# Definition of the bubbles and crashes

can anyone help me to explain how the following model works?

In this formula $P(t)$ is a price at time $t$ and $F(t)$ is the residual noise term.

The $\omega(i;T_i)$ and $P_{o}(i;T_i)$ are uniquely determined from the past $T_{i}$ data points by condition that minimizes the root-mean square of $F(t)$.

Why $(\omega_{1}(i;T_i)-1)$?

• I dont see how to help you without more detailed info. What paper/textbook is this from? What is omega meant to mean? – Jan Sila Sep 18 '16 at 15:07
• The link of the article researchgate.net/publication/…. – user3610659 Sep 18 '16 at 15:13

(1) $\omega>1$ the price is either exponentially increasing or decreasing and $P_{0}$ gives the base line of the exponential divergence. We deﬁne such behavior as a bubble or a crash. In this case the positive feedback from the past price change becomes larger as the time passes.
(2) $\omega=0$ the price follows a random walk.
(3) $\omega<0$ the price is convergent to $P_{0}$.
So the value of the parameter just either magnifies the expression $\{P(t-1)-P_{0}\}$, renders it zero, or makes the price increments revert to $P_{0}$.