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312 CHAPTER 10. TARGET-ZONE MODELSCombine (10.17)-(10.20) to getE t [ ús(t)] − s(t)α= −f(t)α , (10.21)which is a Þrst-order stochastic differential equation. To solve (10.21),mimicthestepsusedtosolvethedeterministicmodeltogetthecontinuoustimeversion of the present-value formulaZ ∞s(t) = 1 e (t−x)/α E t [f(x)]dx. (10.22)α tTo evaluate the expectations in (10.22) you must specify the stochasticprocess governing the fundamentals. For this purpose, we assume thatthe fundamentals process follow the diffusion processdf (t) =ηdt + σdz(t), (10.23)where η and σ are constants, and dz(t) =u √ dt is the standard Wienerprocess. It follows thatf(x) − f(t) ==Z xtZ xtdf (r)drηdr +Z x= η(x − t)+σutσdz(r)q(x − t). (10.24)Take expectations of (10.24) conditional on time t information to getthe prediction ruleE t [f(x)] = f(t)+η(x − t), (10.25)and substitute (10.25) into (10.22) to obtains(t) = 1 αZ ∞t⎡⎢= 1 α ⎣ et/α (f − ηt)e (t−x)α [f(t)+η(x − t)]dxZ ∞Z ∞e −x/α dx +ηe t/α xe −x/α dxt| {z }a⎤⎥t⎦| {z }b= αη + f(t), (10.26)