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# 数学代写|统计计算作业代写Statistical Computing代考|Time discretisation

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## 数学代写|统计计算作业代写Statistical Computing代考|Time discretisation

Compared with the situation of discrete-time processes, for example the Markov chains considered in Section 2.3, simulation in continuous-time introduces new challenges. Consider a stochastic process $\left(X_{t}\right){t \in I}$ where $I \subseteq \mathbb{R}$ is a time interval, for example $I=[0, \infty)$ or $I=[0, T]$ for some time horizon $T>0$. Even if the time interval $I$ is bounded, the trajectory $\left(X{t}\right){t \in I}$ consists of uncountably many values. Since computers only have finite storage capacity, it is impossible to store the whole trajectory of a continuous-time process on a computer. Even computing values for all $X{t}$ would take an infinite amount of time. For these reasons, we restrict ourselves to simulate $X$ only for times $t \in I_{n}$ where $I_{n}=\left{t_{1}, t_{2}, \ldots, t_{n}\right} \subset I$ is finite. This procedure is called time discretisation.

In many cases we can simulate the process $X$ by iterating through the times $t_{1}, t_{2}, \ldots, t_{n} \in I_{n}$ : we first simulate $X_{t_{1}}$, next we use the value of $X_{t_{1}}$ to simulate $X_{t_{2}}$, then we use the values $X_{t_{1}}$ and $X_{t_{2}}$ to simulate $X_{t_{3}}$ and so on. The final step in this procedure is to use the values $X_{t_{1}}, \ldots, X_{t_{n-1}}$ to simulate $X_{t_{n}}$. One problem with this approach is that often the distribution of $X_{t_{k}}$ does not only depend on $X_{t_{i}}$ for $i=1,2, \ldots, k-1$, but also on (unknown to us) values $X_{t}$ where $t \notin I_{n}$. For this reason, most continuous-time processes cannot be simulated exactly on a computer and we have to resort to approximate solutions instead. The error introduced by these approximations is called discretisation error.