Erratum to: Asset price bubbles from heterogeneous beliefs about mean reversion rates

Erratum to: Finance Stoch (2011) 15:221–241 DOI 10.1007/s00780-010-0124-x

Theorem 5.1 of [1] draws correct conclusions, however the proof is incomplete. Indeed, the final paragraph appeals to a “basic comparison theorem for viscosity super- and subsolutions, see e.g. Theorem 5.1 of [2].” Alas, the cited result from [2] concerns equations of the form u(x)+F(Du,D ² u)−f(x)=0, whereas the equation under consideration in [1] does not have this form.

The purpose of that final paragraph was to conclude that Φ(D)≤P _∗(D), where P _∗ is the minimal equilibrium price and Φ is the unique C ² solution of

$$ -\max\bigl\{\kappa_1(\theta - D), \kappa_2( \theta - D)\bigr\}\varPhi' - \frac{1}{2}\sigma^2 \varPhi'' + \lambda \varPhi - D = 0 $$

(1.1)

with linear growth at infinity. Actually, appeal to a general comparison result is unnecessary. The desired conclusion follows easily from the fact that P _∗(D) is a viscosity supersolution, using the asymptotic properties of P _∗(D) and Φ(D) as |D|→∞. Thus Theorem 5.1 of [1] can be replaced with the following:

FormalPara Theorem 5.1

The equilibrium price Φ(D) identified in Sect. 4 and the minimal equilibrium price P _∗(D) discussed in Sect. 3 have the following properties:

• (i) P _∗(D)≤Φ(D), and Φ(D)−P _∗(D)→0 as |D|→∞.

• (ii) P _∗(D) is a lower semicontinuous function.

• (iii) P _∗(D) is a viscosity supersolution of (1.1).

Furthermore, assertions (i)–(iii) imply

$$ \varPhi(D) \leq P_{*}(D) , $$

(1.2)

so Φ=P _∗. Thus, the unique classical solution of the differential equation (1.1) with linear growth at infinity is in fact the minimal equilibrium price.

FormalPara Proof

The assertion P _∗(D)≤Φ(D) is obvious, since Φ is an equilibrium price and P _∗ is the minimal equilibrium price. We also know P _∗(D)≥I(D), where

is the intrinsic value (cf. (2.3) of [1]), since the definition of an equilibrium price (Definition 2.1 of [1]) includes this inequality. Theorem 4.1(b) of [1] shows that Φ(D) − I(D) → 0 as |D| → ∞. This gives (i), since Φ(D) − P _∗(D) ≤ Φ(D) − I(D).

Assertions (ii) and (iii) are stated and proved in Theorem 5.1 of [1].

For the final conclusion (1.2), consider the variational problem

$$\inf_{D \in \mathbb{R}} \bigl\{ P_*(D) - \varPhi(D) \bigr\}. $$

If a minimizing sequence tends to ±∞ then the minimum value is 0 by (i), and (1.2) is true. If on the other hand a minimizing sequence stays bounded, then the minimum is achieved at some D ₁, by (ii). Since P _∗ is a viscosity supersolution we have

$$-\max\bigl\{\kappa_1(\theta - D_1), \kappa_2(\theta - D_1)\bigr\}\varPhi'(D_1) - \frac{1}{2}\sigma^2\varPhi''(D_1) + \lambda P_*(D_1) - D_1 \geq 0. $$

It follows since Φ solves (1.1) that

$$-\lambda \varPhi(D_1) + \lambda P_*(D_1) \geq 0. $$

Since the discount rate λ is positive, we conclude that P _∗(D ₁)−Φ(D ₁)≥0. Thus P _∗(D)−Φ(D)≥P _∗(D ₁)−Φ(D ₁)≥0, completing the proof of (1.2). □