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# 数学代写|统计计算作业代写Statistical Computing代考|Stochastic differential equations

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

In this section we will give a very short introduction to SDEs, mainly by giving an intuitive idea about the properties of processes described by equations such as (6.6). We start by giving an informal explanation of different aspects of equation (6.6).

• The stochastic process $X=\left(X_{t}\right)_{t \geq 0}$ is the ‘unknown’ in equation (6.6). Solving the SDE means to find a stochastic process $X$ such that (6.6) is satisfied. (We will discuss below what this means.) Since the Brownian motion $B$ on the right-hand side of (6.6) is random, the solution $X$ is random, too.
• $x_{0} \in \mathbb{R}^{d}$ is called the initial value of the SDE.The function $\mu:[0, \infty) \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ is called the drift of the SDE. Given the current time $t$ and the current value $X_{t}$, it determines the direction of mean change of the process just after time $t$ :
• $$• \mathbb{E}\left(X_{t+h} \mid X_{t}\right) \approx X_{t}+\mu\left(t, X_{t}\right) h •$$
• as $h \downarrow 0$. The effect of the drift is illustrated in Figure 6.7.
• The matrix-valued function $\sigma:[0, \infty) \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d \times m}$ is called the diffusion coefficient of the SDE. It determines the amount of random fluctuations $X$ is subject to at any given time and place. Conditioned on the value of $X_{t}$, the covariance matrix of $X_{t+h}$ satisfies
• $$• \operatorname{Cov}\left(X_{t+h} \mid X_{t}\right) \approx \sigma\left(t, X_{t}\right) \sigma\left(t, X_{t}\right)^{\top} h •$$
• as $h \downarrow 0$.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Stochastic analysis

Some technical detail is required to give a mathematically rigorous definition of the stochastic integrals in equation (6.7). We omit the rigorous definition here and refer to the references given at the end of this chapter for details. Instead, we restrict ourselves to heuristic explanations of the most important aspects.

The stochastic integral, also called the Ito integral, of the integrand $Y$ with a integrator $X$ is given by the limit
$$\int_{0}^{T} Y_{t} d X_{t}=\lim {n \rightarrow \infty} \sum{i=0}^{n-1} Y_{t_{i}^{(n)}}\left(X_{t_{i+1}^{(n)}}-X_{t_{i}^{(n)}}\right)$$
where $t_{i}^{(n)}=i T / n$ for $i=0,1, \ldots, n$. Here, $X$ and $Y$ are stochastic processes. In (6.7) this relation is used with the integrand $\sigma_{i j}\left(s, X_{s}\right)$ instead of $Y$ and with the integrator $B$ instead of $X$. Since $X$ and $Y$ are random, the value of the stochastic integral (6.9) is a random variable. Equation (6.9) is in analogy to the approximation of the ordinary Riemann integral by Riemann sums:
$$\int_{0}^{T} f(t) d t=\lim {n \rightarrow \infty} \sum{i=0}^{n-1} f\left(t_{i}\right)\left(t_{i+1}-t_{i}\right)$$

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Discretisation schemes

In this section we will discuss methods to simulate solutions of an SDE using a computer. We consider the SDE
\begin{aligned} &d X_{t}=\mu\left(t, X_{t}\right) d t+\sigma\left(t, X_{t}\right) d B_{t} \ &X_{0}=x_{0} \end{aligned}
where $B=\left(B_{t}\right){t \geq 0}$ is an $m$-dimensional Brownian motion, $\mu:[0, \infty) \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ is the drift, and the diffusion coefficient is given by $\sigma:[0, \infty) \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d \times m}$. Our aim is to simulate values of $X$ at times $0=t{0}<t_{1}<\cdots<t_{n}$. We will proceed, starting with $X_{0}=x_{0}$, by successively computing $X_{t_{1}}, X_{t_{2}}, \ldots$ until time $t_{n}$ is reached.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|INTRODUCTION

• 随机过程X=(X吨)吨≥0是等式中的“未知”6.6. 求解 SDE 意味着找到一个随机过程X这样6.6很满意。在和在一世一世一世d一世sC你ssb和一世○在在H一种吨吨H一世s米和一种ns.由于布朗运动乙在右侧6.6是随机的，解X也是随机的。
• X0∈Rd称为 SDE 的初始值。函数μ:[0,∞)×Rd→Rd称为 SDE 的漂移。鉴于当前时间吨和当前值X吨, 它决定了过程在时间之后的平均变化方向吨:
• $$• \mathbb{E}\左X_{t+h} \mid X_{t}\rightX_{t+h} \mid X_{t}\right\约 X_{t}+\mu\leftt, X_{t}\右t, X_{t}\右H •$$
• 作为H↓0. 漂移的影响如图 6.7 所示。
• 矩阵值函数σ:[0,∞)×Rd→Rd×米称为 SDE 的扩散系数。它决定了随机波动的数量X在任何给定的时间和地点都受制于。以价值为条件X吨, 的协方差矩阵X吨+H满足
• $$• \运营商名称{Cov}\左X_{t+h} \mid X_{t}\rightX_{t+h} \mid X_{t}\right\约\西格玛\左t, X_{t}\右t, X_{t}\右\sigma\左t, X_{t}\右t, X_{t}\右^{\top} 小时 •$$
• 作为H↓0.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|STOCHASTIC ANALYSIS

The stochastic integral, also called the Ito integral, of the integrand $Y$ with a integrator $X$ is given by the limit
$$\int_{0}^{T} Y_{t} d X_{t}=\lim {n \rightarrow \infty} \sum{i=0}^{n-1} Y_{t_{i}^{(n)}}\left(X_{t_{i+1}^{(n)}}-X_{t_{i}^{(n)}}\right)$$
where $t_{i}^{(n)}=i T / n$ for $i=0,1, \ldots, n$. Here, $X$ and $Y$ are stochastic processes. In (6.7) this relation is used with the integrand $\sigma_{i j}\left(s, X_{s}\right)$ instead of $Y$ and with the integrator $B$ instead of $X$. Since $X$ and $Y$ are random, the value of the stochastic integral (6.9) is a random variable. Equation (6.9) is in analogy to the approximation of the ordinary Riemann integral by Riemann sums:
$$\int_{0}^{T} f(t) d t=\lim {n \rightarrow \infty} \sum{i=0}^{n-1} f\left(t_{i}\right)\left(t_{i+1}-t_{i}\right)$$

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|DISCRETISATION SCHEMES

dX吨=μ(吨,X吨)d吨+σ(吨,X吨)d乙吨 X0=X0