Choose a parameter so that $\sup_f\operatorname E_\mu\left[\left|\int_0^\tau f(X_t)\:{\rm d}t-\tau\pi f\right|^2\right]$ is minimized

Question

Let

• $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
• $\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ be equipped with the supremum norm;
• $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
• $A$ denote the generator$^1$ of $(\kappa_t)_{t\ge0}$ and $A^\ast$ denote the bounded linear operator on $\mathcal L^1(\lambda)$ with $$\int(Af)g\:{\rm d}\lambda=\int fA^\ast g\:{\rm d}\lambda\tag1$$ for all $f\in\mathcal E_b$ and $g\in\mathcal L^1(\lambda)$;
• $(\Omega,\mathcal A)$ be a measurable space;
• $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
• $(X_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued $(\mathcal F_t)_{t\ge0}$-progressive process on $(\Omega,\mathcal A)$;
• $\operatorname P_x$ be a probability measure on $(\Omega,\mathcal A)$ with $$\operatorname E_x\left[f(X_{s+t})\mid\mathcal F_s\right]=(\kappa_tf)(X_s)\tag1$$ for all $f\in\mathcal E_b$ and $s,t\ge0$ for $x\in E$;
• $\mu$ be a probability measure on $(E,\mathcal E)$ with density $u:E\to[0,\infty)$;
• $\pi$ be a probability measure on $(E,\mathcal E)$ with density $p:E\to(0,\infty)$ and $$c:=\frac{c_0u+A^\ast p}p$$ for some $c_0>0$;
• $$A_t:=\int_0^tc(X_s)\:{\rm d}s$$ and $$M_t:=e^{-A_t}$$ for $t\ge0$;
• $\xi$ be a real-valued random variable on $(\Omega,\mathcal A)$ exponentially distributed with respect to $\operatorname P_x$ for all $x\in E$ and $$\tau:=\inf\left\{t\ge0:A_t\ge\xi\right\}.$$

Question: Let $$Z(f):=\int_0^\tau f(X_t)\:{\rm d}t$$ for $f\in\mathcal E_b$. How should we choose $c_0$ such that $$\sup_{\substack{f\in\mathcal E_b\\f\ge0}}\operatorname E_\mu\left[\left|Z(f)-\tau\pi f\right|^2\right]\tag2$$ is minimized?

We can show that $$\frac{\rm d}{{\rm d}t}\operatorname E_\pi\left[f(X_t);t<\tau\right]=-c_0\operatorname E_\mu\left[f(X_t);t<\tau\right]\tag3$$ for all $t\ge0$ and hence $$\operatorname E_\mu\left[Z(f)\right]=\int_0^\infty\operatorname E_\mu\left[f(X_t);t<\tau\right]\:{\rm d}t=\frac{\pi f}{c_0}\tag4$$ for all $f\in\mathcal E_b$. In particular, $$\operatorname E_\mu\left[\tau\right]=\frac1{c_0}\tag5.$$

Idea: Maybe we can also obtain an expression for $Z(f)^2$ and obtain a bound for $(2)$ that way?

Possible simplifications: If the problem is too hard in general, I'm willing to assume one or all of the following simplifications:

1. $\pi$ is $(\kappa_t)_{t\ge0}$-invariant and hence $$A^\ast p=0\tag6.$$
2. $$\kappa_t=e^{t A}\;\;\;\text{for all }t\ge0$$ and $$A=\alpha\left(\kappa-\operatorname{id}_{\mathcal E_b}\right)\tag7$$ for some Markov kernel $\kappa$ on $(E,\mathcal E)$ and $\alpha\in\mathcal E_b$ with $\alpha\ge0$ and $\pi$ is $\kappa$-invariant.

In the setting of (2.), we may also ask how to choose $c_0$ and $\alpha$ (possibly constant) together such that $(2)$ is minimized.

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$^1$ $(\kappa_t)_{t\ge0}$ is considered as a contraction semigroup on $\mathcal E_b$.