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# 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|MAST90059 The Kolmogorov equations

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## 数学代写|随机微积分代写Stochastic Calculus代考|Explicit examples

Explicit examples. We have already seen examples which canbe solved explicitly, in the form of a well-defined (thus, unique) function of a BM or of its paths. This gives us our first notion of a “solution”, and corresponds to what we shall (in due course) call a strong solution.

Let $\left(t_0, x\right) \in[0, \infty) \times \mathbb{R}$ be given. A solution to (7.1) with initial condition $X_{t_0}=x$ is a process $\left(X_t\right){t \geq t_0}$ satisfying $$X_t=x+\int{t_0}^t a\left(s, X_s\right) \mathrm{d} s+\int_{t_0}^t b\left(s, X_s\right) \mathrm{d} W_s, \quad t \geq t_0$$
which is adapted to the augmented Brownian filtration $\mathbb{F}$.
Here are some (almost trivial) examples.
Example 7.1 (Drifted BM). Take the SDE to be
$$\mathrm{d} X_t=\mu \mathrm{d} t+\mathrm{d} W_t, \quad X_{t_0}=x, \quad(\mu \in \mathbb{R})$$
This has solution
$$X_t=x+\mu\left(t-t_0\right)+W_t-W_{t_0}, \quad t \geq t_0$$
Example $7.2$ (Geometric BM). Take the SDE to be
$$\mathrm{d} X_t=\mu X_t \mathrm{~d} t+\sigma X_t \mathrm{~d} W_t, \quad X_{t_0}=x, \quad(\mu \in \mathbb{R}, \sigma>0)$$
This has solution
$$X_t=x \exp \left(\left(\mu-\frac{1}{2} \sigma^2\right)\left(t-t_0\right)+\sigma\left(W_t-W_{t_0}\right)\right), \quad t \geq t_0$$
Example $7.3$ (The stochastic exponential). Take the SDE to be
$$\mathrm{d} X_t=\theta X_t \mathrm{~d} W_t, \quad X_{t_0}=x, \quad(\theta \in \mathbb{R}) .$$
(We recognise this as Example $7.2$ with $\mu \rightarrow 0$ and $\sigma \rightarrow \theta \in \mathbb{R}$ ).
This has solution
$$X_t=x \exp \left(\theta\left(W_t-W_{t_0}\right)-\frac{1}{2} \theta^2\left(t-t_0\right)\right)=x \frac{\mathcal{E}(\theta W)t}{\mathcal{E}(\theta W){t_0}}, \quad t \geq t_0$$
We also know that we can solve the SDE with an adapted process $\theta$ satisfying $\int_0^* \theta_s^2 \mathrm{~d} s$. The SDE
$$\mathrm{d} X_t=\theta_t X_t \mathrm{~d} W_t, \quad X_0=x,$$
has solution
$$X_t=x \mathcal{E}(\theta \cdot W)_t=\exp \left(\int_0^t \theta_s \mathrm{~d} W_s-\frac{1}{2} \int_0^t \theta_s^2 \mathrm{~d} s\right), \quad t \geq 0$$

## 数学代写|随机微积分代写Stochastic Calculus代考|Notions of solutions of SDE

Notions of solutions of SDEs. We place ourselves on a filtered space $(\Omega, \mathcal{F}, \mathbb{F}=$ $\left.\left(\mathcal{F}t\right){t \geq 0}, \mathbb{P}\right)$, with $\mathbb{F}$ satisfying the usual conditions. We have an $\mathbb{F}$-BM $W$, and an SDE of the form (7.1), repeated below:
(7.3) $\mathrm{d} X_t=a\left(t, X_t\right) \mathrm{d} t+b\left(t, X_t\right) \mathrm{d} W_t, \quad X_0=x$.
In this section, we consider what we mean by a “solution”. Whatever notion we come up with, it will certainly be a continuous, $\mathbb{F}$-adapted process. The question is whether the filtration is specified at the outset, or whether it is part of the solution.

Thus, in broad terms, a solution to (7.3), with given functions $a(\cdot, \cdot), b(\cdot, \cdot)$ (the drift and diffusion (or dispersion) coefficients, respectively) is a pair $(X, W)$ of adapted processes on the stochastic basis $\left(\Omega, \mathcal{F}, \mathbb{F}=\left(\mathcal{F}t\right){t \geq 0}, \mathbb{P}\right)$, such that:

$W$ is a BM.

$X$ is given by the integral version of (7.3):
$$X_t=x+\int_0^t a\left(s, X_s\right) \mathrm{d} s+\int_0^t b\left(s, X_s\right) \mathrm{d} W_s, \quad t \geq 0 .$$
Notice that we only specified the functions $a(\cdot, \cdot), b(\cdot, \cdot)$, and that $W$ as well as $X$ are (in principle) outputs (or solutions) to the problem. The notion of a strong solution applies when $W$ is specified in advance, and the only output is $X$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|EXPLICIT EXAMPLES

$$X_t=x+\int t_0{ }^t a\left(s, X_s\right) \mathrm{d} s+\int_{t_0}^t b\left(s, X_s\right) \mathrm{d} W_s, \quad t \geq t_0$$

$$\mathrm{d} X_t=\mu \mathrm{d} t+\mathrm{d} W_t, \quad X_{t_0}=x, \quad(\mu \in \mathbb{R})$$

$$X_t=x+\mu\left(t-t_0\right)+W_t-W_{t_0}, \quad t \geq t_0$$

$$\mathrm{d} X_t=\mu X_t \mathrm{~d} t+\sigma X_t \mathrm{~d} W_t, \quad X_{t_0}=x, \quad(\mu \in \mathbb{R}, \sigma>0)$$

$$X_t=x \exp \left(\left(\mu-\frac{1}{2} \sigma^2\right)\left(t-t_0\right)+\sigma\left(W_t-W_{t_0}\right)\right), \quad t \geq t_0$$

$$\mathrm{d} X_t=\theta X_t \mathrm{~d} W_t, \quad X_{t_0}=x, \quad(\theta \in \mathbb{R}) .$$
WerecognisethisasExample $\$ 7.2 \$$with \ \mu \rightarrow 0 \$$ and $\$ \sigma \rightarrow \theta \in \mathbb{R} \$$. 这个有解$$
X_t=x \exp \left(\theta\left(W_t-W_{t_0}\right)-\frac{1}{2} \theta^2\left(t-t_0\right)\right)=x \frac{\mathcal{E}(\theta W) t}{\mathcal{E}(\theta W) t_0}, \quad t \geq t_0
$$我们还知道我们可以通过调整流程来解决 \mathrm{SDE} \theta 令人满意 \int_0^* \theta_s^2 \mathrm{~d} s . \mathrm{SDE}$$
\mathrm{d} X_t=\theta_t X_t \mathrm{~d} W_t, \quad X_0=x,
$$有解决办法$$
X_t=x \mathcal{E}(\theta \cdot W)_t=\exp \left(\int_0^t \theta_s \mathrm{~d} W_s-\frac{1}{2} \int_0^t \theta_s^2 \mathrm{~d} s\right), \quad t \geq 0
$$## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|NOTIONS OF SOLUTIONS OF SDE \backslash mathrm {\mathrm{d}} \mathrm{t}+\mathrm{b} \backslash 莁 \mathrm{t}, \mathrm{X}{-} \mathrm{t} \backslash 右 \operatorname{mathrm}{\mathrm{d}} \mathrm{W}{-} \mathrm{t}, \backslash quad \mathrm{X}_{-} 0=\mathrm{x} .Inthissection, weconsiderwhatwemeanbya “solution”. Whatevernotionwecomeupwith, itwillcertainlybeacontinuous, \mathbb{F }-适应过程。问题 是过滤是否在一开始就指定，或者它是否是解决方客的一部分。 因此，从广义上讲，一个解决方案 7.3, 具有给定的功能 a(\cdot, \cdot), b(\cdot, \cdot) thedriftanddiffusion(ordispersion系数，分别) 是一对 (X, W) 随机基础上的适应过程 \\left } \backslash \backslash \text { mega, \mathcal } { \mathrm { F } } , \backslash \text { mathbb } { \mathrm { F } } = \backslash \text { left } ( \backslash \text { mathcal } { \mathrm { F } } t | \text { right } { { \mathrm { t } \backslash \text { geq } 0 } \text { , \mathbb } { \mathrm { P } } \backslash \text { right } ) \ \text { , 这样: } W 是一个BM。 X 由积分版本给出 7.3 :$$
X_t=x+\int_0^t a\left(s, X_s\right) \mathrm{d} s+\int_0^t b\left(s, X_s\right) \mathrm{d} W_s, \quad t \geq 0


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