show that $X_n$ is a supermartingale $\Rightarrow$ $Y_n=\text{min}(X_n,x)$ is a supermartingale

Question

I have to show that: if $X_n$ is a supermartingale then $Y_n=\text{min}(X_n,x)$ is a supermartingale; ($x\in R$)

This what I did:

since we can write : $\text{min}(X_n,x)=X_n{1}_{X_n \leq x} + x{1}_{X_n \geq x}$

we have: $E(Y_{n+1}|F_n)=E(X_{n+1}{1}_{X_{n+1} \leq x}+x{1}_{X_{n+1} \geq x}|F_n)=E(X_{n+1}{1}_{X_{n+1} \leq x}|F_n)+ xE({1}_{X_{n+1} \geq x}|F_n)$

I am stuck here. How can I continue? some help would be appreciated

Answer 1

It's better to split it up the other way round, i.e. to note that it suffices to show that

$$\mathbb{E}(Y_{n+1} \mid \mathcal{F}_n) \leq x \tag{1}$$

and

$$\mathbb{E}(Y_{n+1} \mid \mathcal{F}_n) \leq X_n \tag{2}$$

To prove $(1)$ use that $Y_{n+1} \leq x$; for $(2)$ use $Y_{n+1} \leq X_{n+1}$ and the super-martingale property of $(X_n)_{n \in \mathbb{N}}$.

Remark: If you know Jensen's inequality for conditional expectations, the assertion follows from the fact that $y \mapsto \min\{y,x\}$ is a concave mapping for each fixed $x \in \mathbb{R}$.