6.2 Stochastic Integration

In order to introduce a stochastic process as a solution of a stochastic differential equation as we introduce the concept of the Itô-integral: a stochastic integral with respect to a Wiener process. Formally the construction of the Itô-integral is similar to the one of the Stieltjes-integral. But instead of integrating with respect to a deterministic function (Stieltjes-integral), the Itô-integral integrates with respect to a random function, more precisely, the path of a Wiener process. Since the integrant itself can be random, i.e. it can be a path of a stochastic process, one has to analyze the mutual dependencies of the integrant and the Wiener process.

Let $ \{ Y_t; \, \, t \ge 0\}$ be the process to integrate and let $ \{ W_t ; \, \, t \ge 0 \}$ be a standard Wiener process. The definition of a stochastic integral assumes that
$ \{ Y_t; \, \, t \ge 0\}$ is not anticipating. Intuitively, it means that the process up to time $ s$ does not contain any information about future increments $ W_t - W_s \, , \, \, \, t > s\ ,$ of the Wiener process. In particular, $ Y_s$ is independent of $ W_t - W_s\
.$

An integral of a function is usually defined as the limit of the sum of the suitably weighted function. Similarly, the Itô integral with respect to a Wiener process is defined as the limit of the sum of the (randomly) weighted (random) function $ \{ Y_t; \, \, t \ge 0 \} :$

$\displaystyle I_n = \sum^n_{k=1} \, Y_{(k-1) \Delta t} \, \cdot \, ( W_{k \Delta t} - W_{(k-1) \Delta t} ) \, \, , \quad \Delta t = \frac{t}{n}$ (6.1)

$\displaystyle \int^t_0 \, Y_s\ dW_s = \lim_{n \rightarrow \infty} \, I_n\ ,$

where the limit is to be understood as the limit of a random variable in terms of mean squared error, i.e. it holds

$\displaystyle \mathop{\text{\rm\sf E}}\{\big[ \int^t_0 \, Y_s\ dW _s - I_n \big ] ^2\} \rightarrow 0,
\quad n \rightarrow \infty. $

It is important to note, that each summand of $ I_n$ is a product of two independent random variables. More precisely, $ Y_{(k-1) \Delta t},$ the process to integrate at the left border of the small interval $ [(k-1) \Delta t, \, k
\Delta t ]$ is independent of the increment $ W_{k \Delta t} -
W_{(k-1) \Delta t}$ of the Wiener process in this interval.

It is not hard to be more precise on the non anticipating property of
$ \{ Y_t; \, \, t \ge 0 \}.$

Definition 6.1 (Information structure, non-anticipating)  
For each $ t \ge 0, \, {{\cal F}} _t$ denotes a family of events (having the structure of a $ \sigma$-algebra, i.e. certain combinations of events contained in $ {{\cal F}} _t$ are again in $ {{\cal F}} _t)$ which contain the available information up to time $ t.$ $ {{\cal F}} _t$ consists of events from which is known up to time $ t$ whether they occurred or not. We assume:
$ {{\cal F}} _s \subset {{\cal F}} _t$ for $ s
< t$ (information grows as time
    evolves)
$ \{ a < Y_t < b \} \in {{\cal F}} _t$   ($ Y_t$ contains no information
    about events occurring after
    time $ t$)
$ \{ a < W_t < b \} \in {{\cal F}} _t $    
$ W_t - W_s $ independent of $ {{\cal F}} _s$ for $ s
< t$ (the Wiener process is adapted to
    evolution of information)

Then, we call $ {{\cal F}} _t$ the information structure at time $ t$ and the process $ \{ Y_t; \, \, t \ge 0\}$ non-anticipating with respect to the information structure $ {{\cal F}} _t; \, \, t
\ge 0 .$

The process $ \{ Y_t \}$ is called non-anticipating since due to the second assumption it does not anticipate any future information. The evolving information structure $ {{\cal F}} _t$ and the random variables $ Y_t, W_t$ are adapted to each other.

The integral depends crucially on the point of the interval $ [(k-1)\Delta t, \, \, k \Delta t ]$ at which the random variable $ Y_s$ is evaluated in (5.1). Consider the example $ Y_t =
W_t, \, \, t \ge 0,$ i.e. we integrate the Wiener process with respect to itself. As a gedankenexperiment we replace in (5.1) $ (k-1)\Delta t$ by an arbitrary point $ t(n,k)$ of the interval $ [(k-1) \Delta t, \, \, k \Delta t ].$ If we defined:

$\displaystyle \int ^t_0 W_s \, d W_s = \lim_{n \rightarrow \infty} \,
\sum^n_{k=1} W_{t(n,k)} \, (W_{k \Delta t} - W_{(k-1)\Delta t} )
$

the expected values would converge as well. Hence by interchanging the sum with the covariance operator we get:
  $\displaystyle \mathop{\text{\rm\sf E}}$ $\displaystyle [\sum^n_{k=1} W_{t(n,k)} \, (W_{k \Delta t} - W
_{(k-1)\Delta t} ...
...} \mathop{\text{\rm Cov}}(W_{t(n,k)} , \, W_{k
\Delta t} - W_{(k-1) \Delta t} )$  
  $\displaystyle =$ $\displaystyle \sum^n_{k=1} \{t (n,k) - (k-1)\Delta t\} \rightarrow \mathop{\text{\rm\sf E}}[\int^t_0
W_s \, dW_s] \, .$  

For $ t(n,k)=(k-1) \Delta t$ - which is the case for the Itô-integral - we obtain 0, for $ t(n,k)=k \Delta t$ we obtain $ n \cdot \Delta t = t \, ,$ and for suitably chosen sequences $ t(n,k)$ we could obtain for the expectation of the stochastic integral any value between 0 and $ t.$ In order to assign to $ \int^t_0 W_s\ dW_s$ a unique value, we have to agree on a certain sequence $ t(n,k).$

To illustrate how Itô-integrals are computed, and that other than the usual computation rules have to be applied, we show that:

$\displaystyle \int^t_0 W_s\ dW_s = \frac{1}{2} (W_t^2 - W_0^2 )\ - \, \frac{t}{2} = \frac{1}{2} (W_t ^2 - t)$ (6.2)

Summing the differences $ W_{k\Delta t}^2 - W_{(k-1)\Delta t} ^2 \,
,$ all terms but the first and the last cancel out and remembering that $ n \Delta t = t$ we get
$\displaystyle \frac{1}{2} (W_t^2 - W_0^2 )$ $\displaystyle =$ $\displaystyle \frac{1}{2}\, \sum^n_{k=1}
(W_{k\Delta t}^2 - W_{(k-1)\Delta t}^2 )$  
  $\displaystyle =$ $\displaystyle \frac{1}{2} \, \sum^n_{k=1} (W_{k\Delta t} - W_{(k-1)\Delta
t} )
(W_{k\Delta t} + W_{(k-1)\Delta t} )$  
  $\displaystyle =$ $\displaystyle \frac{1}{2} \, \sum^n_{k=1} (W_{k\Delta t} - W_{(k-1)\Delta
t}
)^2$  
    $\displaystyle + \sum^n_{k=1} (W_{k\Delta t} - W_{(k-1)\Delta t})\
W_{(k-1)\Delta t} \, .$  

While the second term converges to $ \int^t_0 W_s\ dW_s\ ,$ the first term is a sum of $ n$ independent identically distributed random variables and which is thus approximated due to the law of large numbers by its expected value

$\displaystyle \frac{n}{2} \mathop{\text{\rm\sf E}}[(W_{k\Delta t} - W_{k-1)\Delta t} )^2] =
\frac{n}{2} \Delta t = \frac{t}{2} \, . $

For smooth functions $ f_s$, for example continuously differentiable functions, it holds $ \int^t_0 f_s\ df_s = \frac{1}{2} (f_t^2 - f_0^2).$ However, the stochastic integral (5.2) contains the additional term $ -\frac{t}{2}$ since the local increment of the Wiener process over an interval of length $ \Delta t$ is of the size of its standard deviation - that is $ \sqrt{\Delta t}$. The increment of a smooth function $ f_s$ is proportional to $ \Delta t$, and therefore considerably smaller than the increment of the Wiener process for $ \Delta t \rightarrow 0.$