# 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 $\mathbb{R}_{+}$,

 $P_{n}=\left\{0=\tau^{n}_{0}\leq\tau^{n}_{1}\leq\cdots\uparrow\infty\right\}$

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

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

 $I^{n}_{t}(Y,X)\equiv\sum_{k}Y_{\tau^{n}_{k-1}}(X_{\tau^{n}_{k}\wedge t}-X_{% \tau^{n}_{k-1}\wedge t}).$

Since the times $\tau^{n}_{k}$ tend to infinity as $k\rightarrow\infty$, all but finitely many terms are zero. Note that here, the process $Y$ is sampled at $\tau^{n}_{k-1}$, 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,

 $I^{n}_{t}(Y,X)\rightarrow\int_{0}^{t}Y\,dX$

in probability as $n\rightarrow\infty$. 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,

 $I^{n}_{t}(Y,X)\rightarrow\int_{0}^{t}Y_{s-}\,dX_{s}$

in probability as $n\rightarrow\infty$. Furthermore, this converges ucp and in the semimartingale topology.

Title stochastic integration as a limit of Riemann sums StochasticIntegrationAsALimitOfRiemannSums 2013-03-22 18:41:33 2013-03-22 18:41:33 gel (22282) gel (22282) 4 gel (22282) Theorem msc 60H05 msc 60G07 msc 60H10 StochasticIntegration