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# 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|MATH581 Stochastic Differential Equations

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Stochastic Differential Equations

Let us consider the stochastic differential equation (3.5.1) where instead of a Brownian motion as in Chap. 3, here $W=\left(W^1, W^2, \ldots, W^d\right)$ is a amenable semimartingale. The growth estimate (7.2.5) enables one to conclude that in this case too, Theorem $3.30$ is true and the same proof works essentially-using (7.2.5) instead of (3.4.4). Moreover, using random time change, one can conclude that the same is true even when $W$ is any continuous semimartingale. We will prove this along with some results on approximations to the solution of an SDE.

We are going to consider the following general framework for the SDE driven by continuous semimartingales, where the evolution from a time $t_0$ onwards could depend upon the entire past history of the solution rather than only on its current value as was the case in Eq. (3.5.1) driven by a Brownian motion.

Let $Y^1, Y^2, \ldots Y^m$ be continuous semimartingales w.r.t. the filtration $(\mathcal{F}$.). Let $Y=\left(Y^1, Y^2, \ldots Y^m\right)$. Here we will consider an SDE
$$d U_t=b(t, \cdot, U) d Y_t, \quad t \geq 0, \quad U_0=\xi_0$$
where the functional $b$ is given as follows. Recall that $\mathbb{C}d=\mathbb{C}\left([0, \infty), \mathbb{R}^d\right)$. Let $$a:[0, \infty) \times \Omega \times \mathbb{C}_d \rightarrow \mathrm{L}(d, m)$$ be such that for all $\zeta \in \mathbb{C}_d$, $(t, \omega) \mapsto a(t, \omega, \zeta)$ is an r.c.l.l. $(\mathcal{F}$.$) adapted process$ and there is an increasing r.c.l.l. adapted process $K$ such that for all $\zeta_1, \zeta_2 \in \mathbb{C}_d$, $$\sup {0 \leq s \leq t}\left|a\left(s, \omega, \zeta_2\right)-a\left(s, \omega, \zeta_1\right)\right| \leq K_t(\omega) \sup _{0 \leq s \leq t}\left|\zeta_2(s)-\zeta_1(s)\right| .$$
Finally, $b:[0, \infty) \times \Omega \times \mathbb{C}_d \rightarrow \mathrm{L}(d, m)$ be given by
$$b(s, \omega, \zeta)=a(s-, \omega, \zeta)$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Pathwise Formula for Solution of SDE

In this section, we will consider the SDE
$$d V_t=f(t-, H, V) d X_t$$
for an $\mathbb{R}^d$-valued process $V$ where $f:[0, \infty) \times \mathbb{D}r \times \mathbb{C}_d \mapsto \mathrm{L}(d, m), H$ is an $\mathbb{R}^r$-valued r.c.l.l. adapted process, $X$ is a $\mathbb{R}^m$-valued continuous semimartingale. Here $\mathbb{D}_r=\mathbb{D}\left([0, \infty), \mathbb{R}^r\right), \mathbb{C}_d=\mathbb{C}\left([0, \infty), \mathbb{R}^d\right)$. For $t<\infty, \zeta \in \mathbb{C}_d$ and $\gamma \in \mathbb{D}_r$, let $\gamma^t(s)=\gamma(t \wedge s)$ and $\zeta^t(s)=\zeta(t \wedge s)$. We assume that $f$ satisfies \begin{aligned} f(t, \gamma, \zeta) & =f\left(t, \gamma^t, \zeta^t\right), \quad \forall \gamma \in \mathbb{D}_r, \zeta \in \mathbb{C}_d, 0 \leq t<\infty \ t & \mapsto f(t, \gamma, \zeta) \text { is an r.c.l.l. function } \forall \gamma \in \mathbb{D}_r, \zeta \in \mathbb{C}_d \end{aligned} We also assume that there exists a constant $C_T<\infty$ for each $T<\infty$ such that $\forall \gamma \in \mathbb{D}_r, \zeta_1, \zeta_2 \in \mathbb{C}_d, 0 \leq t \leq T$ $$\left|f\left(t, \gamma, \zeta_1\right)-f\left(t, \gamma, \zeta_2\right)\right| \leq C_T\left(1+\sup {0 \leq s \leq t}|\gamma(s)|\right)\left(\sup _{0 \leq s \leq t}\left|\zeta_1(s)-\zeta_2(s)\right|\right)$$
As in Sect. 6.2, we will now obtain a mapping $\Psi$ that yields a pathwise solution to the SDE (7.4.1).

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Stochastic Differential Equations

$$d U_t=b(t, \cdot, U) d Y_t, \quad t \geq 0, \quad U_0=\xi_0$$

$$a:[0, \infty) \times \Omega \times \mathbb{C}d \rightarrow \mathrm{L}(d, m)$$ 对所有人来说 $\zeta \in \mathbb{C}_d,(t, \omega) \mapsto a(t, \omega, \zeta)$ 是一个 $r c |(\mathcal{F}$. adaptedprocess并且有一个越来越多的 rcll 适应过程K这样对于所 有人 $\zeta_1, \zeta_2 \in \mathbb{C}_d$ $$\sup 0 \leq s \leq t\left|a\left(s, \omega, \zeta_2\right)-a\left(s, \omega, \zeta_1\right)\right| \leq K_t(\omega) \sup {0 \leq s \leq t}\left|\zeta_2(s)-\zeta_1(s)\right| .$$

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