The following is from Karakas and Shreve's Brownian Motion and Stochastic Calculus ch5 pp 364-365.
We define:
$$ X_s^{(t, x)}=x+\int_t^s b\left(\theta, X_\theta^{(t, x)}\right) d \theta+\int_t^s \sigma\left(\theta, X_\theta^{(t, x)}\right) d W_\theta ; \quad t \leq s<\infty $$
with the following conditions on the coefficients of $X$
(7.2) $\left\{\begin{array}{l}\text { the coefficients } b_i(t, x), \sigma_{i j}(t, x):[0, \infty) \times \mathbb{R}^d \rightarrow \mathbb{R} \text { are } \\ \text { continuous and satisfy the linear growth condition }(2.13) ;\end{array}\right.$
(7.3) $\left\{\begin{array}{l}\text { the equation }(7.1) \text { has a weak solution }\left(X^{(t, x)}, W\right) \\ (\Omega, \mathscr{F}, P),\left\{\mathscr{F}_s\right\} \text { for every pair }(t, x) \text {; and }\end{array}\right.$
(7.4) this solution is unique in the sense of probability law.
Let $\mathscr{A}$ be elliptic in the open, bounded domain $D$, and consider the continuous functions $k: \bar{D} \rightarrow[0, \infty), g: \bar{D} \rightarrow \mathbb{R}$, and $f: \partial D \rightarrow \mathbb{R}$. The Dirichlet problem is to find a continuous function $u: \bar{D} \rightarrow \mathbb{R}$ such that $u$ is of class $C^2(D)$ and satisfies the elliptic equation $$ \mathscr{A} u-k u=-g ; \quad \text { in } D $$ as well as the boundary condition $$ u=f ; \text { on } \partial D $$
Proposition 7.2. Let $u$ be a solution of the Dirichlet problem (7.5), (7.6) in the open, bounded domain $D$, and let $\tau_D :=\inf \left(t \geq 0 ; X_t \notin D\right)$. If $$ E^x \tau_D<\infty ; \quad \forall x \in D $$
Then we have that $$ u(x)=E^x\left[f\left(X_{\tau_D}\right) \exp \left(-\int_0^{\tau_D} k\left(X_s\right) d s\right)+\int_0^{\tau_D} g\left(X_t\right) \exp \left(-\int_0^t k\left(X_s\right) d s\right) d t\right] $$ for every $u \in \partial D$
Looking at the proof of the proposition, they define $M_t$ as below. However, I feel like I'm hitting my brain against a wall on this proof, I can't see how to show that $M_t$ is a martingale.
$$ M_t := u\left(X_{t \wedge \tau_D}\right) \exp \left(-\int_0^{t \wedge \tau_D} k\left(X_s\right) d s\right) +\int_0^{t \wedge \tau_D} g\left(X_s\right) \exp \left(-\int_0^s k\left(X_\theta\right) d \theta\right) d s ; \quad 0 \leq t<\infty $$
Any help is appreciated.