 Solution

For the Gillespie algorithm, start from $t=0$ and an integer molecular copy number $x\ge 0$. At each step use the current state, without the mean approximation in parts (a)–(b). The reaction propensity functions are $a_{+}(x)=\lambda$ and $a_{-}(x)=\beta \sqrt{x}$, and their sum is $a_0(x)=\lambda +\beta \sqrt{x}$. Draw two independent uniform random variables $U_1,U_2$ on $(0,1)$ and setThe waiting time has an exponential distribution with rate $a_0(x)$. Keep the sample path constant on $[t,t+\Delta t)$, advance $t\leftarrow t+\Delta t$, and choose the reaction byRecord the post-jump state, recompute both reaction propensity functions, and repeat with fresh independent uniforms. At $x=0$ the death propensity is zero, so only a birth is selected and the trajectory remains nonnegative. For a prescribed terminal time $T$, if the next event would occur after $T$, keep the current state to $T$ and stop. This produces exact piecewise constant sample paths of the specified continuous-time Markov chain.