Background: Let $(\Omega, \mathcal{F}, \mathbb{P})$ model a financial market and $T>0$. Denote by $(S_t)_{t\in[0,T]}$ the price process of the risky asset in the financial market. Assume that the financial market is complete, i.e., that there exists a unique state-price density $(\xi_t)_{t\in [0,T]}$ to compute the price of any terminal payoff $X_T$ as $c(X_T) = \mathbb{E}[\xi_T X_T]$ where $\mathbb{E}$ denotes the expectation taken under $\mathbb{P}$.
Consider an agent with a budget $x_0>0$ that maximizes expected utility among all possible derivatives of $S_T$ that cost at most $x_0$, i.e. the agent solves
$\max_{X_T\in \mathcal{X}(x_0)} $ $\mathbb{E}[u(X_T)]$
where
- $\mathcal{X}(x_0):= \{X_T = g(S_T), g: \mathbb{R}_+\rightarrow \mathbb{R},c(X_T) \leq x_0 \}$ is the set of affordable derivatives
- $u: \mathbb{R} \rightarrow \mathbb{R}$ is a utility function that is twice continuously differentiable with $u'(x)>0$ and $u''(x)<0$ that satisfies the Inada-conditions ($\lim_{x\rightarrow 0} u'(x) = \infty$ and $\lim_{x\rightarrow\infty} u'(x) = 0$).
It is well known that a optimal payoff to the expected utility maximization is given by
$X_T^* = I(\lambda \xi_T)$
where $I=(u')^{-1}$ is the inverse of the marginal utility and $\lambda>0$ is chosen such that $\mathbb{E}[\xi_T X_T^*] = x_0$.
Claim: I believe it should be possible to characterize the optimal payoff $X_T^*$ via
$E[u'(X_T^*)\cdot X_T]\leq 0 \quad \forall X_T \in \mathcal{X}(x_0)$
Idea for proof: Take the Gateaux derivative of the functional
$F: \mathcal{X}(x_0) \rightarrow \mathbb{R}, X_T \rightarrow \mathbb{E}[u(X_T)]$.
at $X_T^*$ in the direction of any $X_T\in \mathcal{X}(x_0)$, i.e., compute that
$DF(X_T^*;X_T) = \lim_{\delta \rightarrow 0} \frac{\mathbb{E}[u(X_T^*+\delta X_T)]-\mathbb{E}[u(X_T^*)] }{\delta} = \mathbb{E}[u'(X_T^*)\cdot X_T]$ for all $X_T\in \mathcal{X}(x_0)$.
Thus, $E[u'(X_Tˆ*) \cdot X_T]\leq 0$ for all $X_T \in \mathcal{X}(x_0)$
Questions: What I don't like about my intuitive proof is the following:
1)$\mathcal{X}(x_0)$ is not a vector space as it violates scalar multiplication ($c\cdot X_T \notin \mathcal{X}(x_0)$ as it has a price bigger than $x_0$ for $c>1$). So, I am not sure whether the Gateaux-Derivative is well-defined? (I am new to the subject of Gateaux-Derivatives)
2)Due to the same reason as under 1), $X_T^*+\delta X_T$ is not in $\mathcal{X}(x_0)$,so technically the functional as I defined it above is not defined. Is there an alternative definition of the Gateaux-derivative that uses $(1-\delta)X_T^*+\delta X_T$ instead of $X_T^*+\delta X_T$ in the limit term? (because $(1-\delta)X_T^*+\delta X_T$ would be in $\mathcal{X}(x_0)$).
- Is the Gateaux-Differential (if it exists) in this case the Fréchet Derivative because $u$ is assumed to be continuous?
Any help or pointer towards suitable literature would be most appreciated!