Make Your Martingale III

Question

Let $X_1,X_2,\dots \sim \text{Exp}(\lambda)$ IID. Define the process $\{M_n\}_{n \geq 0}$ by $M_0 = 1$ and $M_n = M_{n-1} \cdot \dfrac{1}{2}e^{\frac{\lambda}{2}X_n}$ with the natural filtration $\mathcal{F}_n = \sigma(X_1,\dots,X_n)$. If $0 < a < p < b$ for some real values $a$ and $b$, then $\{M_n^p\}_{n \geq 0}$ is a sub-martingale. Find $a + b$.