<p><span class="math-container">$\newcommand{\dd}{\, \mathrm{d}}$</span>
If we apply Ito's lemma,
then
<span class="math-container">\begin{align*}
\dd \xi_t &amp;= -\xi_t \dd X_t + \frac 12 \xi_t (\dd X_t)^2\\
  &amp;= -\xi_t \left(\frac 12 \lambda_t^2 \dd t + \lambda_t \dd z_t\right) + \frac 12 \xi_t \lambda_t^2 \dd t \\
  &amp;= -\xi_t \lambda \dd z_t.
\end{align*}</span></p>

CC BY-SA 4.0
2015-12-02T08:23:06.127Z
2019-03-23T04:14:53.327Z
2019-03-23T04:14:53.327Z
jmbejara
<p>$\newcommand{\dd}{\, \mathrm{d}}$
Consider the exponential martingale,
$$
\xi_t^\lambda = \exp \left\{ - \int_0^t \lambda_s \dd z_s - \frac 12 \int_0^T \lambda_s^2 \dd s \right\},
$$
that is used in the statement of Girsanov's theorem (this martingale represents the Radon-Nykodym derivative $\frac{\dd \mathbb Q^\lambda}{\dd \mathbb P}$.).</p>

<p>Exercise 2.4 in Munk's book <em>Financial Asset Pricing Theory</em> deals with applying Ito's lemma to this process.</p>

<p>Suppose
$$
X_t = \frac 12 \int_0^t \lambda _s^2 \dd s + \int_0^t \lambda_s \dd z_s.
$$
Part (a) asks us to argue that $\dd X_t = \frac 12 \lambda_t^2 \dd t + \lambda_t \dd z_t.$ Part (b) asks, "Suppose that the continuous-time stochastic process
$\xi = (\xi_t)$ is defined as $\xi_t = \exp\{-X_t\}$. Show that $\dd \xi_t = -\lambda_t \xi_t \dd z_t$."</p>

<p>Informally, we can argue part (a) by applying Ito's lemma,
\begin{align*}
\dd X_t &amp;= \left( \frac 12 \lambda_t^2 + \lambda_t \dd z_t\right) \dd t + \lambda_t \dd z_t \\
  &amp;= \frac 12 \lambda_t^2 \dd t + \lambda_t \dd z_t,
\end{align*}
and arguing that $\dd z_t \cdot \dd t = 0$.</p>

<p>How do we solve part (b)?</p>

<finance><asset-pricing><stochastic-calculus>
Apply Ito's Lemma to exponential martingale
jmbejara
# Apply Ito's Lemma to exponential martingale

(Asked by: username_0 on 2015-12-02)

$\newcommand{\dd}{\, \mathrm{d}}$
Consider the exponential martingale,
$$
\xi\_t^\lambda = \exp \left\{ - \int\_0^t \lambda\_s \dd z\_s - \frac 12 \int\_0^T \lambda\_s^2 \dd s \right\},
$$
that is used in the statement of Girsanov's theorem (this martingale represents the Radon-Nykodym derivative $\frac{\dd \mathbb Q^\lambda}{\dd \mathbb P}$.).

Exercise 2.4 in Munk's book *Financial Asset Pricing Theory* deals with applying Ito's lemma to this process.

Suppose
$$
X\_t = \frac 12 \int\_0^t \lambda \_s^2 \dd s + \int\_0^t \lambda\_s \dd z\_s.
$$
Part (a) asks us to argue that $\dd X\_t = \frac 12 \lambda\_t^2 \dd t + \lambda\_t \dd z\_t.$ Part (b) asks, "Suppose that the continuous-time stochastic process
$\xi = (\xi\_t)$ is defined as $\xi\_t = \exp\{-X\_t\}$. Show that $\dd \xi\_t = -\lambda\_t \xi\_t \dd z\_t$."

Informally, we can argue part (a) by applying Ito's lemma,
\begin{align\*}
\dd X\_t &= \left( \frac 12 \lambda\_t^2 + \lambda\_t \dd z\_t\right) \dd t + \lambda\_t \dd z\_t \\
&= \frac 12 \lambda\_t^2 \dd t + \lambda\_t \dd z\_t,
\end{align\*}
and arguing that $\dd z\_t \cdot \dd t = 0$.

How do we solve part (b)?

**Tags:** finance, asset-pricing, stochastic-calculus

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## 1 answers

### Answer by username_0 on 2015-12-02. Score: 5 | Accepted answer

$\newcommand{\dd}{\, \mathrm{d}}$
If we apply Ito's lemma,
then
\begin{align\*}
\dd \xi\_t &= -\xi\_t \dd X\_t + \frac 12 \xi\_t (\dd X\_t)^2\\
&= -\xi\_t \left(\frac 12 \lambda\_t^2 \dd t + \lambda\_t \dd z\_t\right) + \frac 12 \xi\_t \lambda\_t^2 \dd t \\
&= -\xi\_t \lambda \dd z\_t.
\end{align\*}


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