the convexity of $\tilde v_*$. From Theorem 5.1 of Mnif and Pham [@mnipham01], we have $$\begin{aligned}
\label{conv1}
\tilde v(t,y)=\Max_{x\geq (\beta-\alpha)(T-t)}[v(t,x)-xy].\end{aligned}$$ We fix $(t,y)\in [0,T]\times (0,\infty)$. From the definition of $\tilde v_*$, there exists a sequence$\big((t_n,y_n)\big)_n$ such that $v(t_n,y_n)\longrightarrow \tilde v_*(t,y)$ and $(t_n,y_n)\longrightarrow (t,y)$ when $n$ goes to infinity. From equation , we have $\tilde v(t_n,y_n)\geq v(t_n,x)-xy_n \geq v_*(t,x)-xy_n $ for all $x\geq (\beta-\alpha)(T-t_n)$. Sending $n$ to $\infty$, we have $\tilde v_*(t,y)\geq v_*(t,x)-xy$ for all $x\geq (\beta-\alpha)(T-t)$ and so $$\begin{aligned}
\label{conv2}
\tilde v_*(t,y)\geq \Max_{x\geq (\beta-\alpha)(T-t)}[v_*(t,x)-xy].\end{aligned}$$ From Theorem 5.1 of Mnif and Pham [@mnipham01], there exists $\hat x$ such that $$\begin{aligned}
\label{conv3}
\tilde v(t,y)= v(t,\hat x)-\hat xy\end{aligned}$$ and $\Dy{\tilde v}(t,y)=-\hat x$. We consider a sequence $\big((t_n,y_n)\big)_n$ such that $v(t_n,y_n)\longrightarrow \tilde v_*(t,y)$ and $(t_n,y_n)\longrightarrow (t,y)$ when $n$ goes to infinity. From equation , there exists $\hat x_n$ such that $\tilde v(t_n,y_n)= v(t_n,\hat x_n)-\hat x_ny_n$. Since $\tilde v \in D_{\gamma}([0,T]\times (0,\infty))$, we have $\hat x_n:=|\Dy{\tilde v}(t_n,y_n)|\leq K(1+y_n^{-(\gamma+1))})$ where $K$ is a positive constant independent of $n$. Since $y_n\longrightarrow y$ when $n$ goes to infinity, the sequence $(\hat x_n)_n$ is bounded, and so long a subsequence $\hat x_n\longrightarrow \hat x $ when $n$ goes to infinity and so we have $$\begin{aligned}
\label{conv4}
\Lim_{n\longrightarrow \infty}v(t_n,\hat x_n)= 