3.4 The Martingale Formulation

Definition 3.4.1 (Martingale Problem).label Let $a: [0, \infty) \times C([0, \infty); \real^{n}) \to L(\real^{n}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{n}) \to \real^{n}$ be previsible path functionals. For each $u \in C_{c}^{\infty}(\real^{n})$, let

then for any $y \in \real^{n}$, filtered probability space $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$, and $X: \Omega \to C([0, \infty); \real^{n})$, $X$ is a solution to the martingale problem for $(a, b)$ starting at $y$ if:

1. (1)

   $X_{0} = y$ almost surely.

2. (2)

   For each $f \in C_{c}^{\infty}(\real^{n})$, the process

   \[C_{t}^{f} = f(X_{t}) - f(X_{0}) - \int_{0}^{t} Lf(s, X_{s})ds\]

   
   

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

If the distribution of $X$ on $C([0, \infty); \real^{n})$ is the unique distribution satisfying the above, then the solution for the martingale problem is unique. If for each $y \in \real^{n}$, such a solution exists, then the martingale problem is well posed.

Theorem 3.4.2 (Equivalence of Formulations).label Let $(\Omega, \bracsn{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-Brownian motion, $\sigma: [0, \infty) \times C([0, \infty); \real^{n}) \to L(\real^{n}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{n}) \to \real^{n}$ be bounded measurable functions such that $\sigma_{t}$ is invertible for all $t \ge 0$.

Let $y \in \real^{n}$ and $X: \Omega \to C([0, \infty); \real^{n})$ be a solution to the martingale problem for $(\sigma^{*}\sigma, b)$ starting at $y$, then there exists a weak solution of the SDE

starting at $y$ whose distribution is the same as $X$.

Theorem 3.4.3.label Let $a: \real^{n} \to L(\real^{n}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be bounded measurable functions such that the martingale problem for $(a, b)$ is well-posed, that is, let

then for each $y \in \real^{n}$, there exists a unique probability measure $\bp^{y}$ on $C([0, \infty); \real^{n})$ such that:

1. (1)

   $\bp^{y}(x_{0} = y) = 1$.

2. (2)

   For each $f \in C_{c}^{\infty}$,

   \[C_{t}^{f} = f(\pi_{t}) - f(x_{0}) - \int_{0}^{t} Lf(x_{s})ds\]

   
   

   is a $\bp^{y}$-martingale.

then

1. (1)

   $\bracs{x_t|t \ge 0}$ is a time-homogeneous strong Markov process with respect to $\bp^{y}$.

2. (2)

   If $a = \sigma \sigma^{*}$ and $(\sigma, b)$ satisfy the Lipschitz condition, then the generator of the above process is $L$.