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

196 ACTA WASAENSIA(4.20), and o(τ 2 ) stands for terms of second and higher order in τ. Inserting (A7.3)into (A7.2) and taking the limit τ → 0 yields the fundamental master equation for theprobability change over infinitesimally small time intervalsP ˙dP(n; t) P (n; t 2 ) − P (n; t 1 )(n; t) := = limdt t 2 →t 1 t 2 − t 1asP ˙ (n; t) = w(n|n )P (n ; t) − w(n |n)P (n; t). (A7.4)n =nn =nIn the next step we confine ourselves to transitions between neighbouring states of theinvestment configuration as a result of the Poisson-type dynamics induced by at mostone trader changing her strategy during any infinitesimal time interval τ. 144 That is,we consider only transitions between configurations n andn ij := {n 1 ,...,(n i − 1),...,(n j +1),...,n L },such thatand definew(n |n) =w(n|n )=0for n = n ij ,w ij (n) :=w(n ij |n) =n i p ij(A7.5)in line with (4.21), (4.23) and (4.24), with p ij denoting the probability for a singletrader to change from strategy i to strategy j. The fundamental master equation(A7.4) reduces then to˙ P (n; t) ==Lw ji (n ij )P (n ij ; t) −i=jLw ji (n ij )P (n ij ; t) −i,j=1Lw ij (n)P (n; t)i=jLw ij (n)P (n; t)i,j=1(A7.6)Note that we need not exclude i = j because n ii = n.Introduce next translation operators T +iand T −ion the configuration space {n} asT ±i F (n 1 ,...,n i ,...,n L ):=F (n 1 ,...,(n i ± 1),...,n L ),(A7.7)144 see the discussion of the herding model by Lux (1995) on pp. 78.