What is $(X_{m}^{n, x})_{n \leq m \leq N}$ in lecture note Optimal Stopping and American Options by Damien Lamberton?

Question

I'm having a problem of understanding Remark 1.5.4 from lecture note Optimal stopping and American options by Damien Lamberton:

Consider $\left(X_{n}\right)_{n \in \mathbb{N}}$, an $\mathbb F$-Markov chain with transition kernels $\left(P_{n}\right)$ and a reward sequence $(Z_{n})$ given by $Z_{n}=f\left(n, X_{n}\right), \quad n \in \mathbb{N}$ where, for every $n \in \mathbb{N}, f(n,\cdot)$ is a nonnegative measurable function such that the random variable $f\left(n, X_{n}\right)$ is integrable.

Proposition 1.5.2 Under the above assumptions, the Snell envelope with horizon $N$ of the sequence $\left(Z_{n}\right)_{n \in \mathbb{N}}$ is given by $U_{n}^{(N)}=V\left(n, X_{n}\right) \quad a . s .$ where the functions $V(n, \cdot)(n=0, \ldots, N)$ are determined by the following dynamic programming algorithm: $$\left\{\begin{array}{l} V(N, x)=f(N, x) \\ V(n, x)=\max \left\{f(n, x), P_{n}[V(n+1, \cdot)](x)\right\}, \quad \text { for } 0 \leq n \leq N-1 \end{array}\right.$$

Because the author uses the same notation $X$ in $X_m^{n,x}$ and $X_n$, I would like to ask if $X_m^{n,x}$ has any relation with our original $X_n$. If there is no such relation, I would interpret $\textbf{Remark 1.5.4}$ as follows:

Fix $n \in\{0, \ldots, N\}$. Assume that $(Y_m)_{n \leq m \leq N}$ is a Markov chain w.r.t the filtration $(\mathcal{F}_{m})_{n \leq m \leq N}$ and has transition kernels $(P_{m})_{n \leq m \leq N}$. If $Y_n=x$ a.s., then $V(n, x)=\sup _{\nu \in \mathcal{T}_{n, N}} \mathbb{E} f\left(\nu, Y_{\nu}\right)$.

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Background information:

1. A transition kernel on a measurable space $(E, \mathcal{E}),$ is a family $(P(x, \cdot))_{x \in E}$ of probability measures on $(E, \mathcal{E})$ such that, for every $A \in \mathcal{E},$ the mapping $x \mapsto P(x, A)$ is measurable. If $P=(P(x, \cdot))_{x \in E}$ is a transition kernel on $(E, \mathcal{E})$ and if $f$ is a nonnegative Borel-measurable function on $E,$ the function $P f,$ defined by $P f(x)=\int_{E} P(x, d y) f(y)$ is measurable and nonnegative.

2. Let $\left(\Omega, \mathcal{F}, \mathbb{F}=\left(\mathcal{F}_{n}\right)_{n \in \mathbb{N}}, \mathbb{P}\right)$ be a filtered probability space and $\left(P_{n}\right)_{n \in \mathbb{N}}$ a sequence of transition kernels on a measurable space $(E, \mathcal{E}) .$ A sequence $\left(X_{n}\right)_{n \in \mathbb{N}}$ of random variables with values in $(E, \mathcal{E})$ is an $\mathbb F$-Markov chain with transition kernels $\left(P_{n}\right)_{n \in \mathbb{N}}$ if $\left(X_{n}\right)_{n \in \mathbb{N}}$ is $\mathbb{F}$ -adapted and, for every nonnegative measurable function $f$ on $(E, \mathcal{E}),$ we have $\mathbb{E}\left(f\left(X_{n+1}\right) | \mathcal{F}_{n}\right)=P_{n} f\left(X_{n}\right),$ for every $n \in \mathbb{N}$

Answer 1

I've just received a reply from Professor Damien Lamberton in which he confirms that mu understanding is correct.

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