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Dependent Increments in Financial Stochastic Processes

Article Quant Q&A · Author: jontwo
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Summary

The discussion distinguishes independent increments from dependence on the current state and from correlated increments. It notes that Wiener motion has independent increments, while geometric Brownian motion’s price changes scale with the current price; the answers disagree about whether geometric Brownian motion has independent increments, reflecting a distinction between price-level and log-price formulations. Ornstein–Uhlenbeck processes are also described as having dependent increments, and ARMA models are suggested for financial series such as returns or interest rates.

Fractional Brownian motion is presented as a process with correlated increments, with the ordinary Brownian case recovered at a parameter value of one half. The discussion also points to stochastic time-changed Lévy models, including Heston, where returns may be uncorrelated while increments are dependent through stochastic, mean-reverting volatility. These are illustrative examples rather than a systematic comparison; the answers provide little detail on estimation, calibration, or model suitability for particular trading tasks.

Key ideas

  • Independent increments differ from changes whose distribution depends on the current process level.
  • Geometric Brownian motion has independent log increments, while price changes scale with the current price.
  • Ornstein–Uhlenbeck and ARMA models are examples used to represent dependence in financial series.
  • Fractional Brownian motion can have correlated increments, with standard Brownian motion as a special case.
  • Stochastic volatility models such as Heston can produce dependent returns despite uncorrelated increments.

Tags

StatisticsVolatilityEquitiesFixed income
Full text
# Stochastic process with non-independent increments


# Stochastic process with non-independent increments












All stochastic process I see always have independent increments. It is true for:

- standard brownian motion

- geometric brownian motion (?)

- Ornstein Uhlenbeck (?)

- in general, Levy process

etc.

What are the most common stochastic process used in quant finance that have dependent increments, i.e. $X(t+h) - X(t)$ not independent from $X(t)$ ?

## Answer by Neeraj (score 6)

https://quant.stackexchange.com/a/24352

First thing, Geometric Brownian motion do not have independent increments. It is only Wiener process or Brownian motion that have independent increment. Under GBM, the increments of process (assume stock prices) show markovian property. It means that changes in the process depend on the current price level. In layman terms, the magnitude of change in stock price atleast depend upon existing price level. The probability that stock price will change by \$5 will be different when stock price is \$100, and when stock price is just \$20.

Same argument also hold true with Ornstein Uhlenbeck. The detail answer of why increments under Ornstein Uhlenbeck are not random is given on Math S.E. (Note: The link was first cited by Leon in his answer.)

Now, your question is:

> What are the most common stochastic process used in quant finance that have dependent increments, i.e. $X(t+h)−X(t)$ not independent from $X(t)$

To model such stochastic process where increments are not random, you can use ARMA class of models. They are very common in finance and used extensively to model stock return and other process(like interest rate, GDP, etc.) too.

## Answer by Leon (score 4)

https://quant.stackexchange.com/a/24350

Geometric Brownian Motion has independent increments but Ornstein-Uhlenbeck doesn't have this property.

For more details you can look here.

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/24349

One of the most famous processes with correlated (thus dependent) increments is Fractional Brownian motion.

In this case $$ E[B_H(t) B_H (s)]=\frac{1}{2} (|t|^{2H}+|s|^{2H}-|t-s|^{2H}), $$ where $H$ is a parameter. For $H=1/2$ we get back at uncorrelated increments, thus Brownian motion.

## Answer by user9403 (score 2)

https://quant.stackexchange.com/a/24355

Stochastic time-changed levy processes have uncorrelated increments (which is consistent with "rational" markets) but not independent. In such a model, volatility is (heuristically) mean reverting while returns are uncorrelated. The simplest such model is Heston's model. For a more comprehensive view, see Carr and Wu's paper.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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