# Complete Financial Market: Integrability condition for Contingent Claims

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:

1. 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)

2. 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!

• @Quantuple Thanks for your comment. I added two references. – Mark Jan 29 '18 at 19:19
• Let's hope a more technical guy can help! Intuition tells me the second formulation guarantees that $H$ can be written as a stochastic Itô integral $H_t = \Bbb{E}_0[H_t] + \int_0^t \phi(s)dX_s$, where $\phi$ is a progressively measurable process and $X$ a semi-martingale adapted to $\mathcal{F}_t$. This is at the heart of arbitrage-free pricing theory (cf. concepts of attainable claims and admissible replicating strategies) and can be used to show that the price of a contingent claim can be written as an expectation under some probability measure. – Quantuple Jan 30 '18 at 11:04
• Still IMO, the first one (which I've hardly met in my readings as far as I can remember) is merely a sufficient condition for the latter expectation to exist but is not enough to develop the underpinning pricing theory and notably show that the price can be obtained as an expectation to begin with. – Quantuple Jan 30 '18 at 11:04