Stochastic Processes and Financial Mathematics
(part two)
Chapter 12 Stochastic integration
In this section we introduce stochastic integrals, through the framework of Ito integration. The mathematical framework for stochastic integration was developed in the 1950s, by the Japenese mathematician Kiyoshi Ito (sometimes written Itô). It has grown into becoming one of the most effective modelling tools of the present day.
In Lemma 11.4.2 we showed that Brownian motion was not differentiable. This is awkward, because mathematical modelling often relies on calculus, which (in its classical form) relies heavily on working with derivatives. However, the difficulty can be overcome by forgetting about differentiation and making integration the central theme.
12.1 Introduction to Ito calculus
In classical calculus, of the sort you are already used to using, we typically deal with objects of the form
\(\seteqnumber{0}{12.}{0}\)\begin{equation} \label {eq:classical_int} \int _a^b f(t)\,dt \end{equation}
where \(f\) is a suitably well behaved function. For simplicity, let’s take \(f\) to be continuous. From an intuitive point of view, we often regard (12.1) as representing the ‘area under the curve’ \(f\) between \(a\) and \(b\). This is justified by the fact that we have
\(\seteqnumber{0}{12.}{1}\)\begin{equation} \label {eq:classical_int_approx} \int _a^b f(t)\,dt=\lim \limits _{\delta \to 0}\,\sum \limits _{i=1}^n f(t_{i-1})[t_{i}-t_{i-1}] \end{equation}
where \((t_i)_{i=0}^n\) is such that \(a=t_0<t_1<\ldots <t_n=b\) and \(\delta =\max _i|t_t-t_{t-1}|\). Note that sending \(\delta \to 0\) means that the \(t_i\) change position and get closer together, and consequently \(n\to \infty \); this is a mild abuse of notation that is commonly used.
Note that if \(f\) is a random continuous function then (12.1) still makes sense: now, \(\int _a^b f(t)\,dt\) is just the area under a random curve; itself a random quantity. This is one way to involve random variables in calculus. There is another:
In Ito calculus we are interested in integrals that are written
\[\int _a^b f(t)\,dB_t\]
where \(B_t\) is a Brownian motion. Let us begin by discussing what this new type of integral represents; it is not the area under a curve.
In (12.2), the \(dt\) on the left side corresponds to the \(t_i-t_{i-1}\) on the right. By analogy to (12.2), our new \(dB_t\) term corresponds to \(B_{t_{i}}-B_{t_{i-1}}\), giving
\(\seteqnumber{0}{12.}{2}\)\begin{equation} \label {eq:stoch_int_approx} \int _a^b f(t)\,dB_t=\lim \limits _{\delta \to 0}\,\sum \limits _{i=1}^n f(t_{i-1})[B_{t_{i}}-B_{t_{i-1}}]. \end{equation}
Graphically, this means that we measure the widths of the bars using increments of Brownian motion, instead of side-length. For now, let us not worry about which mode of convergence will be used for the limit, or how to choose the \(t_i\)s.
In order to understand why this is a useful idea, from the point of view of stochastic modelling, we need to think about \(\sigma \)-fields and filtrations. In particular, let us take \(f(t)\) to be a stochastic process, and let us assume that \(f(t)\) is adapted, with respect to the filtration \(\mc {F}_t=\sigma (B_s\-s\leq t)\). Now, consider the term
\[f(t_{i-1})[B_{t_i}-B_{t_{i-1}}].\]
This formula represents a generic model of taking a decision that then has a random effect. The value of \(f(t_{i-1})\) is chosen, based only on information known at time \(t_{i-1}\), then during \(t_{i-1}\mapsto t_i\) the world evolves randomly around us, and the effect of our decision combined with this random evolution is represented by \(f(t_{i-1})[B_{t_i}-B_{t_{i-1}}]\).
The sum,
\(\seteqnumber{0}{12.}{3}\)\begin{equation} \label {eq:ito_mart_transform} \sum \limits _{i=1}^n f(t_{i-1})[B_{t_{i}}-B_{t_{i-1}}] \end{equation}
corresponds to the cumulative result of multiple decision making steps, at times \(t_0\mapsto t_1\mapsto t_2\mapsto \ldots \mapsto t_n\). At each time \(t_{i-1}\) a decision is taken for the value of \(f(t_i)\), based only on previously available information, then the world changes randomly during \(t_{i-1}\mapsto t_i\), and at time \(t_i\) we receive and add the random effect of our decision: \(f(t_{i-1})[B_{t_i}-B_{t_{i-1}}]\).
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Remark 12.1.1 We’ve seen this idea before, in Section 7.2 when we modelled roulette using the martingale transform. If we set \(t_i=i\), \(C_n=f(t_n)\) and \(M_n=B_{t_n}=B_n\), then \(M_n\) is a discrete time martingale (take \(\mc {F}_n=\sigma (B_i\-i\leq n)\)), and (12.4) is precisely the martingale transform \((C\circ M)_n=\sum _{i=1}^n C_{i-1}(M_i-M_{i-1})\).
The final stage of this intuition is to understand the limit in (12.3). Now we take decisions at times
\[a=t_0\mapsto t_1\mapsto t_2\ldots \mapsto t_n=b\]
as \(\delta =\max _{i}|t_i-t_{i-1}|\to 0\). The corresponds to taking a continuous stream of decisions during the time interval \([a,b]\), each based on previously available information, each of which has an (infinitesimally) small effect. The stochastic integral,
\[\int _a^b f(t)\,dB_t\]
corresponds to the cumulative effect of all these decisions.
Of course, the situation we are most interested in, within this course, is that of managing a portfolio. In continuous time we can continually take decisions to buy and sell based on the information that is currently available to us. Our \(f(t_i)\) will be a process relating to the stocks that we hold, and the Brownian motion \(B_t\) will provide the randomness that moves stock prices up and down. Developing the details of this modelling effort, and the pricing results that come out of it, will take up the rest of the course.
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Remark 12.1.3 As an alternative approach to defining the meaning of Ito integrals it might be tempting to try and write
\[\int _a^b F_t\,d B_t=\int _a^b F_t\,\frac {d B_t}{dt}\, dt\]
and use this idea to relate stochastic integrals to classical integrals. Unfortunately, the right hand side of the above expression does not make sense - we have shown in Lemma 11.4.2 that \(B_t\) is not differentiable, so \(\frac {d B_t}{dt}\) does not exist.