Let $\Omega$ be a set of elements $\omega\in\Omega$ and let $A(\omega)$ be a random variable. Consider the function $$ K(\omega,A(\omega),P) $$ where $P>0$. Assume that $K$ is strictly positive, and nondecreasing and continuously differentiable in $P$.
Consider the mapping $T(P)$ defined as $$ T(P) = K(\omega_P,A(\omega_P),P) $$ where $\omega_P=\arg\min_\omega E[K(\omega,A(\omega),P)]$.
I am looking for a way to show that $T$ has a (unique?) fixed point $P^*=T(P^*)$ (note that $P^*$ will be a random variable in general). To do so, my plan is to show that $T$ is a contraction: there is an $0\leq r<1$ such that, for any $P$ and $P'$, we have $$ ||T(P)-T(P')||\leq r ||P-P'||. $$
I'm happy to assume additional restrictions on $K$ and $A$ to get to that result if needed. Any help would be much appreciated!
What I've tried so far
I haven't been able to make much progress. It seems that using the norm $||x||=E[|x|]$ might be useful. For instance, we can write \begin{align} ||T(P)-T(P')|| &= E\left[\left|K(\omega_P,A(\omega_P),P)-K(\omega_{P'},A(\omega_{P'}),P')\right|\right] \end{align} If we make the (restrictive) assumption that $K(\omega_P,A(\omega_P),P)\geq K(\omega_{P'},A(\omega_{P'}),P')$ we can write $$ ||T(P)-T(P')|| = \min_\omega E\left[K(\omega,A(\omega),P)\right]-\min_\omega E\left[K(\omega,A(\omega),P')\right]$$ But I'm not sure how to make $E[|P-P'|]$ appear in there.
It might also be possible to use Tarski's fixed point theorem but I haven't made much progress on that front either.
Use the envelope theorem on $T$ with respect to the parameter $p$, letting $P$ be the convex space of $p$'s, and $\mathcal{C}$ a smooth curve from $p'$ to $p$: \begin{eqnarray} |T(p)-T(p')| &=& \left| \int_{p'}^p \nabla_p K(w(z),A(z),z) d\mathcal{C} \right|\\ &\le& ||p-p'|| \sup_{z \in \mathcal{C}} \left|\left| \nabla_p K(w(z),A(z),z)\right|\right|\\ &\le& ||p-p'|| \sup_{z \in P} \left|\left| \nabla_p K(w(z),A(z),z)\right|\right| \end{eqnarray} And assume (or provide sufficient conditions) that $$ r = \sup_{z \in P} \left|\left| \nabla_p K(w(z),A(z),z)\right|\right| <1. $$ Does that work? See Milgrom and Segal, "Envelope Theorems for Arbitrary Choice Sets."