Posted on Categories:Ordinary Differential Equations, 常微分方程, 数学代写

# 数学代写|常微分方程代考Ordinary Differential Equations代写|Stochastic differential equations

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Stochastic differential equations

The interpretation of the variable $x$ as the time variable implies that we aim at modelling a possible causality relation between the random variables that define a stochastic process. This aim eventually leads to the formulation of an evolution equation, which is the purpose of this section.
Consider two events $A_1$ and $A_2$ in the sample space $\Omega$ of the probability space $(\Omega, \mathcal{F}, P)$. We define the conditional probability that the event $A_2$ occurs, given that the event $A_1$ occurs with probability $P\left(A_1\right)>0$, as follows:
$$P\left(A_2 \mid A_1\right)=\frac{P\left(A_1 \cap A_2\right)}{P\left(A_1\right)} .$$
Now, let us define a partition of $\Omega$ as the family of events $\left{B_j: j \in J\right}$ such that $B_i \bigcap B_j=\emptyset, i \neq j$, and $\Omega=\bigcup_{j \in J} B_j$. Then it holds that for any event $A$ and any partition $\left{B_j: j \in J\right}$, we have
$$P(A)=\sum_{j \in J} P\left(A \mid B_j\right) P\left(B_j\right) .$$
This is the so-called law of total probability.

In the case of two discrete-space random variables $y$ and $w$, the notion of conditional probability can be defined as above. We have
$$P\left(w=v_2 \mid y=v_1\right)=\frac{P\left(y=v_1 \text { and } w=v_2\right)}{P\left(y=v_1\right)},$$
where $v_1$ and $v_2$ are elements of $\operatorname{Range}(y)$ and $\operatorname{Range}(w)$, respectively. On the other hand, in the case of continuous-space random variables, it is possible to define the following conditional PDF of $w$ given the occurrence of the value $v$ of $y$. We have
$$f_w(z \mid y=v)=\frac{f_{w y}(z, v)}{f_y(v)}$$
where $f_{w y}(z, v)$ represents the joint PDF (assuming it exists) of $y$ and $w$, and $z \in \operatorname{Range}(w), v \in \operatorname{Range}(y)$ with $f_y(v)>0$. Notice that $f_{w y}(z, v)=f_w(z \mid y=$ v) $f_y(v)=f_y(v \mid w=z) f_w(z)$. In particular, if $y$ and $w$ are independent, then $f_w(z \mid y=v)=f_w(z)$ and $f_y(v \mid w=z)=f_y(v)$.

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Euler-Maruyama method

This section is devoted to the discussion of a numerical integration scheme for stochastic differential equations. Our focus is the Euler-Maruyama (EM) method, which was developed by Gisiro Maruyama . However, as in the ODE case, there are many other numerical methods for SDE problems available; see $[101,120]$.
We discuss the EM scheme to solve the following jump-diffusion SDE
$$d y(x)=a(x, y(x)) d x+b(x, y(x)) d W(x)+c\left(x, y\left(x^{-}\right)\right) d Y(x), \quad y(0)=y_0$$
Our aim is to determine the sample paths of the solution to (13.13) on a time interval $I=[0, T]$ that is subdivided in $M$ subintervals. Therefore we define the mesh size $h=T / M$ and the following time grid:
$$I_h:=\left{x_i=i h, i=0, \ldots, M\right} \subset I$$
The values of a sample path of our SDE are specified at the points of the time grid $I_h$. We denote with $y_i$ the value of the numerical approximation to $y\left(x_i\right)$ on the grid point $x_i$, where $y(\cdot)$ denotes the solution to (13.13).

To construct the EM approximation to (one realisation of) (13.13), we start from its original integral formulation given by
$$y(x)=y(0)+\int_0^x a(s, y(s)) d s+\int_0^x b(s, y(s)) d W(s)+\int_0^x c\left(s, y\left(s^{-}\right)\right) d Y(s) .$$
This equation is considered in the interval $\left[x_i, x_{i+1}\right]$ (replace 0 by $x_i$ and $x$ by $\left.x_{i+1}\right)$ and approximated by quadrature to define the EM scheme as follows:
$$y_{i+1}=y_i+a\left(x_i, y_i\right) h+b\left(x_i, y_i\right) \Delta W_{i+1}+c\left(x_i, y_i\right) \Delta Y_{i+1}$$
where $\Delta W_{i+1}$ and $\Delta Y_{i+1}$ denote the increments of the Brownian and compound Poisson processes over $\left(x_i, x_{i+1}\right]$. We have $i=0, \ldots, M-1$, and $y_0$ corresponds to the value of the initial condition that can be fixed or given with a normal distribution. We also have $W_0=0$ and $N_0=0$. The EM scheme defined by (13.15) resembles the explicit Euler scheme for the numerical solution of ODE problems. However, in the EM method, we need to determine the random increments $\Delta W_i$ and $\Delta Y_i, i=1, \ldots, M$.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Stochastic differential equations

$$P\left(A_2 \mid A_1\right)=\frac{P\left(A_1 \cap A_2\right)}{P\left(A_1\right)} .$$

$$P(A)=\sum_{j \in J} P\left(A \mid B_j\right) P\left(B_j\right) .$$

$$P\left(w=v_2 \mid y=v_1\right)=\frac{P\left(y=v_1 \text { and } w=v_2\right)}{P\left(y=v_1\right)},$$

$$f_w(z \mid y=v)=\frac{f_{w y}(z, v)}{f_y(v)}$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Euler-Maruyama method

$$d y(x)=a(x, y(x)) d x+b(x, y(x)) d W(x)+c\left(x, y\left(x^{-}\right)\right) d Y(x), \quad y(0)=y_0$$

$$I_h:=\left{x_i=i h, i=0, \ldots, M\right} \subset I$$

$$y(x)=y(0)+\int_0^x a(s, y(s)) d s+\int_0^x b(s, y(s)) d W(s)+\int_0^x c\left(s, y\left(s^{-}\right)\right) d Y(s) .$$

$$y_{i+1}=y_i+a\left(x_i, y_i\right) h+b\left(x_i, y_i\right) \Delta W_{i+1}+c\left(x_i, y_i\right) \Delta Y_{i+1}$$

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