I am trying to calculate the maximum value of the following expected value :
$$\mathbb{E}_{\mathbb{Q}_v}\big[(K-(1+R)s_0)^+\big]$$
Now, the function $x^+$ is defined as $x^+ := \max\{x,0\}$. Thus, I need to find the maximum for :
$$\mathbb{E}_{\mathbb{Q}_v}\big[\max\{K-(1+R)s_0,0\}\big]$$
Note that $K>0$ and $s_0 > 0$ are just two constants, while $R$ is a random variable depending on the probability measure $\mathbb{Q}$. The probability measure $\mathbb{Q}$ is depended on $v$.
To find this maximum, is it enough finding the maximum for the expression $K-(1+R)s_0$ ? Considering that $R$ is a random variable, what would essentialy change ? Would I need to implement the expected value somehow ?
You are looking at
$$E(Z),\;\;\; Z=\max\{X,0\}$$
Then
$$Z= \begin{cases} X & X \geq 0 \\ 0 & X < 0 \end {cases}$$
and so
$$E(Z) = E(X\mid X\geq 0)\cdot \Pr(X\geq 0) + 0\cdot \Pr(X< 0) = E(X\mid X\geq 0)\cdot \Pr(X\geq0)$$
which in your case translates into
$$\Big[ K-s_0[1+E(R \mid R \leq (K/s_0) - 1)]\Big]\cdot \Pr(K-(1+R)s_0\geq0)$$
Your expression contains constants and the truncated expected value of a random variable. It is a number, not a function. In order to contemplate the concept of the maximum, you should either treat the constants as decision variables, or have a functional expression for $R = h(...)$ in which some components are considered as decision variables.