The jumps of a process \(X : T \to \Omega \to E\) is the process \(\Delta X : T \to \Omega \to E\) defined by \((t, \omega ) \mapsto X_t(\omega ) - X_{t^-}(\omega )\) .

The jump part (or purely discontinuous part) of a process \(X : T \to \Omega \to E\) is the process \(X^d : T \to \Omega \to E\) defined by \((t, \omega ) \mapsto \sum _{0 {\lt} s \le t} \Delta X_s(\omega )\) .

The continuous part of a process \(X : T \to \Omega \to E\) is the process \(X^c : T \to \Omega \to E\) defined by \((t, \omega ) \mapsto X_t(\omega ) - X^d_t(\omega )\) .

The large jump part of a process \(X : T \to \Omega \to E\) at level \(\varepsilon \) is the process \(X^{d, \varepsilon } : T \to \Omega \to E\) defined by \((t, \omega ) \mapsto \sum _{0 {\lt} s \le t} \Delta X_s(\omega ) \mathbb {1}_{\{ \Vert \Delta X_s(\omega ) \Vert {\gt} \varepsilon \} }\) .

The (extended-real-valued) variation of a function \(f : T \to E\) on a set \(s\) inside a linear order is the supremum of \(\sum _i \mathrm{edist}(f(u_{i+1}, f(u_i)))\) over all finite increasing sequences \(u : N \to T\) in \(s\). We denote it by \(V_f(s)\) .

A function \(f : T \to E\) is of bounded variation on a set \(s\) if its variation \(V_f(s)\) is finite.

A function \(f : T \to E\) is of locally bounded variation on a set \(s\) if for every \(a, b\) in \(s\), the variation \(V_f(s \cap [a, b])\) is finite.

A process \(X : T \to \Omega \to E\) is of finite variation if it is right-continuous and for every \(\omega \in \Omega \), the path \(t \mapsto X_t(\omega )\) is of locally bounded variation on \(T\) .

We denote by \(\mathcal{V}\) the class of finite variation processes.

The variation of a process \(X : T \to \Omega \to E\) is the process \(V_X : T \to \Omega \to \mathbb {R}\) defined by \((t, \omega ) \mapsto V_{X(\omega )}([0,t])\) .

We say that a stochastic process \(X\) has integrable variation if its variation process \(V_X\) (Definition 13.11) is integrable (Definition 9.31).

We denote by \(\mathcal{A}\) the class of adapted processes with integrable variation.

Let \(T\) be a linear order with bottom element 0, on which we have a filtration \(\mathcal{F}\) satisfying the usual conditions. We say that a martingale \(M : T \to \Omega \to E\) is square integrable if it is càdlàg and \(\sup _{t \in T} \Vert M_t \Vert _{L^2} {\lt} \infty \) (Lean remark: use eLpNorm (M t) 2).

We denote by \(\mathcal{M}^2(E)\) or simply \(\mathcal{M}^2\) the space of equivalence classes with respect to indistinguishability of square integrable martingales \(T \to \Omega \to E\) .

We define a norm on \(\mathcal{M}^2\) by

We define an inner product on \(\mathcal{M}^2\) by

A process is locally square-integrable if it locally satisfies the square-integrable martingale property. We denote that class of processes by \(\mathcal{M}^2_{\mathrm{loc}}\) .

For \(M \in \mathcal{M}^2_{\mathrm{loc}}\) with càdlàg paths, the predictable quadratic variation of \(M\) is defined as the predictable part of the Doob-Meyer decomposition of the local submartingale \(\Vert M \Vert ^2\) . We denote it by \(\langle M \rangle \) .

For \(M, N \in \mathcal{M}^2_{\mathrm{loc}}\), the predictable covariation \(\langle M, N \rangle \) is a stochastic process defined by polarization of the predictable quadratic variation:

A process \(X\) is a semimartingale if it can be decomposed as \(X = M + A\), where \(M\) is a local martingale (\(M \in \mathcal{M}_{\mathrm{loc}}\)) and \(A\) is an adapted process with finite variation.

TODO. Denoted by \(X^c\).

Let \(X\) and \(Y\) be semimartingales. Their quadratic covariation \([X, Y]\) is defined as

The quadratic variation of a semimartingale \(X\) is defined as \([X] = [X, X]\) .

A semimartingale \(X\) is a special semimartingale if it can be decomposed as \(X = M + A\), where \(M\) is a local martingale and \(A\) is an adapted process with locally integrable variation.