Background: Hello everybody. I'm reading the paper Optimal Portfolio under Fractional Stochastic Environment by Fouque and Hu. The proof of Proposition 2.2 contains an estimation (Abschätzung) of an expected value I cannot deduce myself. Thanks a lot for your help!
Setup: Let $[0,T]$ be a time-index set and $(\Omega,\mathcal{F},\mathbb{P})$ a probability space supporting an $\mathbb{R}$-valued Brownian motion $W$. Let $\mathbb{F}$ be the augmented filtration generated by $W$. Let $\xi\in L^{2}(W)$. Let $c\in\mathbb{R}_{>0}$ and $k\in\mathbb{R}_{>1}$. We assume that $$\mathbb{E}\Big[e^{c\int_{0}^{T}\xi_{s}^{2} ds}\Big] < \infty.\tag{1}$$
Question: Can we deduce that $$\mathbb{E}\bigg[\int_{0}^{T}\xi_{s}^{2k}ds\bigg] < \infty? \tag{2}$$
My thoughts: (1) implies that the moment generating function of the random variable $\int_{0}^{T}\xi_{s}^{2}ds$ is well-defined in a neighborhood of $0$. Hence, $$\forall l\in\mathbb{N}:\, \mathbb{E}\Bigg[\bigg(\int_{0}^{T}\xi_{s}^{2}ds\bigg)^{l}\Bigg] < \infty. \tag{3}$$ However, I cannot prove that $$\exists l\in\mathbb{N}:\, \mathbb{E}\bigg[\int_{0}^{T}\xi_{s}^{2k}ds\bigg] \leq \mathbb{E}\Bigg[\bigg(\int_{0}^{T}\xi_{s}^{2}ds\bigg)^{l}\Bigg],$$ so (3) is not of much help.
No, $(1)$ does, in general, not imply $(2)$. Define
$$\xi(s,\omega) := \frac{1}{s^{1/4}} 1_{(0,\infty)}(s) .$$
Clearly,
$$\mathbb{E} \exp \left( \int_0^T \xi_s^2 \, ds \right) = \exp \left( \int_0^T s^{-1/2} \, ds \right)<\infty$$
for any $T>0$ but
$$\mathbb{E} \left( \int_0^T \xi_s^{2k} \, ds \right) = \int_0^T s^{-k/2} \, ds = \infty$$
for all $k \geq 2$.