Prove it is a markov chain

Question

I'm given a sequence of random variables $(X_n)$ that verifies the following recursion $X_{n+1} = (X_n + Y_n - K)^+ +Z_n$ where:

$Y_n$ is a sequence of iid poisson random variables of paramater $\lambda_1 d_1$

$(Z_n)$ is a sequence of iid random variables of parameter $\lambda_1(d_2 + 2L/v)$

$K $ is an integer

Both sequences $(Y_n) and (Z_n)$ are independent.

I have to show that $(X_n)$ is a markov chain.

Any help or hints would be greatly appreciated

Answer 1

For a Markov chain $P(X_{n+1}=x_{n+1} | X_n=x_n, X_{n-1}=x_{n-1}, \ldots) = P(X_{n+1}=x_{n+1} | X_n=x_n)$, i.e. you should be able to compute the probability distribution of $X_{n+1}$ conditioned on history by simply knowing $X_n$, independent of what happened to $X_{n-1}, X_{n-2}, ... \ldots$.

In your problem, $X_{n+1} = f(X_n; Y_n, K, Z_n)$, i.e. $X_{n+1}$ depends on $X_n$ and a bunch of other constants/ RVs that are independent of $X$ and have identical distributions at all time steps. Thus, you can compute the probability distribution of $X_{n+1}$ conditioned on history without knowing anything about $X_{n-1}, X_{n-2}, \ldots$, i.e. $X_n$ is a Markov process.