Semimartingale

Alternative definition

First, the simple predictable processes are defined to be linear combinations of processes of the form H_t = A1_{{t > T}} for stopping times T and F_T -measurable random variables A. The integral H ⋅ X for any such simple predictable process H and real valued process X is

${\displaystyle H\cdot X_t:=1_{\{t>T\}}A(X_t-X_T).}$

This is extended to all simple predictable processes by the linearity of H ⋅ X in H.

A real valued process X is a semimartingale if it is càdlàg, adapted, and for every t ≥ 0,

${\displaystyle \left\{H\cdot X_t:H\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{d}|H|\le 1\right\}}$

is bounded in probability. The Bichteler–Dellacherie Theorem states that these two definitions are equivalent (Protter 2004, p. 144).

Examples

• Adapted and continuously differentiable processes are continuous, locally finite-variation processes, and hence semimartingales.
• Brownian motion is a semimartingale.
• All càdlàg martingales, submartingales and supermartingales are semimartingales.
• Itō processes, which satisfy a stochastic differential equation of the form dX = σdW + μdt are semimartingales. Here, W is a Brownian motion and σ, μ are adapted processes.
• Every Lévy process is a semimartingale.

Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case.

• Fractional Brownian motion with Hurst parameter H ≠ 1/2 is not a semimartingale.

Properties

• The semimartingales form the largest class of processes for which the Itō integral can be defined.
• Linear combinations of semimartingales are semimartingales.
• Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the Itō integral.
• The quadratic variation exists for every semimartingale.
• The class of semimartingales is closed under optional stopping, localization, change of time and absolutely continuous change of probability measure (see Girsanov's Theorem).
• If X is an R^m valued semimartingale and f is a twice continuously differentiable function from R^m to Rⁿ, then f(X) is a semimartingale. This is a consequence of Itō's lemma.
• The property of being a semimartingale is preserved under shrinking the filtration. More precisely, if X is a semimartingale with respect to the filtration F_t, and is adapted with respect to the subfiltration G_t, then X is a G_t-semimartingale.
• (Jacod's Countable Expansion) The property of being a semimartingale is preserved under enlarging the filtration by a countable set of disjoint sets. Suppose that F_t is a filtration, and G_t is the filtration generated by F_t and a countable set of disjoint measurable sets. Then, every F_t-semimartingale is also a G_t-semimartingale. (Protter 2004, p. 53)

Semimartingale decompositions

By definition, every semimartingale is a sum of a local martingale and a finite-variation process. However, this decomposition is not unique.

Continuous semimartingales

A continuous semimartingale uniquely decomposes as X = M + A where M is a continuous local martingale and A is a continuous finite-variation process starting at zero. (Rogers & Williams 1987, p. 358)

For example, if X is an Itō process satisfying the stochastic differential equation dX_t = σ_t dW_t + b_t dt, then

${\displaystyle M_t=X_0+{\int }_0^t{\sigma }_{s}d{W}_s,A_t={\int }_0^t{b}_{s}ds.}$

Special semimartingales

A special semimartingale is a real valued process ${\displaystyle X}$ with the decomposition ${\displaystyle X=M^X+B^X}$, where ${\displaystyle M^X}$ is a local martingale and ${\displaystyle B^X}$ is a predictable finite-variation process starting at zero. If this decomposition exists, then it is unique up to a P-null set.

Every special semimartingale is a semimartingale. Conversely, a semimartingale is a special semimartingale if and only if the process X_t^* ≡ sup_{s ≤ t} |X_s| is locally integrable (Protter 2004, p. 130).

For example, every continuous semimartingale is a special semimartingale, in which case M and A are both continuous processes.

Multiplicative decompositions

Recall that ${\displaystyle \mathcal{E}(X)}$ denotes the stochastic exponential of semimartingale ${\displaystyle X}$. If ${\displaystyle X}$ is a special semimartingale such that ${\displaystyle \mathrm{\Delta }{B}^X\ne -1}$, then ${\displaystyle \mathcal{E}(B^X)\ne 0}$ and ${\displaystyle \mathcal{E}(X)/\mathcal{E}(B^X)=\mathcal{E}\left({\int }_0^{\cdot }\frac{M_u^X}{1+\mathrm{\Delta }{B}_u^X}\right)}$ is a local martingale. Process ${\displaystyle \mathcal{E}(B^X)}$ is called the multiplicative compensator of ${\displaystyle \mathcal{E}(X)}$ and the identity ${\displaystyle \mathcal{E}(X)=\mathcal{E}\left({\int }_0^{\cdot }\frac{M_u^X}{1+\mathrm{\Delta }{B}_u^X}\right)\mathcal{E}(B^X)}$ the multiplicative decomposition of ${\displaystyle \mathcal{E}(X)}$.

Purely discontinuous semimartingales / quadratic pure-jump semimartingales

A semimartingale is called purely discontinuous (Kallenberg 2002) if its quadratic variation [X] is a finite-variation pure-jump process, i.e.,

${\displaystyle [X{]}_t=\sum _{s\le t}(\mathrm{\Delta }{X}_s{)}^2}$.

By this definition, time is a purely discontinuous semimartingale even though it exhibits no jumps at all. The alternative (and preferred) terminology quadratic pure-jump semimartingale for a purely discontinuous semimartingale (Protter 2004, p. 71) is motivated by the fact that the quadratic variation of a purely discontinuous semimartingale is a pure jump process. Every finite-variation semimartingale is a quadratic pure-jump semimartingale. An adapted continuous process is a quadratic pure-jump semimartingale if and only if it is of finite variation.

For every semimartingale X there is a unique continuous local martingale ${\displaystyle X^c}$ starting at zero such that ${\displaystyle X-X^c}$ is a quadratic pure-jump semimartingale (He, Wang & Yan 1992, p. 209; Kallenberg 2002, p. 527). The local martingale ${\displaystyle X^c}$ is called the continuous martingale part of X.

Observe that ${\displaystyle X^c}$ is measure-specific. If ${\displaystyle P}$ and ${\displaystyle Q}$ are two equivalent measures then ${\displaystyle X^c(P)}$ is typically different from ${\displaystyle X^c(Q)}$, while both ${\displaystyle X-X^c(P)}$ and ${\displaystyle X-X^c(Q)}$ are quadratic pure-jump semimartingales. By Girsanov's theorem ${\displaystyle X^c(P)-X^c(Q)}$ is a continuous finite-variation process, yielding ${\displaystyle [X^c(P)]=[X^c(Q)]=[X]-\sum _{s\le \cdot }(\mathrm{\Delta }{X}_s{)}^2}$.

Continuous-time and discrete-time components of a semimartingale

Every semimartingale ${\displaystyle X}$ has a unique decomposition $${\displaystyle X=X_0+X^{\mathrm{q}\mathrm{c}}+X^{\mathrm{d}\mathrm{p}},}$$where ${\displaystyle X_0^{\mathrm{q}\mathrm{c}}=X_0^{\mathrm{d}\mathrm{p}}=0}$, the ${\displaystyle X^{\mathrm{q}\mathrm{c}}}$ component does not jump at predictable times, and the ${\displaystyle X^{\mathrm{d}\mathrm{p}}}$ component is equal to the sum of its jumps at predictable times in the semimartingale topology. One then has ${\displaystyle [X^{\mathrm{q}\mathrm{c}},X^{\mathrm{d}\mathrm{p}}]=0}$. Typical examples of the "qc" component are Itô process and Lévy process. The "dp" component is often taken to be a Markov chain but in general the predictable jump times may not be isolated points; for example, in principle ${\displaystyle X^{\mathrm{d}\mathrm{p}}}$ may jump at every rational time. Observe also that ${\displaystyle X^{\mathrm{d}\mathrm{p}}}$ is not necessarily of finite variation, even though it is equal to the sum of its jumps (in the semimartingale topology). For example, on the time interval ${\displaystyle [0,\mathrm{\infty })}$ take ${\displaystyle X^{\mathrm{d}\mathrm{p}}}$ to have independent increments, with jumps at times ${\displaystyle \{{\tau }_n=2-1/n{\}}_{n\in \mathbb{N}}}$ taking values ${\displaystyle \pm 1/n}$ with equal probability.

Semimartingales on a manifold

The concept of semimartingales, and the associated theory of stochastic calculus, extends to processes taking values in a differentiable manifold. A process X on the manifold M is a semimartingale if f(X) is a semimartingale for every smooth function f from M to R. (Rogers & Williams 1987, p. 24) Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.