Hence, EP(Vn ··· VN log Vn ··· VN |Fnâ1)
462
A. S. CHERNY AND V. P. MASLOV
for P-a.e. ω (here we used (5.17)). Hence, EP (Vn · · · VN log Vn · · · VN | Fn−1 ) = EP (Vn · · · VN log Vn+1 · · · VN | Fn−1 ) + EP (Vn · · · VN log Vn | Fn−1 ) = EP (Vn Yn | Fn−1 ) + EP (Vn log Vn | Fn−1 ) ∗ = − log EP eτn , ∆Xn −Yn Fn−1 = Yn−1 . Thus, we have proved (5.6) and (5.7) for k = 0, . . . , N . Note that Q∗ = Q−1 . Using (5.17), we get EQ∗ (∆Xn | Fn−1 ) =
EP (V0 · · · VN ∆Xn | Fn−1 ) EP (V0 · · · VN | Fn−1 )
=
EP (EP (Vn · · · VN ∆Xn | Fn ) | Fn−1 ) EP (Vn · · · VN | Fn−1 )
=
EP (Vn ∆Xn | Fn−1 ) = 0. EP (Vn · · · VN | Fn−1 )
Consequently, Q∗ ∈ Ma . Due to the equality Q = QN , we have H(Q∗ , P) H(Q, P) and N τn∗ , ∆Xn . H(Q∗ , P) = EP (V0 · · · VN log V0 · · · VN ) = Y−1 = − log EP exp n=1
Thus, the minimum of H(Q, P) over M is attained at the measure Q∗ . The uniqueness of the minimizing measure follows from the convexity of Ma and the strict convexity of the function x → x log x. Remark. The method of constructing the minimal entropy martingale measure described above is somewhat similar to the construction of a martingale measure through the conditional Esscher transform; see [14]. Corollary 5.1. Consider an arbitrage-free model of the form (5.1). Suppose that ∆Xn is independent of Fn−1 for any n = 1, . . . , N , and EP eτ,∆Xn < ∞ for any n = 1, . . . , N , τ ∈ Rd . Then, for each n = 1, . . . , N , there exists a point τn∗ ∈ Rd at which the function ϕn (τ ) = EP eτ,∆Xn attains its minimum. The minimum of H(Q, P) over Ma is attained at a unique measure N τn∗ , ∆Xn P. Q∗ = const. exp a
n=1
Proof. It is sufficient to note that the conditional expectations in (5.2) coincide with the usual ones. The result now follows from Theorem 5.2. Corollary 5.2. Let us consider a one-dimensional arbitrage-free model of the form (5.1). Suppose that Xn = eYn , where ∆Yn = Yn − Yn−1 is independent of ∆Yn −1) Fn−1 for any n = 1, . . . , N , and EP eτ (e < ∞ for any n = 1, . . . , N , τ ∈ R. Then, for each n = 1, . . . , N , there exists a point λ∗n ∈ R at which the function ∆Yn −1) attains its minimum. The minimum of H(Q, P) over Ma is ϕn (λ) = EP eλ(e attained at a unique measure N λ∗ n Q∗ = const. exp ∆Xn P. Xn−1 n=1
A. S. CHERNY AND V. P. MASLOV
for P-a.e. ω (here we used (5.17)). Hence, EP (Vn · · · VN log Vn · · · VN | Fn−1 ) = EP (Vn · · · VN log Vn+1 · · · VN | Fn−1 ) + EP (Vn · · · VN log Vn | Fn−1 ) = EP (Vn Yn | Fn−1 ) + EP (Vn log Vn | Fn−1 ) ∗ = − log EP eτn , ∆Xn −Yn Fn−1 = Yn−1 . Thus, we have proved (5.6) and (5.7) for k = 0, . . . , N . Note that Q∗ = Q−1 . Using (5.17), we get EQ∗ (∆Xn | Fn−1 ) =
EP (V0 · · · VN ∆Xn | Fn−1 ) EP (V0 · · · VN | Fn−1 )
=
EP (EP (Vn · · · VN ∆Xn | Fn ) | Fn−1 ) EP (Vn · · · VN | Fn−1 )
=
EP (Vn ∆Xn | Fn−1 ) = 0. EP (Vn · · · VN | Fn−1 )
Consequently, Q∗ ∈ Ma . Due to the equality Q = QN , we have H(Q∗ , P) H(Q, P) and N τn∗ , ∆Xn . H(Q∗ , P) = EP (V0 · · · VN log V0 · · · VN ) = Y−1 = − log EP exp n=1
Thus, the minimum of H(Q, P) over M is attained at the measure Q∗ . The uniqueness of the minimizing measure follows from the convexity of Ma and the strict convexity of the function x → x log x. Remark. The method of constructing the minimal entropy martingale measure described above is somewhat similar to the construction of a martingale measure through the conditional Esscher transform; see [14]. Corollary 5.1. Consider an arbitrage-free model of the form (5.1). Suppose that ∆Xn is independent of Fn−1 for any n = 1, . . . , N , and EP eτ,∆Xn < ∞ for any n = 1, . . . , N , τ ∈ Rd . Then, for each n = 1, . . . , N , there exists a point τn∗ ∈ Rd at which the function ϕn (τ ) = EP eτ,∆Xn attains its minimum. The minimum of H(Q, P) over Ma is attained at a unique measure N τn∗ , ∆Xn P. Q∗ = const. exp a
n=1
Proof. It is sufficient to note that the conditional expectations in (5.2) coincide with the usual ones. The result now follows from Theorem 5.2. Corollary 5.2. Let us consider a one-dimensional arbitrage-free model of the form (5.1). Suppose that Xn = eYn , where ∆Yn = Yn − Yn−1 is independent of ∆Yn −1) Fn−1 for any n = 1, . . . , N , and EP eτ (e < ∞ for any n = 1, . . . , N , τ ∈ R. Then, for each n = 1, . . . , N , there exists a point λ∗n ∈ R at which the function ∆Yn −1) attains its minimum. The minimum of H(Q, P) over Ma is ϕn (λ) = EP eλ(e attained at a unique measure N λ∗ n Q∗ = const. exp ∆Xn P. Xn−1 n=1
