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# 数学代写|统计计算作业代写Statistical Computing代考|Monte Carlo estimates

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## 数学代写|统计计算作业代写Statistical Computing代考|Basic Monte Carlo

Assume that $X=\left(X_{t}\right){t \in[0, T]}$ is given as the solution of an SDE and that we want to compute $\mathbb{E}(f(X))$. In order to estimate this quantity using Monte Carlo integration, we generate independent samples $X^{(n, 1)}, X^{(n, 2)}, \ldots, X^{(n, N)}$, using numerical approximations to $X$, obtained by repeatedly solving a discretised SDE with discretisation parameter $n$. Then we can use the approximation $$\mathbb{E}(f(X)) \approx \frac{1}{N} \sum{j=1}^{N} f\left(X^{(n, j)}\right)=Z_{n, N}$$
Here we allow for the function $f$ to depend on the whole path of $X$ until time $T$. We can choose, for example,
$$f(X)=\sup {t \in[0, T]} X{t}$$
to get the maximum of a one-dimensional path, or $f(X)=\left|X_{T}\right|^{2}$ to get the second moment of the final point of the path.

As for all Monte Carlo methods, there is a trade-off between accuracy of the result and computational cost. One notable feature of Monte Carlo estimation for SDEs is that the result is not only affected by the Monte Carlo error, but also by discretisation error.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Variance reduction methods

The error in Monte Carlo estimates such as (6.28) can be significantly reduced by employing the variance reduction techniques from Section 3.3.
6.5.2.1 Antithetic paths
Pairs of antithetic paths can be easily generated for SDEs by using the fact that, if $B$ is a Brownian motion, $-B$ is also a Brownian motion (see lemma 6.5): thus, solving the SDE using the Brownian motions $B$ and $B^{\prime}=-B$ gives rise to two paths $X$ and $X^{\prime}$ of the SDE. Since $B_{t}$ and $B_{t}^{\prime}$ are negatively correlated, typically $X_{t}$ and $X_{t}^{\prime}$ will also be negatively correlated for all $t \in[0, T]$.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Multilevel Monte Carlo estimates

The variance reduction methods we have discussed so far are applications of the general methods from Section $3.3$ to the problem of estimating expectations for paths of SDEs. In contrast, the multilevel Monte Carlo approach, discussed in the rest of this section, is specific to situations where discretisation error is involved.

Multilevel Monte Carlo methods, by cleverly balancing the effects of discretisation error and Monte Carlo error, allow us to reduce the computational cost required to compute an estimate with a given level of error. Let $X$ be the solution of an SDE and let $\varepsilon>0$. In equation (6.34) we have seen that the basic Monte Carlo estimate for $\mathbb{E}\left(f\left(X_{T}\right)\right)$ requires computational cost of order $\mathcal{O}\left(1 / \varepsilon^{3}\right)$ in order to bring the root-mean squared error down to $\varepsilon>0$. We will see that multilevel Monte Carlo methods can reduce this cost to nearly $\mathcal{O}\left(1 / \varepsilon^{2}\right)$.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|BASIC MONTE CARLO

Here we allow for the function $f$ to depend on the whole path of $X$ until time $T$. We can choose, for example,
$$f(X)=\sup {t \in[0, T]} X{t}$$

6.5.2.1