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310 CHAPTER 10. TARGET-ZONE MODELSthat you can treat the deviation of dz(t) 2 from its mean dt as an errorterm of order O(dt 2 ). 2 Write it asdz(t) 2 = dt + O(dt 2 ).Taking a second-order Taylor expansion of G(x(t),t)gives∆G = ∂G∂x ∆x(t)+∂G ∂t ∆t+ 1 "∂ 2 #G2 ∂x 2 ∆x(t)2 + ∂2 G∂t 2 ∆t2 +2 ∂2 G∂x∂t [∆x(t)∆t]+ O(∆t 2 ), (10.8)where O(∆t 2 ) are the ‘higher-ordered’ terms involving (∆t) k with k>2. You can ignore those terms when you send ∆t → 0.If x(t) evolves according to the diffusion process, you know that∆x(t) = η∆t + σ∆z(t), with ∆z(t) = u √ ∆t, and(∆x) 2 = η 2 (∆t) 2 + σ 2 (∆z) 2 +2ησ(∆t)(∆z) =σ 2 ∆t + O(∆t 3/2 ). Substitutethese expressions into the square-bracketed term in (10.8) toget,∆G = ∂G ∂G ∂ 2 G(∆x(t)) +∂x ∂t (∆t)+σ2 2 ∂x 2 (∆t)+O(∆t3/2 ). (10.9)As ∆t → 0, (10.9) goes to (10.6), because the O(∆t 3/2 )termscanbeignored. The result is Ito’s lemma.10.2 The Continuous—Time Monetary ModelA deterministic setting. To see how the monetary model works in continuoustime, we will start in a deterministic setting. As in chapter 3,all variables except interest rates are in logarithms. The money marketequilibrium conditions at home and abroad arem(t) − p(t) = φy(t) − αi(t), (10.10)m ∗ (t) − p ∗ (t) = φy ∗ (t) − αi ∗ (t). (10.11)2 An O(dt 2 ) term divided by dt 2 is constant.