Schauder basis of the Hardy space of semi-martingales

Question

Fix $p\in [1,2]$, a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$, and let $\mathcal{H}_{\mathscr{S}}^p$ denote the space of semimartingales $X$ such that the norm $$ \|X\|_{\mathcal{H}_{\mathscr{S}}^p}:= \inf\big\{ \mathbb{E}\big[\big( [M]_{\infty}^{1/2} + \int_0^{\infty} |dA|_s \big)^p\big]^{1/p} \big\} $$ is finite, where the infimum is taken over all decompositions of $X$ into a local martingale with $M_0=0$ and a process $A$ of finite variation with $A_0=X_0=\Delta A_0$. It is known (Lemma 16.2.25 of this book) that every semi-martingale in ${\mathcal{H}_{\mathscr{S}}^p}$ is special and the above norm can be written as $$ \|X\|_{\mathcal{H}_{\mathscr{S}}^p} = \mathbb{E}[([\tilde{M}]^{1/2})^p]^{1/p} + \mathbb{E}[(\int_0^{\infty} |d\tilde{A}_s|)^p]^{1/p}, $$ where $X=\tilde{M}+\tilde{A}$ is the canonical decomposition of $X$.

We call $\mathcal{H}_{\mathscr{S}}^p$ the semi-martingale-hardy space.

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Clearly, $\mathcal{H}_{\mathscr{S}}^p$ is a separable Banach space and it is a Hilbert space when $p=2$.

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Question: What is an explicit Schauder basis of this space?

Answer 1

There can be many possibilities. Let me focus on the martingale part. For each different Martingale one can construct a basis using the difference functions eg see Theorem 4.2.6. in Schauder Bases and the Factorization Property where they also reference chapter 7 in "Introduction to Banach spaces: analysis and probability".

So there can be many possibilities. In terms of standard construction, you can push the Schauder basis from here Proof of existence of Schauder basis for $L^p(\Omega)$? by mapping it to the probability space $\Omega$.

For simplicity I focused only the martingale part since it is a subspace of the above. By Burkholder–Davis–Gundy inequality, the norm in terms of quadratic variation is comparable to the Lp of the sup of the martingale.