BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

ACTA WASAENSIA 197such thatT + j T −i F (n) =F (n ij ) and T +i T − j F (n) =F (n ji).(A7.8)We may therefore writeLi,j=1T +i T − j w ij(n)P (n; t) =Lw ij (n ji )P (n ji ; t) =i,j=1Lw ji (n ij )P (n ij ; t),i,j=1such that the master equation (A7.6) may be rewritten as˙ P (n; t) =L(T +i T − j − 1)w ij (n)P (n; t). (A7.9)i,j=1Note the following properties of the translation operators:T ±i n k F (n) =(n k ± δ ik )T ±i F (n)⇒ n k T i ± F (n) =T i ± n k F (n) ∓ δ ik T i ± F (n) =T i ± (n k ∓ δ ik )F (n)⇒ n k T +i T − j+F (n) =T i n k Tj − F (n) − T i + δ ik Tj − F (n)= T +i T − j (n k + δ jk )F (n) − T +i T − j δ ikF (n)= T i + Tj − (n k + δ jk − δ ik )F (n)(A7.10a)(A7.10b)(A7.10c)Property (A7.10c) may be used in conjunction with (A7.9) in order to calculate thedynamics of the expected number of traders using strategy k, t := {n}n k P (n; t),