Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is some terminal time horizon and $\mathbb{Q}$ is the (unique) risk-neutral measure. Moreover, let $B=\{B_{t}\}_{t\,\in\,[0,T]}$ denote the (risk-less) bank account.
I have seen two definitions for a contingent claim:
A contingent claim with payoff at time $t\in[0,T]$ is an $\mathcal{F}_{t}$-measureable random variable $H$ with $\frac{H}{B_{t}}\in\mathcal{L}^{1}$ (see e.g. here)
A contingent claim with payoff at time $t\in[0,T]$ is an $\mathcal{F}_{t}$-measureable random variable $H$ with $H\in\mathcal{L}^{2}$ (see e.g. here, page 4).
I understand that integrability is required for the existence of the (conditional) expectation under the risk-neutral measure, but why do some authors require square-integrability of $H$ (and not even $\frac{H}{B_{t}}$)?
To add some context: the first definition is usually given in texts that consider a Brownian financial market model. The second definition is given in more general semi-martingale models. I guess the reason for the two different definitions is a technical one, but I don't see where this is needed.
Thanks for your time and effort!