Riemann surface
matlab

For an arbitrary real parameter $\beta$, we denote by $\mathcal{L}^\beta$ the Dynkin operator associated to the process $\left(Y^b, Z^b\right)$ :
$$\mathcal{L}^\beta:=D_t+\frac{1}{2} \beta^2 D_{y y}^2+\frac{1}{2} y^2 D_{z z}^2+\beta y D_{y z}^2 .$$
In this step, we intend to prove that for all $t \in[0, T]$ and $y, z \in \mathbb{R}$ :
$$\max {|\beta| \leq 1} \mathcal{L}^\beta v(t, y, z)=\mathcal{L}^1 v(t, y, z)=0 .$$ The second equality follows from the fact that $\left{v\left(t, Y_t^1, Z_t^1\right), t \leq T\right}$ is a martingale. As for the first equality, we see from (1.18) and (1.19) that 1 is a maximizer of both functions $\beta \longmapsto \beta^2 D{y y}^2 v(t, y, z)$ and $\beta \longmapsto \beta y D_{y z}^2 v(t, y, z)$ on $[-1,1]$.

Let $b$ be some given predictable process valued in $[-1,1]$, and define the sequence of stopping times
$$\tau_k:=T \wedge \inf \left{t \geq 0:\left(\left|Y_t^b\right|+\left|Z_t^b\right| \geq k\right}, \quad k \in \mathbb{N} .\right.$$
By Itô’s lemma and (1.20), it follows that :
\begin{aligned} v\left(0, Y_0, Z_0\right)= & v\left(\tau_k, Y_{\tau_k}^b, Z_{\tau_k}^b\right)-\int_0^{\tau_k}\left[b D_y v+y D_z v\right]\left(t, Y_t^b, Z_t^b\right) d W_t \ & -\int_0^{\tau_k} \mathcal{L}^{b_t} v\left(t, Y_t^b, Z_t^b\right) d t \ \geq & v\left(\tau_k, Y_{\tau_k}^b, Z_{\tau_k}^b\right)-\int_0^{\tau_k}\left[b D_y v+y D_z v\right]\left(t, Y_t^b, Z_t^b\right) d W_t . \end{aligned}
Taking expected values and sending $k$ to infinity, we get by Fatou’s lemma :
\begin{aligned} v\left(0, Y_0, Z_0\right) & \geq \liminf {k \rightarrow \infty} E\left[v\left(\tau_k, Y{\tau_k}^b, Z_{\tau_k}^b\right)\right] \ & \geq E\left[v\left(T, Y_T^b, Z_T^b\right)\right]=E\left[e^{2 \lambda Z_T^b}\right], \end{aligned}
which proves the lemma.

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