"Technical Analysis in Financial Markets"

6.3.5 The dynamical system

Using equations (6.36), (6.25) and (6.52) and setting MA_t=MA_t-1 and F_t^h=F_t-1^h the following dynamical system for s=0 is obtained

0.1cm P_t(P_t-1, MA_t-1, F_t^fund, F_t^ma);
MA_t=� P_t-1 + (1-�) MA_t-1;
F_t^h=r^F + y_t-2^h (r_t-1^P-r^F) + η F_t-1^h, for h=(fund, ma).

Introducing new variables P_i,t-1=P_t-i and MA_i,t-1=MA_t-i the following eight dimensional dynamical system is derived from the above equations

0.1cm P_1,t=q_t^fund/q_t^fund R - a σ² q_t^MA y_t^MA (P^*+v(P_1,t-1-P^*)+D);
P_2,t=P_1,t-1;
P_3,t=P_1,t-2=P_2,t-1;
MA_1,t=� P_1,t-1 + (1-�) MA_1,t-1;
MA_2,t=MA_1,t-1;
MA_3,t=MA_1,t-2=MA_2,t-1;
F_t^fund=r^F + y_t-2^fund (P_1,t-1+D_t-1/P_2,t-1-R)- C^fund + η F_t-1^fund;
F_t^ma=r^F + y_t-2^ma (P_1,t-1+D_t-1/P_2,t-1 -R)- C^ma + η F_t-1^ma,

where

0.1cm D_t=D+δ_t; δ_t ~ N(0, σ_δ²);
P^*=D/r^F;
q_t^ma=1-q_t^fund , q_t^fund=m^fund+(1-m^fund-m^ma) q_t^fund,
q_t^fund=exp(β F_t^fund)/exp(β F_t^fund)+exp(β F_t^ma);
y_t^ma=2 γ x_t-1/1+x_t-1² , x_t-1=1/λP_1,t-1- MA_1,t-1/MA_1,t-1;
y_t-2^fund=1/P_t-2(P^*+v(P_t-3-P^*)+D)-R/a σ²=1/P_2,t-1(P^*+v(P_3,t-1-P^*)+D)-R/a σ²;
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