Consider a market model with risk-free rate $r > -1$ and one risky asset that is such that $\pi^{1} > 0 $. Assume that the distribution of $S^1$ has a strictly positive density function $f:(0,\inf) \to (0,\inf)$ that is, $P(S^1 \leq x) = $ $\int_{0}^{x} f(y) dy$ for $x>0$. Find an equivalent risk-neutral measure $P^*$
I am currently struggling with this exercise. I know that for a risk neutral measure it needs to hold that:
$\pi^i = E_{P^*}[\frac{S^i}{1+r}] $
and
$P \approx P^{*} \leftarrow (P^*(A) = 0$ iff $P(A) = 0$ $A\in F $ on $(\Omega,F)$
so if I start out I go with:
$0<\pi^1 = E_{P^*}[\frac{S^i}{1+r}] = \frac{1}{1+r} E_{P^*}[S^i] = \frac{1}{1+r} E_{P}[\frac{dP^*}{dP}S^i] $ while $\frac{dP^*}{dP}$ is the density function $f$
I guess that I can write: $\frac{1}{1+r} E_{P}[\frac{dP^*}{dP}S^i] = \frac{1}{1+r} (x*P(S^1 \leq x)) $ but I am not sure about that.
How can I continue from here?