Assumptions and notations
=========================
In this section, we list and interpret all assumptions used to prove our main results in Theorem \[Main Thm\].
We now introduce the model of diffusion under the following stochastic differential equation $$\label{Eq:Main diffusion}
\left\{\begin{array}{ll}
dr_t = \Big[\gamma_1 (r_t - q_t)+ \gamma_2\sigma_1(r_t-q_t)W_t^{\mathbb{P},\mathbb{Q}_1}\Big]dt + \sigma_2 (r_t - q_t) dW_t^{\mathbb{P},\mathbb{Q}_2}, \\
d q_t = b(q_t)dt,
\end{array}\right.$$ where $\sigma_1, \sigma_2$ are positive constants, $\gamma_1, \gamma_2$ are positive constant such that $\gamma_1<2\gamma_2$. $W_t^{\mathbb{P},\mathbb{Q}_1}$ and $W_t^{\mathbb{P},\mathbb{Q}_2}$ are Brownian motions under $\mathbb{P}$ and $\mathbb{Q}_1$, respectively. We recall that $\mathbb{P}$ and $\mathbb{Q}_1$ are equivalent. This means that we consider the diffusion $(q_t, r_t)$ which is under $\mathbb{P}$ the weak solution of (\[Eq:Main diffusion\]) (see Remark \[Rmk:Conditions equivalence between P and Q1\]). The drift and diffusion coefficients $b$ and $\sigma$ are of the class $\mathcal{C}^{1}$ on $I$, and $\sigma$ is of class $\mathcal{C}^{3}$. Furthermore, $\sigma$ is strictly positive on $I$. The generator of the diffusion is given by $$\mathcal{A}u = \sigma^{-2}\Delta u + \frac{1}{2}\sigma^{-1}\sigma'Du, \quad \hbox{ on } I.$$ and the domain of $\mathcal{A}$ is $\mathcal{D}(A) = \{u\in H^2(I):\mathcal{B}u\in L^2(I)\}$ where $$\mathcal{B}u = \frac{\sigma'u}{\sigma}-\frac{\sigma''\sigma^{-1}u}{\sigma^2}.$$ We also set $A = -\mathcal{B}$ on $L^2(I)$ with domain $D(A) = \mathcal{D}(A)$. We say that $f \in \mathcal{H}_2$ if $f$ is twice continuously differentiable with respect to time and twice continuously differentiable with respect to the state variable. In addition, $$\langle f,u\rangle_t = \int_{I} f(s)u(s) ds.$$ The set of admissible control processes is $\mathcal{U}:=\{u:\Omega\times [0,T]\rightarrow \mathbb{R}\}_{t\in [0,T]}$ such that $u$ is a control adapted to the natural filtration $\mathbb{F}$ and $$u\in\mathcal{U}\;\mbox{if and only if}\; u(\cdot)\in L^2(\Omega\times [0,T];R^m)\;\mbox{with}\; \mathbb{E}[\int_{0}^{T}|u_s|^2ds]<+\infty.$$ Note that in our paper, the control $u$ acts on both Brownian motions $W_t^{\mathbb{P},\mathbb{Q}_1}$ and $W_t^{\mathbb{P},\mathbb{Q}_2}$. For a given control process $u$ and corresponding trajectory $(q_t,r_t)$, the cost functional is defined by $$\label{Eq:Objective functional}
J(u):=\frac{1}{2}\mathbb{E}\Big[\int_{0}^{T}\Big(\Big|\frac{dq_s}{ds}\Big|^2 + |r_s|^2\Big)ds +
\Big| q_T-\xi\Big|^2 \Big].$$ The corresponding value function of (\[Eq:Main stochastic control problem\]) is defined by $$V(q) = \inf_{u\in\mathcal{U}} J(u).$$ We also set $A_1 := -\mathcal{B}$ with domain $D(A_1)$ defined on $\mathcal{H}_2$, where $\mathcal{B}$ is defined as above. Let $B$ be the linear bounded operator given by $$Bf = -\gamma_1 f + \gamma_2 f' ,$$ with domain $D(B)=\mathcal{H}_3$ where $$\mathcal{H}_3 = \{f\in\mathcal{H}_2\; :\; f(q)=f'(q)=0\; \forall\; q=q_1,q_2\}.$$ For this operator, the second condition of Theorem \[Main Thm\] holds. Now we define the linear operator $A$ on $\mathcal{H}_3$ by $$\begin{aligned}
\label{Eq:A}
\mathcal{A}f &=&\sigma^{-2}\Delta f + \frac{1}{2}\sigma^{-1}\sigma'f' +
b(q)f' \\
&=&\mathcal{A}_1f + b(q)f', \quad \forall f\in D(A).\end{aligned}$$ The condition (\[Eq:Main\]) holds for the two operators $A$ and $B$ if we have $$\label{Eq:Conditions on B and A}
\begin{cases}
\gamma_1\leq \alpha \beta,\\
\gamma_2\alpha^2\leq \gamma_1,\\
\gamma_2\leq \alpha \beta,\\
\gamma_2\alpha\geq 2\gamma_1,\\
\sigma \beta^2 \alpha > \sqrt{\alpha \beta}.
\end{cases}$$ Under these assumptions, Theorem \[Main Thm\] guarantees that $V$ is the unique solution of the following p.d.e. $$\label{Eq:V PIDE}
\left\{\begin{array}{ll}
\partial_t V(q,t) + \mathcal{A} V(q,t) =0, \;&\; t\in [0,T]\\
V(q,T) = \xi(q).
\end{array}\right.$$ Similarly, we can define $$\begin{aligned}
\mathcal{H}_1 &=& \{f\in \mathcal{H}_2\; : \;f(q)=f'(q)=0\; \forall\;q=q_1, q_2, \;\; f(q_3)=0\},\\
\mathcal{H}_2 &=& \{f\in \mathcal{H}_1\; : \; f(q) >0\; \forall\; q\in I\}.\end{aligned}$$
Now, we introduce the following assumptions on $b, \sigma$ and $\xi$:
**H1:**
: $\sigma,b\in \mathcal{C}^{3}(I)$ and $b(q)>0$.
**H2:**
: $(\gamma_1\vee\gamma_2) (\alpha\vee\sigma^{-2})\leq \frac{1}{2}\gamma_1$, $(\gamma_1\vee\gamma_2)^2(\alpha^2\vee\sigma^{-2})\leq \frac{1}{2}\gamma_1$.
**H3:**
: $\gamma_2\alpha\geq \gamma_1$.
**H4:**
: $c^2<\frac{1}{4\sigma^4}\frac{4\gamma_1\gamma_2}{\alpha^3}(\gamma_1\vee\gamma_2)^2$.
We now recall the definition of the set $\tilde{\mathcal{M}}_A$ and some useful lemmas (for proofs, see the appendix of [@Agram2009]).
For given $t>0$, $\theta>0$, let $\mathcal{G}_{t,\theta}$ denote the family of stopping times such that $$\mathcal{G}_{t,\theta}:=\{\tau\;:\;
\mathbb{E}[e^{\frac{1}{2}\int_{t}^{\tau}A_1h(s)ds}|\mathcal{F}_{t}]
\leq
e^{\frac{1}{2}\theta\int_{t}^{\tau}h(s)ds},\forall\; h\in D(A_1)\}.$$
\[D\] Let $t>0$ and $\theta>0$ be fixed. $\mathcal{G}_{t,\theta}$ is a nonempty, closed and convex subset of the Borel sigma field $\mathcal{B}([0,+\infty))$ under the sup norm $\|\cdot\|_{\infty}$. In addition, for each $h\in \mathcal{H}_3$, we have $$e^{\frac{1}{2}\int_{t}^{\tau}Ah} \leq e^{\frac{1}{2}\theta \int_{t}^{\tau}h},
\quad \forall \tau \in \mathcal{G}_{t,\theta}.$$
Next, we define the function $H_2(q)$ given by
