→ h)|;
if zth=0 then Volth=|#longt(→ h)|+|#shortt(→ h)|.
    (41)
The total turnover of shares is equal to
Volt=
H
∑
h=1
Volth,
doubly counted. For example, if belief h advises to hold a long position in the market at time t, then first it is determined how many long positions are transferred to belief h from the other beliefs at time t. The change in long positions is given by |zth-#longt(→ h)|. This gives the (minimum) number of stock positions sold or bought to reach the new long position from the old long position. Because a long position in the market is held, the short positions transferred to belief h must be closed, which gives an extra volume of |#shortt(→ h)|. Adding the two turnovers together yields the total trading volume in belief h, in the case zth>0. The other cases in (6.41) have a similar explanation.

6.5  Stability analysis

6.5.1  Steady state

For the system to be in the steady state it is required that the memory parameter is restricted to 0 ≤ η <1, thus we assume finite memory. A variable x at its steady state will be denoted by x. A steady state for the map Φ is a point z for which Φ(z)=z. Hence in the steady state MA=µ P + (1-µ) MA, implying P=MA. Furthermore, P=MA, implies yMA=0 and hence Fma=rF-Cma/1-η. Thus the steady state price of the risky asset is equal to the steady state exponential moving average of the price. This relation implies that the steady state demand of the moving average belief is equal to zero.

The steady state price must satisfy P=1/R(P*+v(P-P*)+D), where P*=D/rf is the fundamental price. This implies that P=P*, yfund=0 and Ffund=rF-Cfund/1-η. As for the moving average belief, also the steady state demand of the fundamental belief is equal to zero.

In the steady state the difference in probability with which both beliefs are chosen is equal to

qfund-qma=(mfund-mma) + (1-mfund-mma)tanh ⎛
⎜
⎜
⎝
-β
2(1-η)
(Cfund-Cma) ⎞
⎟
⎟
⎠		 .
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