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

ACTA WASAENSIA 79dependent probability of a single pessimist to become an optimist per unit time withp −+ (n), such that the event of a pessimist becoming an optimist within the time interval∆t is Bernoulli distributed with probability p −+ (n)∆t. The total number of pessimistschanging to an optimistic view of the market within that time interval is then binomiallydistributed with parameters n − and p −+ (n)∆t, which in the limit of a large populationand a small time interval becomes Poisson distributed with parameter n − p −+ (n)∆t.The probability of an integer increase ∆n =1, 2,... of the state variable n over thetime interval ∆t is therefore given byP (n + ∆n; t + ∆t|n; t) = (n −p −+ (n)∆t) ∆ne (n −p −+ (n)∆t) , (4.19)∆n!such that defining the per unit time transition probability from state n to state n +∆nasP (n + ∆n; t + ∆t|n; t)w(n + ∆n|n) := lim, (4.20)∆t→0 ∆tone obtains for the transition rate from state n to n + ∆n:w(n + ∆n|n) =n − p −+ (n)δ ∆n,1 , ∆n =1, 2,..., (4.21)where δ denotes the Kronecker delta function1, if x = x;δ x,x :=0, otherwise.(4.22)In a similar way it can be shown that the transition rate from state n to state n − ∆nis given byw(n − ∆n|n) =n + p +− (n)δ ∆n,1 , ∆n =1, 2,..., (4.23)where p +− (n) denotes the state dependent transition probability per unit time of anindividual agent to move from the optimist to the pessimist subgroup. The masterequation contains therefore only transitions between nearest neighbour states n andn = n ± 1. Abbreviatingw −+ (n) :=n − p −+ (n), w +− (n) :=n + p +1 (n), (4.24)the master equation (4.18) reduces todP(n; t)dt= w −+ (n − 1)P (n − 1; t)+w +− (n +1)P (n +1;t)− w −+ (n)P (n; t) − w +− (n)P (n; t). (4.25)