**Exercise 3.1 (antithetic sampling)**

**Exercise 3.2 (antithetic sampling, Cauchy distribution)**

Let $Y$ be a standard Cauchy variable. Write a simulation program to compare the antithetic transformation $Y\rightarrow -Y$ and the semi-antithetic transformation $Y \rightarrow 1/Y$ for the computations of $\mathbb{E}(f(Y))$ in the three cases

- $f(y)=\sin(y)$,
- $f(y)=\cos(y)$,
- $f(y)=(y)_+^{1/4}$.

**Exercise 3.3 (stratification, optimal allocation)**

The stratification estimator for computing $\mathbb{E}(X)$ is defined by

$$I^{\rm strat.}_{M_1,\dots,M_k}=\sum_{j=1}^{k}p_j\frac 1{M_j}\sum_{m=1}^{M_j} X_{j,m}$$ where $(p_j)_{1\leq j\leq k}$ are the probabilities of strata and $(X_{j,m}:1\leq m\leq M_j)$ are independent copies of $X$ conditionally to the strata $j$.

- Prove that the optimal allocation is indeed given by $M^*_j=M\frac{ p_j \sigma_j }{\sum_{i=1}^k p_i \sigma_i }$.
- For such a choice, derive a central limit theorem for the estimator $I^{\rm strat., opt. alloc.}_{M_1,\dots,M_k}$ as $M\to+\infty$.

**Solution.**

**Exercise 3.4 (stratification of Gaussian vectors)**

We aim at describing in detail the generation of random variables in steps 1 and 2 of Example 3.2.2. We assume that $Y$ is a standard $d$-dimensional Gaussian vector and that $\beta$ is normalized to 1, i.e. $|\beta|=1$.

- What is the distribution of $Z=\beta\cdot Y$ and its parameters?
- Assume that ${\cal S}_j=[-x_{j-1},x_j)$ with $-\infty:=x_0\leq ...\leq x_j\leq ...\leq x_k=+\infty$. Compute $p_j=\mathbb{P}(Z\in {\cal S}_j)$ as a function of the $(x_i)_i$. Derive the cumulative distribution function of $Z$ given $\{Z\in {\cal S}_j \}$. Deduce an algorithm to generate such a distribution (assuming that ${\cal N}^{-1}(\cdot)$ is known).
- Compute explicitly the distribution of $Y-\beta Z$. Show that it is independent on $Z$.
- Deduce a generation scheme of $Y$ given $\{Z\in {\cal S}_j\}$.

**Exercise 3.5 (control variates)**

In the Monte-Carlo computation of $\mathbb{E}(X)$ we use extra centered random variables $(Z_j)_j$ (the control variates) and we set $$Z(\beta):=\beta\cdot Z=\sum_{j=1}^d \beta_i Z_i.$$ Establish a central limit theorem for the control variate estimator $I^{\rm Cont.Var.}_{\beta^{*}_M,M}$ that uses the optimal empirical weights $$\beta^{*}_M=\Big[\frac 1M\sum_{m=1}^M Z_mZ_m^T\Big]^{-1}\frac 1M\sum_{m=1}^M X_m Z_m.$$ Check that the limit variance is the lowest possible variance $\mathbf{V}ar(X-Z(\beta^*))$.

**Exercise 3.6 (importance sampling, Gaussian vectors)**

Extend the formula of Corollary 3.4.9 to the multidimensional case, by changing both the mean and the covariance.

**Solution.**

**Exercise 3.7 (Esscher transform, Gaussian and exponential distributions)**

In Proposition 3.4.15, we consider a random sum $$Y=\sum_{n=1}^N Z_n$$ where $N$ has a Poisson distribution and $(Z_n)_n$ are independent, in a model defined under the probability $\mathbb{P}$. For a given function $\varphi:\mathbb{R}\mapsto \mathbb{R}$ with linear growth at most, $$L:=\exp\Big(\sum_{n=1}^N \varphi(Z_n)-\lambda \mathbb{E}_\mathbb{P}(e^{\varphi(Z)}-1)\Big)$$ defines the likelihood of a new probability measure $\mathbb{Q}$ equivalent to $\mathbb{P}$. Explicitly characterize the distributions of $Z$ under $\mathbb{Q}$, when we take $\varphi(z)=z$ and when under $\mathbb{P}$, $Z$ has either a Gaussian distribution or an exponential one.

**Exercise 3.8 (importance sampling, Poisson distribution)**

Write a simulation program for computing $\mathbb{P}(Y\geq x)$, by importance sampling, when $Y$ has a Poisson distribution with parameter 1 and $x$ is large (see Example 3.4.16).

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