stochastic integration as a limit of Riemann sums

As with the Riemann (http://planetmath.org/RiemannIntegral) and Riemann-Stieltjes integrals, the stochastic integral can be calculated as a limit of approximations computed on partitions (http://planetmath.org/Partition3), called Riemann sums.

Let $P_n$ be a sequence of partitions of ${ℝ}_{+}$,

$$P_n=\{0={\tau }_0^n\le {\tau }_1^n\le \mathrm{\cdots }\uparrow \mathrm{\infty }\}$$

where, ${\tau }_k^n$ can, in general, be stopping times. Suppose that the mesh $|P_n^t|={sup}_k({\tau }_k^n\wedge t-{\tau }_{k-1}^n\wedge t)$ tends to zero in probability (http://planetmath.org/ConvergenceInProbability) as $n\to \mathrm{\infty }$, for each time $t>0$.

The stochastic integral of a process $Y$ with respect to $X$ can be defined on each of the partitions,

$$I_t^n(Y,X)\equiv \sum _k{Y}_{{\tau }_{k-1}^n}(X_{{\tau }_k^n\wedge t}-X_{{\tau }_{k-1}^n\wedge t}).$$

Since the times ${\tau }_k^n$ tend to infinity as $k\to \mathrm{\infty }$, all but finitely many terms are zero.Note that here, the process $Y$ is sampled at ${\tau }_{k-1}^n$, which are the left hand points of the intervals. For this reason, the stochastic integral is sometimes called the forward integral. Alternatively, the backward integral can be computed by sampling $Y$ at time $t_k$ and the Stratonovich integral takes the average of $Y_{t_{k-1}}$ and $Y_{t_k}$. However, these alternative integrals are less general and may not exist even when $Y$ is a continuous and adapted process.

For left-continuous integrands, the approximations do indeed converge to the stochastic integral.

Theorem 1.

Suppose that $X$ is a semimartingale and $Y$ is an adapted, left-continuous and locally bounded process. Then,

$$I_t^n(Y,X)\to {\int }_0^{t}Y𝑑X$$

in probability as $n\mathrm{\to }\mathrm{\infty }$. Furthermore, this converges ucp and in the semimartingale topology.

Similarly, convergence is also obtained for cadlag integrands. However, in this case, it is necessary to use the left limit $Y_{s-}$ in the integral. The integral of $Y$ does not even exist when it is a general cadlag adapted process, as it might not be predictable.

Theorem 2.

Suppose that $X$ is a semimartingale and $Y$ is a cadlag adapted process. Then,

$$I_t^n(Y,X)\to {\int }_0^t{Y}_{s-}𝑑X_s$$

in probability as $n\mathrm{\to }\mathrm{\infty }$. Furthermore, this converges ucp and in the semimartingale topology.