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Semimartingale

Definition (Semimartingale)

A continuous semimartingale is a process (St)t≥0(S_{t})_{t\ge 0} s.t. St=S0+Mt+At∀t≥0S_{t}=S_{0}+M_{t}+A_{t}\quad\forall t\ge 0where

  1. S0S_{0} is F0\mathcal{F}_{0}-measurable
  2. (Mt)t≥0(M_{t})_{t\ge 0} is a continuous local martingale s.t. M0=0M_{0}=0
  3. (At)t≥0(A_{t})_{t\ge 0} is a continuous adapted process of Finite Variation s.t. A0=0A_{0}=0

Proposition (Semimartingale properties)

The space of semimartingales is closed under:

  1. Multiplication
  2. Absolute change of Measure
  3. Composition with C2C^{2} functions on R\mathbb{R}
  4. i.e. given (St)t≥0(S_{t})_{t\ge 0} semimartingale, ϕ∈C2(R;R)  ⟹  (ϕ(St))t≥0\phi\in C^{2}(\mathbb{R};\mathbb{R})\implies(\phi(S_{t}))_{t\ge 0} is a semimartingale

Linked from

Bichteler-Dellacherie Theorem

Semimartingale Integral

Semimartingale

Solution to SDE