last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part two)

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12.4 Ito processes

We are now ready to define precisely the types of stochastic process that we will be interested in for most of the remainder of this course.

  • Definition 12.4.1 A stochastic process \(X\) is known as an Ito process if \(X_0\) is \(\mc {F}_0\) measurable and \(X\) can be written in the form

    \begin{equation} \label {eq:ito_proc} X_t=X_0+\int _0^t F_u\,du + \int _0^t G_u\, dB_u \end{equation}

    Here, \(G\in \mc {H}^2\) and \(F\) is a continuous adapted process.

The first integral is a classical integral: the area under the random curve \(F_t\). The second integral is an Ito integral. All Ito processes are continuous, adapted stochastic processes.

Note that, to fully specify \(X_t\), we also need to know both \(F_u\), \(G_u\), and the initial value \(X_0\). Since this means we’ll be dealing with integrals of the form \(\int _0^t F_u\,du\), it is helpful for us to know a fact from integration theory:

  • Lemma 12.4.2 For a continuous stochastic process \(F\), if one (which \(\ra \) both) of the two sides is finite, then we have \(\E \l [\int _0^t F_u\,du\r ]=\int _0^t\E [F_u]\,du.\)

In words, we can swap \(\int \,du\) and \(\E \)s as long as we aren’t dealing with \(\infty \) (warning: it doesn’t work for Ito integrals, see exercise 12.8!). We won’t include a proof in this course, but you can find one which works for both \(\int \)s and \(\sum \)s in MAS61022.

We can calculate the expectation of \(X_t\) using Lemma 12.4.2.

\begin{align*} \E [X_t]&=\E [X_0]+\E \l [\int _0^t F_u\,du\r ]+\E \l [\int _0^t G_u\,dB_u\r ].\\ &=\E [X_0]+\int _0^t \E \l [F_u\r ]\,du. \end{align*} Here, Lemma 12.4.2 allows us to swap \(\int \) and \(\E \) for the \(du\) integral. The term \(\E [\int _0^t G_u\,dB_u]\) is zero because Theorem 12.2.1 told us that \(\int _0^t G_u\,dB_u\) was martingale with zero mean.

  • Example 12.4.3 Let \(X_t\) be the Ito process satisfying

    \[X_t=1+\int _0^t2B_u^2\,du+\int _0^t3 u\,dB_u.\]

    Then

    \begin{align*} \E [X_t]&=\E [1]+\E \l [\int _0^t2B_u^2\,du\r ]+\E \l [\int _0^t 3u\,dB_u\,\r ]\\ &=1+2\int _0^t\E [B_u^2]\,du\\ &=1+2\int _0^t u\,du\\ &=1+t^2. \end{align*}

  • Example 12.4.4 Brownian motion is an Ito process. It satisfies \(B_t=B_0+\int _0^t0\,du+\int _0^t 1\,dB_u\).

A useful fact about Ito processes is that we can ‘equate coefficients’ in much the same way as we equate the coefficients of terms in polynomials. To be precise, we have

  • Lemma 12.4.5 Suppose that

    \begin{align*} X_t&=X_0+\int _0^t F^X_u\,du+\int _0^tG^X_u\,dB_u\\ Y_t&=Y_0+\int _0^t F^Y_u\,du+\int _0^tG^Y_u\,dB_u \end{align*} are Ito processes and that \(\P [\text {for all }t, X_t=Y_t]=1\). Then,

    \[\P [\text {for all }t, F^X_t=F^Y_t\text { and }G^X=G^Y_t]=1.\]

Proof of this lemma is outside of the scope of our course. However, the result will be very important to us, since it is key to the argument that will allow to hedge financial derivatives in continuous time.