Let me introduce the objects:
0) $(\Omega, \mathcal{F},\Bbb{P})$ is a probability space
1)$S_N $ is the set of symmetric, non-negative definite $N\times N$ matrices
2)$a:[0, \infty) \times \Omega \to S_N $ is bounded and progressively measurable
3)$b:[0,\infty)\times \Omega \to \mathbb{R}^N$ bounded and progressively measurable
4)$ K_u (f) = \frac{1}{2}\sum_{i,j} a_{ij}(u) \partial_i \partial_j f + \sum_i b_i(u) \partial_i f $
5) $\xi: [0,\infty) \times \Omega \to \Bbb{R}$ continuous non decreasing and progressively measurable.
5-i)$\int_0^t \langle \theta, a(u) \theta \rangle \, d \xi(u) $
5-ii) $\Bbb{E}^P [\exp\{\lambda \big(\xi(t) - \xi(0)\big)\}]< \infty\quad \forall\; \lambda >0\; t>0$
6) $\alpha: [0, \infty) \times \Omega \to \Bbb{R}^N$
Theorem The following conditions are equivalent:
$(a) \, \forall f \in C^\infty_c(\Bbb{R}^N)$ $$f(\alpha(t)) - \int_0^t K_u f(\alpha(u))\, d\xi(u)$$ is a $P$ - martingale
$(b) \, \forall f \in C^{1,2}_b([0,\infty)\times \Bbb{R}^N)$ $$f(t,\alpha(t)) - \int_0^t f_u(u,\alpha(u))\, du - \int_0^t K_u f(u,\alpha(u))\, d\xi(u)$$ is a $P$ - martingale
$(c) \, \forall f \in C^{1,2}_b([0,\infty)\times \Bbb{R}^N)$ uniformly positive $$f(t,\alpha(t))\exp\bigg\{ - \int_0^t \frac{f_u(u,\alpha(u))}{f(u,\alpha(u))}\, du - \int_0^t \frac{K_uf(u,\alpha(u))\, }{f(u,\alpha(u))} d\xi(u)\bigg\} $$ is a $P$ - martingale
$(d) \, \forall g \in C^{1,2}_b([0,\infty)\times \Bbb{R}^N)$ $\theta \in \Bbb{R}^N$ $$\exp\bigg\{ g(t,\alpha(t))-g(0,\alpha(0))+ \langle \theta,\alpha(t) - \alpha(0)\rangle -\int_0^t g_u(u,\alpha(u))\, du - \int_0^t K_ug(u,\alpha(u))\, d\xi(u) \, \\- \frac{1}{2}\int_0^t\langle \nabla g(u,\alpha(u)), a(u)\nabla g(u,\alpha(u)) \rangle d\xi(u)- \int_0^t\langle \theta, a(u)\nabla g(u,\alpha(u)) \rangle d\xi(u) \\-\frac{1}{2} \int_0^t\langle \theta, a(u)\theta \rangle d\xi(u) - \int_0^t\langle \theta, b(u)\rangle d\xi(u) \bigg\} $$ is a $P$ - martingale
$(e) \forall \, \theta \in \Bbb{R}^N,$ $$X_\theta(t) = \exp\bigg\{\big\langle \theta, \alpha(t) - \alpha(0)\big\rangle -\int_0^t\langle \theta, b(u)\rangle d\xi(u) -\frac{1}{2} \int_0^t\langle \theta, a(u)\theta \rangle d\xi(u)\bigg\}$$ is a $P$ - martingale
$(f) \forall \, \theta \in \Bbb{R}^N,$ $$X_{i\theta}(t) = \exp\bigg\{i\big\langle \theta, \alpha(t) - \alpha(0)\big\rangle - i \int_0^t\langle \theta, b(u)\rangle d\xi(u) +\frac{1}{2} \int_0^t\langle \theta, a(u)\theta \rangle d\xi(u)\bigg\}$$ is a $P$ - martingale
The problem begins now:
Let $g\in C^{1,2}_b([0, \infty) \times \Bbb{R}^N)$. Define $\tilde{\alpha}:[0, \infty) \times \Omega \to \Bbb{R}^{N+1}$ as:
$$\tilde{\alpha}_i = \alpha_i(t) -\int_0^t b_i(u)d\xi(u) \qquad \qquad 1 \leq i \leq N \\ \tilde{\alpha}_{N+1} = g(t,\alpha(t)) - \int_0^t g_u(u,\alpha(u))\, du - \int_0^t K_u g(u,\alpha(u))\, d\xi(u)$$
Now define $\tilde{a}:[0, \infty) \times \Omega \to S_{N+1}$ by
$\tilde{a}_{ij}(t) = a_{ij}(t) \qquad \qquad 1 \leq i,j\leq N$
$\tilde{a}_{N+1_i}(t) = \sum_j a_{ij}(t)\partial_j g(t,\alpha(t))\qquad 1 \leq i \leq N$
$\tilde{a}_{N+1_i}(t) = \sum_{ij} \partial_i g(t,\alpha(t))a_{ij}(t)\partial_j g(t,\alpha(t))$
Assume that $(e)$ holds, and that $$\int_0^t\langle \tilde{\theta},\tilde{ a}(u)\tilde{\theta }\rangle d\xi(u) = \int_0^t\langle \tilde{\theta},\tilde{ a}(u)\tilde{\theta }\rangle d(u) \qquad \forall \theta \in \Bbb{R}^{N}\; t \geq 0$$
Question why is $$\tilde{X}_{\tilde{\theta}}(t) = exp\bigg\{\big\langle \tilde{\theta}, \tilde{\alpha(t)} - \tilde{\alpha(0)}\big\rangle -\frac{1}{2} \int_0^t\langle \tilde{\theta},\tilde{ a}(u)\tilde{\theta }\rangle d(u)\bigg\}$$a $P$ - martingale for every $\tilde{\theta} \in \Bbb{R}^{N+1} $?
remark I believe every background on the question is provided, but in case you might be looking for a reference this construction is from Stroock and Varadhan 1971 - Diffusion processes with boundary conditions pgs 149,150, 154,155
It suffices to apply $(d)$ with $\tilde{g}= \theta_{N+1} g$ to the expression in $\tilde{X}_{\tilde{\theta}}$. $$\tilde{X}_{\tilde{\theta}}(t) = \exp \bigg\{\langle \theta, \alpha(t)- \alpha(0) \rangle - \int_0^t \langle \theta, b(u) \rangle \, d\xi(u) -\frac{1}{2} \int_0^t \langle \theta, a(u) \theta \rangle \, du + \\ \theta_{N+1} g(t,\alpha(t)) - \theta_{N+1} g(0,\alpha(0)) - \int_0^t \theta_{N+1} g_u(u,\alpha(u))\, du -\int_0^t K_u\theta_{N+1} g(u,\alpha(u))\, d\xi(u) - \int_0^t \langle \theta, a(u)\nabla \theta_{N+1}g(u,\alpha(u)) \rangle \, du - \frac{1}{2}\int_0^t\langle \nabla \theta_{N+1}g(u,\alpha(u)), a(u)\nabla \theta_{N+1}g(u,\alpha(u)) \rangle\, du \bigg\} $$