Pembahasan Ujian PAI: F10 – No. 06 – November 2025

• Harga saham di waktu ke-t (\(X_{t}\)) mengikuti model arithmetic Brownian motion:
  $$dX_{t} = \alpha\,dt + \sigma\,dB_{t}$$

• \(y = \frac{1}{X}\)
• \(z = \sqrt{X}\)

• Lemma Itô untuk suatu fungsi \(Y_{t}\):
  $$dY_{t} = Y_{t}’\,dX_{t} + \tfrac{1}{2}(Y_{t}”)\left(dX_{t}\right)^{2}$$
• Aturan Itô:
  • \((dt)^{2} = 0\)
  • \(d_{t}dB_{t} = 0\)
  • \((dB_{t})^{2} = dt\)

• \(Y_{t} = \frac{1}{X_{t}}\)
• \(Y_{t}’ = -\frac{1}{X_{t}^{2}}\)
• \(Y_{t}” = \frac{2}{X_{t}^{3}}\)
• \(\left(dX_{t}\right)^{2} = \left(\alpha\,dt + \sigma\,dB_{t}\right)^{2}\);
  \(= \alpha^{2}(dt)^{2} + \sigma^{2}\left(dB_{t}\right)^{2} + 2\alpha\sigma\,dt\,dB_{t}\);
  \(= \sigma^{2}\,dt\)
• \(Z_{t} = \sqrt{X_{t}}\)
• \(Z_{t}’ = \frac{1}{2\sqrt{X_{t}}}\)
• \(Z_{t}” = -\frac{1}{4}\left(\frac{1}{X_{t}\sqrt{X_{t}}}\right)\)

Selanjutnya, substitusi setiap komponen ke dalam persamaan Itô tersebut:

$$\begin{aligned}
dY_{t} &= Y_{t}’\,dX_{t} + \tfrac{1}{2}Y_{t}”\left(dX_{t}\right)^{2} \\
&= -\frac{1}{X_{t}^{2}}\left(\alpha\,dt + \sigma\,dB_{t}\right) + \tfrac{1}{2}\left(\frac{2}{X_{t}^{3}}\right)\left(\sigma^{2}dt\right) \\
&= -\frac{\alpha}{X_{t}^{2}}\,dt – \frac{\sigma}{X_{t}^{2}}\,dB_{t} + \frac{\sigma^{2}}{X_{t}^{3}}\,dt \\
&= \left(-\frac{\alpha}{X_{t}^{2}} + \frac{\sigma^{2}}{X_{t}^{3}}\right)dt – \frac{\sigma}{X_{t}^{2}}\,dB_{t}
\end{aligned}$$

Terakhir, nyatakan ruas kanan yang masih dalam variabel \(X_{t}\) menjadi \(Y_{t}\):

Diketahui \(y = \frac{1}{x}\)

$$dy = \left(-\alpha y^{2} + \sigma^{2}y^{3}\right)dt – \sigma y^{2}\,dB_{t}$$

Lakukan hal yang sama untuk fungsi ke-2, yakni \(Z_{t}\):

Substitusi setiap komponen ke dalam persamaan Itô tersebut:

$$\begin{aligned}
dZ_{t} &= Z_{t}’\,dX_{t} + \tfrac{1}{2}Z_{t}”\left(dX_{t}\right)^{2} \\
&= \frac{1}{2\sqrt{X_{t}}}\left(\alpha\,dt + \sigma\,dB_{t}\right) – \tfrac{1}{2} \times \tfrac{1}{4}\left(\frac{1}{X_{t}\sqrt{X_{t}}}\right)\left(\sigma^{2}dt\right) \\
&= \frac{\alpha}{2\sqrt{X_{t}}}\,dt + \frac{\sigma}{2\sqrt{X_{t}}}\,dB_{t} – \frac{\sigma^{2}}{8\,X_{t}\sqrt{X_{t}}}\,dt \\
&= \left(\frac{\alpha}{2\sqrt{X_{t}}} – \frac{\sigma^{2}}{8\,X_{t}\sqrt{X_{t}}}\right)dt + \frac{\sigma}{2\sqrt{X_{t}}}\,dB_{t}
\end{aligned}$$

Terakhir, nyatakan ruas kanan yang masih dalam variabel \(X_{t}\) menjadi \(Z_{t}\):

Diketahui \(z = \sqrt{x}\)

$$dz = \left(\frac{\alpha}{2z} – \frac{\sigma^{2}}{8z^{3}}\right)dt + \frac{\sigma}{2z}\,dB_{t}$$

Jadi bentuk dari \(dy\) dan \(dz\) dengan menggunakan lemma Itô adalah:

\(dy = \left(-\alpha y^{2} + \sigma^{2}y^{3}\right)dt – \sigma y^{2}\,dB_{t}\) dan \(dz = \left(\frac{\alpha}{2z} – \frac{\sigma^{2}}{8z^{3}}\right)dt + \frac{\sigma}{2z}\,dB_{t}\)