Initial boundary value problem for fractional $p$-Laplacian Kirchhoff type equations with logarithmic nonlinearity

In this note we show that the result of Theorem 4.16 of ^[1] is false by constructing a sequence of simple predictable strategies achieving Free-Lunch-with-Vanishing-Risk $(FLVR)$ whose existence contradicts the conclusions of the theorem. The fault in the proof in ^[1] comes from the improper use of a bound on the compensator $\alpha^G$. Indeed the bound holds only ${\boldsymbol{P}}$-almost surely, that is not strong enough to assure the required Novikov condition.

Using the notation introduced in ^[1], we consider the initial enlargement $\mathbb{G} \supset \mathbb{F}$ obtained by extending the natural filtration by the random variable

(1)

Assuming a constant proportional volatility $\xi > 0$, it follows that

and the random variable $G$ can be rewritten as , where $c_k : = \tilde s_0 e^{\xi k}$. The length of each interval is $\lambda_k: = c_{2k}-{c_{2k-1}} = \tilde s_0 e^{\xi 2k}\left(1-e^{-\xi}\right) > 0$. To simplify the computations, we assume that the interest rate $r = 0$.

Proposition 1. Let $G$ be as in (1), the condition ${\textrm{(FLVR)}} _{ \mathcal{H}(\mathbb{G})}$ is satisfied.

Proof. Without loss of generality, we assume that, for some $t_0 < T$ there exists $k_0\in \mathbb{Z}$ such that $c_{2k_0}-\lambda_{k_0}/4\geq S_{t_0}\geq c_{2k_0-1}+\lambda_{k_0}/4$. We reason for the case $G = 0$, that in particular implies that $S_T\not\in(c_{2k_0-1}, c_{2k_0})$, the case $G = 1$ is equivalent by symmetry. We define the following finite sets

together with the following sequence of stopping times, $\tau_0 = t_0$ and for $n\geq 1$

where we define $\inf\emptyset = T$. With some abuse of notation, we construct a sequence of strategies $\{\Theta_n\}_n$ with $\Theta_0 = 0$ and for $n\geq1$, being $C_n$ the following $\mathcal{F}_{\tau_{2n-1}}$-measurable random variable

We prove that the sequence of strategies $\{\Theta_n\}_n$ achieves a gain greater than $\dfrac{\lambda_{k_0}}{4}-\delta(k_0)$, and by appropriately choosing $\delta(k_0)$ we can get (FLVR)$_{ \mathcal{H}(\mathbb{G})}$. To short the notation, we introduce the family

We need to verify that $\lim_{m\to\infty} H_m = 0$, ${\boldsymbol{P}}(\cdot\, \vert\, G = 0){\text{-a.s.}}$ By definition of convergence a.s., it is equivalent to

The sequence of indicator functions is strictly decreasing by construction, so we need to check that

where the last condition is satisfied.

Remark. By using an analogous technique, it can be proved that any random variable generates (FLVR) when $B$ is a subset of positive probability less than one.

Since the result of Theorem 4.16 in ^[1] is false, we prove here a weaker result by showing that the strategies of type buy-and-hold do not generate arbitrage, (NA), as it is shown in the following proposition.

Proposition 2. Let $G$ be as in (1), the condition $\textrm{(NA)$ _{ \mathcal{H}(\mathbb{G})} $}$ is satisfied with strategies of the type $\Theta =$ , being $\sigma$ any $\mathbb{G}$-stopping time and $C$ a $\mathcal{G}_\sigma$-measurable random variable not identically zero.

Proof. We claim that there exists some achieving arbitrage and we look for a contradiction. We start by computing the following conditional probabilities

We introduce the event $A: = \{C(S_T-S_\sigma) < 0\}$, by the definition of arbitrage we have ${\boldsymbol{P}}({\boldsymbol{A}}) = 0$ and jointly with the law of total probability we find the following contradiction,

which is positive because ${\boldsymbol{P}}(C\not = 0) > 0$ and the conditional probabilities given by (2).

Conflict of interest

The authors declare there is no conflict of interest.