1.2 Equivalence of Certain Martingales

Definition 1.2.1 (Random Differential Operator). Let $(\Omega, \cf, \bp)$ be a filtered probability space, $s \ge 0$, and $k \in \natp$. For each multi-index $\alpha \in \nat_{0}^{d}$ with $\abs{\alpha}\le k$, let $a_{\alpha}: [s, \infty) \times \Omega \to \real$ be measurable function. For each $f \in C^{k}(\real^{d})$ and $t \ge s$, let

then $L = \sum_{|\alpha| \le k}a_{\alpha} \partial^{\alpha}$ is a random differential operator of order $k$ with coefficients $\bracs{a_\alpha}_{|\alpha| \le k}$.

Definition 1.2.2 (Progressively Measurable Random Differential Operator). Let $(\Omega, \cf, \bp)$ be a filtered probability space and $L = \sum_{|\alpha| \le k}a_{\alpha}$ be a random differential operator on $\Omega$, then $L$ is progressively measurable with respect to $\bracs{\mathcal{F}_t}$ if for each $\alpha \in \nat_{0}$ with $|\alpha| \le k$, $a_{\alpha}: [s, \infty) \times \Omega \to \real$ is progressively measurable with respect to $\bracs{\mathcal{F}_t}$.

Lemma 1.2.3. Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $s \ge 0$, $k \in \natp$, $L = \sum_{|\alpha| \le k}a_{\alpha} \partial^{\alpha}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, $f \in C^{k}(\real^{d})$, and $\bracs{X_t|t \ge 0}$ be a $\real^{d}$-valued progressively measurable process, then

is a progressively measurable process.

Theorem 1.2.4 (Integration by Parts). Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $\bracs{X_t}$ be a $\bracs{\mathcal{F}_t}$-martingale, and $\phi: [0, \infty) \times \Omega \to \complex$ be a continuous, progressively measurable function. If:

1. For every $\omega \in \Omega$, $\phi(\cdot, \omega) \in BV_{\text{loc}}([0, \infty))$.

2. For all $t \ge 0$,

   \[\ev\braks{\sup_{0 \le s \le t}|X_s|(|\phi|(s) + |\phi|(t))}< \infty\]

then the process

is a $\bracs{\mathcal{F}_t}$-martingale.

Theorem 1.2.5 (Equivalence of Martingales). Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space,

be a second-order $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, and $\bracs{X_t|t \ge 0}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable process with continuous sample paths, then the following are equivalent:

1. For every $f \in C_{c}^{\infty}(\real^{d})$, the process

   \[E_{t} = f(X_{t}) - \int_{0}^{t} (L_{r}f)(X_{r})dr\]

   is a $\bracs{\mathcal{F}_t}$-martingale.

2. For every $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$,

   \[H_{t} = f(t, X_{t}) - \int_{0}^{t} [(\partial_{r} + L_{r})f](r, X_{r})dr\]

   is a $\bracs{\mathcal{F}_t}$-martingale.

3. For each uniformly positive $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$,

   \[X_{t} = f(t, X_{t})\exp\braks{-\int_0^t \frac{(\partial_{r} + L_{r})f}{f}(r, X_r)dr}\]

   is a $\bracs{\mathcal{F}_t}$-martingale.

4. For each $x \in \real^{d}$ and $g \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$, the process

   \begin{align*}Y_{t}^{x, g}&= \exp\braks{\dpn{x, X_t - X_0 - \int_0^t b(r)dr}{\real^d} + g(t, X_t)}\\&\cdot \exp\braks{-\frac{1}{2}\int_0^t \dpn{x + Dg, A(r)(x + Dg)}{\real^d}(r, X_r)dr}\\&\cdot \exp\braks{\int_0^t [(\partial_r + L_r)g](r, X_r)}dr\end{align*}

   is a $\bracs{\mathcal{F}_t}$-martingale.

5. For each $\theta \in \real^{d}$,