"Technical Analysis in Financial Markets"

W_t^h=∑_j=1^N W_j,t^h. Finally, total market wealth is equal to W_t=∑_h=1^H W_t^h. The expected wealth transferred from agent j to belief h conditioned on the wealth of agent j and the information set I_t={P_t-i, D_t-i; i≥ 0} is equal to:

E(W_j,t^h|I_t, W_j,t)=W_j,t E(X_j,t^h|I_t)= W_j,t (q_t^h 1 + (1-q_t^h) 0)=q_t^h W_j,t.

The expectation of wealth transferred from agent j to belief h conditioned only on I_t is equal to:

E(W_j,t^h|I_t)=E(E(W_j,t^h|I_t, W_j,t)|I_t)=q_t^h E(W_j,t|I_t).

According to (6.28) the wealth of agent j at time t depends on the fraction of the wealth invested at time t-1 and this chosen fraction depends on the agent's belief at time t-1. Hence the expected wealth of agent j at time t conditioned on his wealth at time t-1 is equal to:

E(W_j,t|I_t, W_j,t-1)=

∑

h=1

⎛
⎜
⎜
⎝

R W_{j, t-1} + (P_t+D_t-R P_t-1)

y_t-1^h W_{j, t-1}

P_t-1

⎞
⎟
⎟
⎠

q_t-1^h

⎞
⎟
⎟
⎠

R W_{j, t-1} + (P_t+D_t-R P_t-1)

E(y_t-1) W_{j, t-1}

P_t-1

where E(y_t-1)=∑_h=1^H y_t-1^h q_t-1^h. The expected wealth of agent j at time t only conditioned on I_t is equal to:

E(W_j,t|I_t)=E_{W_j,t-1}(E(W_j,t|I_t, W_j,t-1)|I_t)=

∑

{W_j,t-1}

E(W_j,t|I_t, W_j,t-1) P(W_j,t-1)=

R E(W_{j, t-1}|I_t-1) + (P_t+D_t-R P_t-1)

E(y_t-1) E(W_{j, t-1}|I_t-1)

P_t-1

. (53)

In the end:

E(W_j,t|I_t)=

⎛
⎜
⎜
⎝

R +

(P_t+D_t-R P_t-1)

P_t-1

E(y_t-1)

⎞
⎟
⎟
⎠

E(W_{j, t-1}|I_t-1)=

(1+r^F + (r_t^P-r^F) E(y_t-1) ) E(W_{j, t-1}|I_t-1),

which is a recursive formula for the expected wealth of agent j at time t given the dividends paid and given the equilibrium prices {P_t-i: i ≥ 0 } the auctioneer did set. Given the wealth of agent j at time 0, the expected wealth of agent j at time t is equal to:

E(W_j,t|I_t)= W_j,0

t-1

∏

i=0

(

1+r^F + (r_t-i^P-r^F) E(y_t-1-i)

)

279