For the customer, the utility premium $H^{\star}$ is defined as the premium making the homeowner indifferent to insure or not, e.g. $H^{\star}$ solves $$u(x-H^{\star})=\mathbb{E}u(x-X).$$
How does it look for the insurer (insurance company)? Is it just that the insurance company offers insurance if $H>\mathbb{E}[X]$? Or how does one calculate the utility premium for the insurer?