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

ACTA WASAENSIA 201A9ProofofProposition2Local stability with respect to regime-specificdynamicswillbeconsideredbyinspectingthe Jacobian of the system of differential equations for the trader populations andprices. 145 We will for that purpose reformulate the population dynamics (A8.3) interms of the new variablesc 1 := n c1N ,c 2 := n c2N ,f 1 := n f1N ,f 2 := n f2N(A9.1)as c˙1 = v B c 1−c1 −c 2 −f 1 −f 21 c 1 +c 2 +f 1 +f 2e α B(2(c 1 +c 2 +f 1 +f 2 )−1) − e −α B(2(c 1 +c 2 +f 1 +f 2 )−1) + v (c 1 + c 2 ) tanh(α(C 1 − C 2 )) − c 1 − c 2cosh(α(C 1 − C 2 ))c 1 + c2+(c 1 + f 1 ) tanh(α(C 1 − F 1 )) − c 1 − f 1cosh(α(C 1 − F 1 ))c 1 + f1+(c 1 + f 2 ) tanh(α(C 1 − F 2 )) − c 1 − f 2cosh(α(C 1 − F 2 ))c 1 + f 2 1−cc˙2 = v B c 1 −c 2 −f 1 −f 22 c 1 +c 2 +f 1 +f 2e α B(2(c 1 +c 2 +f 1 +f 2 )−1) − e −α B(2(c 1 +c 2 +f 1 +f 2 )−1) + v (c 2 + c 1 ) tanh(α(C 2 − C 1 )) − c 2 − c 1cosh(α(C 2 − C 1 ))c 2 + c1+(c 2 + f 1 ) tanh(α(C 2 − F 1 )) − c 2 − f 1cosh(α(C 2 − F 1 ))c 2 + f1+(c 2 + f 2 ) tanh(α(C 2 − F 2 )) − c 2 − f 2cosh(α(C 2 − F 2 ))c 2 + f 2(A9.2a)(A9.2b)145 see e.g. Gandolfo (1996).