# On the Inception of Financial Representative Bubbles

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## Abstract

**:**

## 1. Introduction

## 2. The Inception and the Representativeness

#### 2.1. The Rational Case

#### 2.2. A Brief Digression: The Representativeness

#### Diagnostic Expectations

**Assumption**

**1.**

**Definition**

**1.**

#### 2.3. The Representative Case

#### 2.3.1. Diagnostic Expectations: Extrapolation and Neglect

- (i)
- the first term on the right-hand side is:$$\begin{array}{cc}\hfill \frac{1}{{F}_{t}^{d}\left(s\right)}\frac{\partial {F}_{t}^{d}\left(s\right)}{\partial {\mathbb{E}}_{t}^{d}\left({y}_{t+1}\right)}& ={\int}_{\overline{s}}^{\infty}\frac{{y}_{t+1}-{\mathbb{E}}_{t}^{d}\left({y}_{t+1}\right)}{{\sigma}^{2}}\xb7exp\left[-\frac{{({y}_{t+1}-{\mathbb{E}}_{t}^{d}\left({y}_{t+1}\right))}^{2}}{2{\sigma}^{2}}\right]\frac{d{y}_{t+1}}{{F}_{t}^{d}\left(s\right)}\hfill \\ & =\frac{1}{{\sigma}^{2}}{\int}_{\overline{s}}^{\infty}{y}_{t+1}exp\left(-\frac{1}{2{\sigma}^{2}}\left[{({y}_{t+1}-{\mathbb{E}}_{t}^{d}\left({y}_{t+1}\right))}^{2}\right]\right)\frac{d{y}_{t+1}}{\partial {F}_{t}^{d}\left(s\right)}-\frac{{\mathbb{E}}_{t}^{d}\left({y}_{t+1}\right)}{{\sigma}^{2}}\hfill \end{array}$$$$\frac{\partial ln{F}_{t}^{d}\left(s\right)}{\partial {y}_{t}}=[{\mathbb{E}}_{t}^{d}({y}_{t+1}\mid {y}_{t+1}\ge \overline{s})-{\mathbb{E}}_{t}^{d}\left({y}_{t+1}\right)]\frac{1}{{\sigma}^{2}};$$
- (ii)
- as for the second term, assume normal densities and an autoregressive model of order one, AR (1) for ${\mathbb{E}}_{t}\left({y}_{t+1}\right)=a+b{y}_{t}$, rewriting (13) in these terms, then:$$\frac{\partial {\mathbb{E}}_{t}^{d}\left({y}_{t+1}\right)}{\partial {y}_{t}}=b(1+\theta )>0.$$

#### 2.3.2. The Representative Bubble

**Assumption**

**2.**

#### 2.3.3. When Does It Start?

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Conclusions

- -
- From Proposition 1, a representative bubble can start at any time due to displacements that follow from extrapolative expectations and neglect of tail risks. In short, from diagnostic expectations;
- -
- As rational bubbles, the representative dynamics cannot be negative due to the free disposal;
- -
- Following Proposition 1, representative bubbles can arise, burst at zero and then grow again in the same asset;
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- A representative bubble is not dependent on the failure of the transversality condition; hence, it can be persistent and distorted.

## Author Contributions

## Conflicts of Interest

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Ferrara, M.; Pansera, B.A.; Strati, F. On the Inception of Financial Representative Bubbles. *Mathematics* **2017**, *5*, 64.
https://doi.org/10.3390/math5040064

**AMA Style**

Ferrara M, Pansera BA, Strati F. On the Inception of Financial Representative Bubbles. *Mathematics*. 2017; 5(4):64.
https://doi.org/10.3390/math5040064

**Chicago/Turabian Style**

Ferrara, Massimiliano, Bruno A. Pansera, and Francesco Strati. 2017. "On the Inception of Financial Representative Bubbles" *Mathematics* 5, no. 4: 64.
https://doi.org/10.3390/math5040064