3.1 Definitions

Definition 3.1.1 (Previsible $\sigma$-Algebra).label Let $(\Omega, \cf)$ be a measurable space and $\bracs{\cf_t|t \ge 0}$ be a filtration on $\Omega$, then the previsible $\sigma$-algebra $\mathscr{J}_{\omega}$ on $(0, \infty) \times \Omega$ associated with $\bracs{\mathcal{F}_t}$ is the $\sigma$-algebra generated by

In other words, it is the smallest $\sigma$-algebra on $(0, \infty) \times \Omega$ on which every $\bracs{\mathcal{F}_t}$-adapted process with left-continuous sample paths is measurable.

Definition 3.1.2 (Previsible Path Functional).label Let $C([0, \infty); \real^{d})$ be the space of $\real^{d}$-valued continuous functions on $[0, \infty)$, equipped with the topology of uniform convergence. For each $t \ge 0$, let

and $\mathscr{J}_{C([0, \infty); \real^d)}$ be the previsible $\sigma$-algebra on $(0, \infty) \times C([0, \infty); \real^{d})$, then a previsible path functional is a $\mathscr{J}$-measurable mapping on $(0, \infty) \times C([0, \infty); \real^{d})$.

Definition 3.1.3 (Augmentation).label Let $(\Omega, \bracs{\cf_t}, \bp)$ be a filtered probability space and $\mathcal{N}$ be the collection of all $\bp$-null sets in $\cf = \sigma(\bracs{\cf_t|t \ge 0})$. For each $t \ge 0$, let $\ol{\cf}_{t} = \sigma(\cf_{t} \cup \mathcal{N})$, then the filtration $\bracsn{\ol{\cf}_t|t \ge 0}$ is the $\bp$-augmentation of $\bracs{\mathcal{F}_t}$.

Lemma 3.1.4.label Let $(\Omega, \bracs{\cf_t})$ be a filtered space and $X: \Omega \to C([0, \infty); \real^{d})$ be a $\bracs{\mathcal{F}_t}$-adapted process with continuous sample paths, then

1. (1)

   $X$ is $(\mathscr{J}_{\Omega}, \mathscr{J}_{C([0, \infty); \real^d)})$-measurable.

2. (2)

   For any previsible path functional $\alpha: (0, \infty) \times C([0, \infty); \real^{d})$, $\alpha(t, X)$ is $\bracs{\mathcal{F}_t}$-previsible.

Definition 3.1.5 (Diffusion Type SDE).label Let $\sigma: \real^{d} \to L(\real^{d}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be measurable functions, then a SDE of diffusion type is the equation

under the constraint

for all $t > 0$, where

• $B$ is a standard Brownian motion on a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$.

• $\xi$ is a $\cf_{0}$-measurable random variable.

Definition 3.1.6 (Pathwise Uniqueness).label Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals, then the SDE

has pathwise uniqueness if given

• A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,

• $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,

• Continuous $\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^{d})$ satisfying Equation 3.2 and Equation 3.1,

then $X = Y$ almost surely.

Definition 3.1.7 (Pathwise Exact).label Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals, then the SDE

is pathwise exact if given

• A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,

• $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,

• $\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^{d})$ satisfying Equation 3.2 and Equation 3.1,

then for every $t \ge 0$, $X_{t} = Y_{t}$ almost surely.

Definition 3.1.8 (Strong Solution).label Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals, then a mapping

is a strong solution to the SDE

if

1. (1)

   For each $t \ge 0$, let

   \[\mathcal{F}_{t} = \sigma(\bracs{\pi_s: C([0, \infty); \real^d) \to \real^d|0 \le s \le t}) \quad \mathcal{G}_{t} = \sigma(\bracs{\pi_s: C([0, \infty); \real^n) \to \real^n|0 \le s \le t})\]

   
   

   Let $\wien^{d}$ and $\wien^{n}$ be the classical Wiener measure on $C([0, \infty); \real^{d})$ and $C([0, \infty); \real^{n})$, respectively, and $\bracsn{\ol{\mathcal{G}}_t|t \ge 0}$ be the $\wien^{n}$-augmentation of $\bracs{\mathcal{G}_t|t \ge 0}$, then

   \[F^{-1}(\mathcal{F}_{t}) \subset \mathcal{B}(\real^{n}) \times \overline{\mathcal{G}_t}\]

   
   

   for all $t \ge 0$.

2. (2)

   For any filtered probability space $(\Omega, \bracs{\mathcal{H}_t}, \bp)$, random variable $\xi: \Omega \to \real^{n}$, and $\bracs{\mathcal{H}_t}$-Brownian motion $B$, the process $X = F(\xi, B)$ solves Equation 3.3.