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

80 ACTA WASAENSIAWe wish to obtain information about the dynamics of the average market opinionN t := nP (n; t), (4.26)n=−Nwhose evolution through time is due to (4.25): t := d dt t ===Nn=−N+1N−1−n=−NNn=−NNn=−NdP(n; t)ndtnw −+ (n − 1)P (n − 1; t) +nw −+ (n)P (n; t) −Nn=−N+1N−1n=−Nnw +− (n +1)P (n +1;t)nw +− (n)P (n; t)[w −+ (n) − w +− (n)] P (n; t), (4.27)wherewehaveshiftedthesummationindexinthefirst two terms by −/+ 1 and madeuse of the boundary conditionsw −+ (N) =w +− (−N) =0. (4.28)Solving (4.27) requires knowledge of the full probability distribution of n. Itishoweverdesirable to have an equation for the average market opinion that depends on meanvalues only. For that purpose an opinion index x,x := n/N, −1 ≤ x ≤ 1, (4.29)is introduced, which in the limit N →∞may be regarded as a continuous randomvariable with associated probability measure P (x; t) normalized as 11P (x; t) dx ≈ P (x; t)∆x =1,−1x=−1∆x = ∆nN = 1 N . (4.30)AbbreviatingK(x) :=w −+ (x) − w +− (x), (4.31)(4.27) may be reexpressed as t := d dt 1P (x; t) dx = 1−1−1K(x)P (x; t) dx = t . (4.32)