Continous Semimartingales
Defintion (Radon Nikodym Derivative): Let $\mu,\nu$ be measures such that $\mu « \nu$ i.e. $\mu$ is absolutely continous w.r.t. $\nu$. The Radon-Nikodym derivate is given as : \(\frac{d\mu}{d\nu}\)
A signed measure is the difference of two finite positive measures.
Definition (Finite Variation): Let $T \ge 0$ . A continous function $a : [0,T] \rightarrow \R$ such that $a(0) = 0$ is said to have a finite variation if there exists a signed measure $\mu$ on $[0,T]$ such that $a(t) = \mu([0,t])$ for every $t \in [0,T]$.
For $\mu$, the total variation is given as $\abs{\mu} = \mu_{+} + \mu_{-} $.
The measure $\abs{\mu}$ is called the total variation of $a$.
Let $f : [0,T] \rightarrow \R$ be a measurable function such that $ \int_{[0,T]} \abs{f(s)}\abs{\mu}(ds) < \infty$. We set: \(\int_{0}^{T} f(s)da(s) = \int_{[0,T]} f(s)\mu(ds)\) \(\int_{0}^{T} f(s)\abs{da(s)} = \int_{[0,T]} f(s)\abs{\mu}(ds)\)
Proposition 4.2 . For every $t \in (0,T]$, \(\int_{0}^T \abs{da(s)} = \{ \sum_{i=1}^p (\abs{a(t_i)} - a(t_{i-1})\}\)
Lemma 4.3 If $f : [0,T] \rightarrow \R$ is a continuous function, and if $0 = t_0^n < t_1^n < \cdots $ is a sequence of subdivisions of $[0,T]$ whose mesh values tends to $0$, we have \(\int_{0}^T f(s)da(s) = \lim_{n \rightarrow \infty} \sum_{i=1}^p f(t_{i-1}^n)\)
Definition 4.4 An adapted process $A = (A_t){t \ge 0}$ is called a finite variation process if all its sample paths are finite variation functions on $\R+$. If in addition the sample paths are nondecreasing functions, the process $A$ is called an increasing process.
If $A$ is a finite variation process, the process: \(V_t = \int_{0}^t \abs{dA_s}\) is an increasing process.
Remark: For each $\omega$ $V_t(\omega)$ is a sample path, and then I know what above means. \(V_t(\omega) = \int_0^t \abs{dA_s(\omega)}\)
Proposition 4.5 Let $A$ be a finite variation process, and let $H$ be a progressive process such that \(\forall t \ge 0, \forall \omega \in \Omega, \int_{0}^t \abs{H_s(\omega)} \abs{dA_s(\omega)} < infty.\) Then the process $H \cdot A = ((H\cdot A)t){t\ge 0}$ defined by: \((H \cdot A)_t = \int_{0}^t H_s dA_s\) is also a finite variation proces.
A good example is when $A_t = t$.
Definition An adapted process $M = (M_t){t \ge 0}$ with continuous sample paths and such that $M_0 = 0$ a.s. is called a continuous local martingale if there exists a nondecreasing sequence $(T_n){n \ge 0}$ of stopping times such that $T_n \uparrow \infty$ and for every $n$ the stopped process $M^{T_n}$ is a uniformly integrable martingale.
Some examples of continous local martingales that are not (true) martingales.
Proposition 4.7:
- A nonnegative continuous local martingale $M$ such that $M_0 \in L^1$ is a supermartingale.
- A continuous local martingale $M$ such that there exists a random variable $Z \in L^1$ with $\abs{M_t} \le Z$ for every $t \ge 0$ is a uniformly integrable martingale.
Theorem 4.9 Let $M = (M_t){t\ge 0}$ be a continuous local martingale. There exists an increasing process denoted by $(\langle M,M\rangle_t){t\ge 0}$ which is unique upto indistinguishability such that $M_t^2 - \langle M,M \rangle_t$ is a continous local martingale. Furthermore, \(\langle M,M \rangle_t = \lim_{n \rightarrow \infty} \sum_{i=1}^{p_n} (M_{t_i^n} - M_{t_{i-1}^n})^2\) in probability. For Brownian Motion \(\langle B,B \rangle_t = t\)