# 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 $X(t)$ ?

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. • It depends on how you define increments though. GBM, under a transformation, can be described by$X(t+h)-X(t)$where$X(t+h)-X(t)$is independent of$X_t$. There is no such transformation where an OU process can be described in such a way. – user9403 Feb 17 '16 at 12:09 • @user9403: Of course independence and increment refers to the original and NOT the transformed variable. Otherwise, it would be a truism that renders the concept and distinction of dependence and independence meaningless, because any two distinct random variables can be made independent by a transformation. – Hans Feb 24 '16 at 21:00 Geometric Brownian Motion has independent increments but Ornstein-Uhlenbeck doesn't have this property. For more details you can look here. • increments under GBM are not independent. – Neeraj Feb 17 '16 at 8:32 • Ok, sorry for my mistake. – Leon Feb 17 '16 at 8:44 • It is ok @Leon. We all here to lean from each other. – Neeraj Feb 17 '16 at 8:50 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.

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.