Simultaneous Stochastic Diﬀerential Equations

6.1 Correlated Brownian Motion

Let w1 (t), w2 (t) be two independent Brownian Motions, that is, for each value of t, w1 , w2 are independent random variables. Then w1 (0) = w2 (0) = 0, E(w1 ) = E(w2 ) = 0 and

2 2 E(w1 ) = E(w2 ) = t

E(w1 w2 ) = E(w2 w1 ) = 0 Let W = Then WWT =

2 w1 w1 w2 2 w2 w1 w2 2 E(w1 ) E(w1 w2 ) 2 E(w2 w1 ) E(w2 )

w1 w2

E(W W T ) = =

1 0 t = It 0 1

Now let z1 (t), z2 (t) be two (possibly) correlated Brownian Motions, that is, for each value of t,

2 2 E(z1 ) = E(z2 ) = t

E(z1 z2 ) = E(z2 z1 ) = ρ12 t = ρ21 t 1

2CHAPTER 6. SIMULTANEOUS STOCHASTIC DIFFERENTIAL EQUATIONS Let Z= Then ZZ T =

2 z1 z1 z2 2 z2 w1 z2 2 E(z1 ) E(z1 z2 ) 2 E(z2 z1 ) E(z2 )

z1 z2

E(ZZ T ) = = =

1 ρ12 t ρ21 1 1 ρ t ρ 1 (ρ12 = ρ21 = ρ)

This can clearly be generalized to vectors with n components (n ≥ 1), so that w1 z1 . . W = . ,Z = . . . wn zn E(W W T ) = I 1 ρ12 · · · ρ21 1 · · · E(ZZ T ) = . . . . . . . . . ρn1 ρn2 · · · where ρij t = ρji t = E(zi zj ). Correlated BM’s are necessary when describing several simultaneous processes governed by SDE’s, where upward (downward) movements in one generally result in upward (downward) movements of the other. For example, two foreign currencies may be modelled whose countries have closely linked economies. Correlations could be negative, in which case upward movement of one generally implies downward movement of the other. Construction of Correlated BM’s Let W be an n-vector of independent BM’s, and let C be a constant n × n matrix. Then Z = CW can be shown to also be a BM, in that increments are normal and independent, although the variances of each of the zi may not be 1. Correlations between diﬀerent zi may not be 0.

ρ1n ρ2n . t . . 1

6.2. CHOLESKY FACTORIZATION

3

Suppose that we are given this W and a correlation matrix ρ = (ρij ) where ρij = ρji and ρii = 1 for all i, j (and −1 ≤ ρij ≤ 1). If this is to be a correlation matrix, it must be positive semi-deﬁnite as well. Problem: To ﬁnd a matrix C so that if Z = CW then E(ZZ T ) = ρt. We need E(ZZ T ) = E(CW (CW )T ) = CE(W W T )C T = CIC T t = CC T t = ρt Thus, any (constant) matrix C satisfying CC T = ρ will have a corresponding vector Z with correlation matrix tρ. C is called a pseodo-squareroot of ρ. In the next section, we will see how to construct C under the assumption that ρ is positive deﬁnite. In this case, a C can be found which has the additional property that it is lower triangular, that is, the elements above the main diagonal are 0. This simpliﬁes things since we have fewer elements in C. Later, we will see how to ﬁnd C even if it is positive semi-deﬁnite, but the method will not yield lower triangular values of C. The lower triangular factorization is called a Cholesky factorization.

6.2

Cholesky Factorization

Problem: Given a symmetric positive deﬁnite N × N matrix A. To ﬁnd a lower triangular N × N matrix C such that CC T = A Let C = (cij ) where cij = 0 for i > j and let A = (aij ). We need

N N

aij = (CC T )ij

= k=1 i

cik cT = kj k=1 cik cjk (i ≤ j)

= k=1 cik cjk

Note: We will use the convention

0

xk = 0 k=1 4CHAPTER 6. SIMULTANEOUS STOCHASTIC DIFFERENTIAL EQUATIONS In writing this into a loop in VBA, we can use For k = 1 To (i-1) ··· Next k and this will not do anything if i ≤ 1. Then CC T = A iﬀ for i ≤ j we have i aij

= k=1 i−1

cik cjk

= k=1 cik cjk + cii cji

iﬀ for i ≤ j i−1 cii = (aii − 1 (aij − cii

c2 ) 2 ik k=1 i−1

1

(i = 1..N )

cji =

cik cjk ) k=1 i−1 2 k=1 cik

(j = (i + 1)..N ) < 0 for some i.

Note that this will not work if aii −

This can be implemented in VBA using extensions to the code c(1, 1) = a(1, 1) For i = 1 To n For j = i To n sum = a(i, j) For k = 1 To i - 1 sum = sum - c(i, k) * c(j, k) Next k If i = j Then If sum 0 It can be shown that a symmetric matrix