4.5

(a)

フォワードレートの公式(p.97)より
e^s(t₂)t₂ = e^s(t₁)t₁ e^{f_t1,t2(t₂ - t₁)}
s(t₂)t₂ = s(t₁)t₁ + f_t1,t2(t₂ - t₁)
∴ f_t1,t2 = (s(t₂)t₂ - s(t₁)t₁) / (t₂ - t₁)

(b)

r(t) = lim_t2→tf(t,t₂)
= lim_t2→t(s(t₂)t₂ - s(t₁)t₁)/(t₂ - t₁)
分子分母をt₂で微分する
= lim_t2→t(s′(t₂)t₂ + s(t₂)) = s′(t)t + s(t)

(c)

dx(t)/dt = r(t)x(t)
dx(t)/x(t) = r(t)dt
∫dx(t)/x = ∫r(t)dt
log x(t) = ∫(s(t)+s′(t)t)dt = s(t)t + C
x(t) = e^{s(t)t + C}
t = 0の時、x(0) = e^C
∴ x(t) = x(0)e^s(t)t、但しx(0) = e^C

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