Show that if {Z n ; n ≥ 1} is a martingale (sub martingale or super martingale) relative to a joint process {Z n , X n ; n ≥ 1} and if J is a stopping trial for {Z n ; n ≥ 1} relative to {Z n , X n ;...

Show that if {Z_n; n ≥ 1} is a martingale (sub martingale or super martingale) relative to a joint process {Z_n, X_n; n ≥ 1} and if J is a stopping trial for {Z_n; n ≥ 1} relative to {Z_n, X_n; n ≥ 1}, then the stopped process satisfies respectively.