 c

Take an integer $c\ge \max \{a,b\}$. On $I=\{-c,\dots ,c\}$, the modified transition matrix $Q=(q_{i,j})$ agrees with $P$. Above $c$, it stays put with probability $a/(a+b)$ and jumps to each of $c-b+1,\dots ,c$ with probability $1/(a+b)$. Below $-c$, it stays put with probability $b/(a+b)$ and jumps to each of $-c,\dots ,-c+a-1$ with probability $1/(a+b)$. All unspecified entries are zero. These rows sum to one. The integer qualification is necessary: if $c$ were nonintegral, $i=\lfloor c\rfloor +1$ would fit none of the printed row conditions.