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

198 ACTA WASAENSIAas t = n k P ˙ (n; t){n}L = n k (T +i T − j − 1)w ij (n)P (n; t)=====i,j=1 {n}L i,j=1 {n}T +i T − j (δ jk − δ ik )w ij (n)P (n; t)L w ik (n ki )P (n ki ; t) −i=1{n}L w ik (n)P (n; t) −i=1{n}L t −i=1L w kj (n jk )P (n jk ; t)j=1{n}L w kj (n)P (n; t)j=1{n}L tj=1L t . (A7.11)i=1Equation (A7.11) is the exact mean value equation for trader population n k .Inordertoobtain the approximate quasi-meanvalue equations for the trader populations, expandall transition rates w ij (n)tofirst order around their values at the expected configuration t at time t,L ∂ w ij ( t )w ij (n) ≈ w ij ( t )+∆n l , (A7.12)∂n ll=1and insert into the exact mean value equations (A7.11):L t ≈ [ t ]=i=1+Li=1Ll=1 ∂ wik ( t )− ∂ w ki( t ) t∂n l ∂n lL[w ik ( t ) − w ki ( t )] (A7.13)i=1Inserting the individual transition probablilites (A7.5) yields the quasi-meanvalue equationsin the form of the main text:L t = ( t p ik − t p ki ) .(A7.14)i=1