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Converting a Discrete GBM Sum into an Itô Integral

Article Quant Q&A · Author: texmex
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Summary

The document considers the limit of a discrete sum involving two geometric Brownian motions, with a constant offset applied to one process. It defines a partition of the time interval and rewrites the sum using increments of the second process. As the partition becomes finer, the sum converges to a stochastic integral with respect to that process, with the integrand expressed as the first process minus the constant.

This identifies the continuous-time object but does not provide a method for evaluating the integral or discuss conditions for convergence. The source is a short answer to a mathematical question, so it offers no worked solution, numerical evidence, or treatment of dependence between the Brownian motions. Further analysis would require specifying the processes’ parameters and joint dynamics.

Key ideas

  • A partitioned sum of process values multiplied by increments can converge to a stochastic integral.
  • The stated limit integrates the first process minus the constant with respect to the second process.
  • The answer identifies the integral but does not solve it or state convergence conditions.
  • The joint behavior of the two Brownian motions is not addressed.

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# Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions


# Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions












Could someone please suggest with detailed steps and/or a reference,

1) How to convert the below discrete time summation to continuous time form and write it as an integral?

2) Any methods to solve it?

$$ \sum_{t=0}^{T}\left[\left(K-X_{t}\right)\right]\left(Y_{t}-Y_{t+1}\right) $$

Here, $K$ is a constant. $X$ is a geometric brownian motion. $Y$ is another geometric brownian motion.

Please let me know if anything is not clear.

Posted initially on Math Forum with no response; hence posting here. Please let me know if I should delete the posting on the other forum.

https://math.stackexchange.com/questions/1899002/discrete-time-to-continuous-time-and-summation-of-two-geometric-brownian-motions

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/29956

By definition

let $t^n_k=\frac{k}{n}T$, $$I^n = \sum_{k=0}^{n-1} (K-X_{t_k})(Y_{t_k}-Y_{t_{k+1}}) \to_{n\to\infty} I = \int_0^T (X_t-K)dY_t $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.