# 数学代写|数值分析代写numerical analysis代考|Stratonovich Integrals and Differential Equations

## 数学代写|数值分析代写numerical analysis代考|Stratonovich Integrals and Differential Equations

The Itô integral is based on approximations by finite sums of the form
$$\int_a^b f\left(X_t\right) d Y_t \approx \sum_{k=0}^{n-1} f\left(X_{a+k h}\right)\left(Y_{a+(k+1) h}-Y_{a+k h}\right)$$
As was noted in Section 7.5.2, the stochastic integral $\int_0^t W_s d W_s$ has expectation zero when interpreted in the sense of Itô. There is another interpretation called the Stratonovich integral, which is based on the trapezoidal rule:
$(7.5 .7)$
$$\int_a^b f\left(X_t\right) \circ d Y_t \approx \sum_{k=0}^{n-1} \frac{1}{2}\left[f\left(X_{a+k h}\right)+f\left(X_{a+(k+1) h}\right)\right]\left(Y_{a+(k+1) h}-Y_{a+k h}\right) .$$
Because of the nature of Wiener processes, the limits as $n \rightarrow \infty$ are different for integrals like \begin{aligned} \int_0^t W_s \circ d W_s & \approx \sum_{k=0}^{n-1} \frac{1}{2}\left(W_{h(k+1)}+W_{h k}\right)\left(W_{h(k+1)}-W_{h k}\right) \ & =\sum_{k=0}^{n-1} \frac{1}{2}\left(W_{h(k+1)}^2-W_{h k}^2\right)=\frac{1}{2}\left(W_t^2-W_0^2\right), \end{aligned}
matching the naive application of standard calculus rules.
Stratonovich stochastic differential equations have the form
$$d \boldsymbol{X}_t=\boldsymbol{f}\left(\boldsymbol{X}_t\right) d t+\sigma\left(\boldsymbol{X}_t\right) \circ d \boldsymbol{W}_t, \quad \boldsymbol{X}_0=\boldsymbol{x}_0 .$$
If integrals are interpreted in the Stratonovich sense,
$$\boldsymbol{X}_t=\boldsymbol{x}_0+\int_0^t \boldsymbol{f}\left(\boldsymbol{X}_s\right) d s+\int_0^t \sigma\left(\boldsymbol{X}_t\right) \circ d \boldsymbol{W}_t$$
then the solutions are solutions in the sense of Stratonovich.

## 数学代写|数值分析代写numerical analysis代考|Euler–Maruyama Method

The Euler-Maruyama method is essentially the Euler method applied to the stochastic differential equation
$$d \boldsymbol{X}t=\boldsymbol{f}\left(t, \boldsymbol{X}_t\right) d t+\sigma\left(t, \boldsymbol{X}_t\right) d \boldsymbol{W}_t, \quad \boldsymbol{X}_0=\boldsymbol{x}_0 .$$ For a step size $h>0$, the method consists of the iteration (7.5.9) $\quad \widehat{\boldsymbol{X}}{h(k+1)}^h=\widehat{\boldsymbol{X}}{h k}^h+h \boldsymbol{f}\left(t, \widehat{\boldsymbol{X}}{h k}^h\right)+\sigma\left(h k, \widehat{\boldsymbol{X}}{h k}^h\right)\left(\boldsymbol{W}{h(k+1)}-\boldsymbol{W}{h k}\right)$ where the numerical solution is $\widehat{\boldsymbol{X}}{h k}^h \approx \boldsymbol{X}_{h k}, k=0,1,2, \ldots$, with initial value $\widehat{\boldsymbol{X}}_0=\boldsymbol{x}_0$.

An example of the result of the Euler-Maruyama method is shown in Figure 7.5.1. This shows solutions for
$$d X_t=r X_t d t+s X_t d W_t, \quad X_0=1,$$
with $r=s=1$. This is a model for price evolution with a natural interest or inflation rate of $r$ and volatility $s$. The trajectories in Figure 7.5.1(a) use the same underlying Wiener process. The details, in case you want to reconstruct this solution, are as follows: the approximate Wiener process was created using Matlab’s randn function based on the Mersenne Twister generator with seed 95324965 to generate $2^{20}$ pseudorandom distributed according to the $\operatorname{Normal}\left(0,2^{-20}\right)$ distribution; then cumulative sums were used to give the values of $W_{h k}$ with $h=2^{-20}$ and $k=0,1,2, \ldots, 2^{20}$.
The convergence of the trajectories, rather than just the statistical properties of the trajectories, illustrates strong convergence of the approximate trajectories for this method. That is, there are positive constants $C$ and $h_0$ where
$$\max {0 \leq h k \leq T} \mathbb{E}\left[\left|\widehat{\boldsymbol{X}}{h k}^h-\boldsymbol{X}_{h k}\right|\right] \leq C h^\alpha \quad \text { for all } 0<h \leq h_0 \text {. }$$

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Stratonovich Integrals and Differential Equations

Itô 积分基于以下形式的有限和的近似值
$$\int_a^b f\left(X_t\right) d Y_t \approx \sum_{k=0}^{n-1} f\left(X_{a+k h}\right)\left(Y_{a+(k+1) h}-Y_{a+k h}\right)$$

$$\int_a^b f\left(X_t\right) \circ d Y_t \approx \sum_{k=0}^{n-1} \frac{1}{2}\left[f\left(X_{a+k h}\right)+f\left(X_{a+(k+1) h}\right)\right]\left(Y_{a+(k+1) h}-Y_{a+k h}\right)$$

$$\int_0^t W_s \circ d W_s \approx \sum_{k=0}^{n-1} \frac{1}{2}\left(W_{h(k+1)}+W_{h k}\right)\left(W_{h(k+1)}-W_{h k}\right) \quad=\sum_{k=0}^{n-1} \frac{1}{2}\left(W_{h(k+1)}^2\right.$$

Stratonovich 随机微分方程具有以下形式
$$d \boldsymbol{X}_t=\boldsymbol{f}\left(\boldsymbol{X}_t\right) d t+\sigma\left(\boldsymbol{X}_t\right) \circ d \boldsymbol{W}_t, \quad \boldsymbol{X}_0=\boldsymbol{x}_0 .$$

$$\boldsymbol{X}_t=\boldsymbol{x}_0+\int_0^t \boldsymbol{f}\left(\boldsymbol{X}_s\right) d s+\int_0^t \sigma\left(\boldsymbol{X}_t\right) \circ d \boldsymbol{W}_t$$

## 数学代写|数值分析代写numerical analysis代考|Euler–Maruyama Method

Euler-Maruyama方法本质上是将Euler方法应用于随机微分方程
$$d \boldsymbol{X} t=\boldsymbol{f}\left(t, \boldsymbol{X}t\right) d t+\sigma\left(t, \boldsymbol{X}_t\right) d \boldsymbol{W}_t, \quad \boldsymbol{X}_0=\boldsymbol{x}_0 .$$ 对于步长 $h>0$, 该方法由迭代 (7.5.9) 组成 $$\widehat{\boldsymbol{X}} h(k+1)^h=\widehat{\boldsymbol{X}} h k^h+h \boldsymbol{f}\left(t, \widehat{\boldsymbol{X}} h k^h\right)+\sigma\left(h k, \widehat{\boldsymbol{X}} h k^h\right)(\boldsymbol{W} h(k+1)-\boldsymbol{W} h k)$$ 其中数值解是 $\widehat{\boldsymbol{X}} h k^h \approx \boldsymbol{X}{h k}, k=0,1,2, \ldots$, 有初值 $\widehat{\boldsymbol{X}}0=\boldsymbol{x}_0$. Euler-Maruyama 方法的结果示例如图 7.5.1 所示。这显示了解决方案 $$d X_t=r X_t d t+s X_t d W_t, \quad X_0=1,$$ 和 $r=s=1$. 这是一个价格演变模型，自然利率或通货膨胀率为 $r$ 和波动性 $s$. 图 7.5.1(a) 中的轨迹使用相同的底层维纳过程。详细信息，如果你想重建这个解决方案， 如下所示: 近似维纳过程是使用 Matlab 的 randn 函数创建的，该函数基于 Mersenne Twister 生成器，种子为 95324965 以生成 $2^{20}$ 伪随机分布 $\operatorname{Normal}\left(0,2^{-20}\right)$ 分配; 然后 累积总和被用来给出的值 $W{h k}$ 和 $h=2^{-20}$ 和 $k=0,1,2, \ldots, 2^{20}$.

$$\max 0 \leq h k \leq T \mathbb{E}\left[\left|\widehat{\boldsymbol{X}} h k^h-\boldsymbol{X}_{h k}\right|\right] \leq C h^\alpha \quad \text { for all } 0<h \leq h_0$$

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