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

$${P}_{n}=\{0={\tau}_{0}^{n}\le {\tau}_{1}^{n}\le \mathrm{\cdots}\uparrow \mathrm{\infty}\}$$ |

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,

$${I}_{t}^{n}(Y,X)\to {\int}_{0}^{t}Y\mathit{d}X$$ |

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,

$${I}_{t}^{n}(Y,X)\to {\int}_{0}^{t}{Y}_{s-}\mathit{d}{X}_{s}$$ |

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

Title | stochastic integration as a limit of Riemann sums |
---|---|

Canonical name | StochasticIntegrationAsALimitOfRiemannSums |

Date of creation | 2013-03-22 18:41:33 |

Last modified on | 2013-03-22 18:41:33 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 60H05 |

Classification | msc 60G07 |

Classification | msc 60H10 |

Related topic | StochasticIntegration |