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# 数学代写|随机分析代写Stochastic Calculus代考|GRA6550 Sufficient and Necessary Condition

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## 数学代写|随机分析代写Stochastic Calculus代考|Sufficient and Necessary Condition

We first introduce the generator of the controlled state process. Given $\mathbf{u} \in \mathbf{U}$ and interval $[a, b] \subseteq[0, T]$, consider $\xi$ that maps $(t, x) \in[a, b] \times \mathbb{X}$ to $\xi(t, x) \in \mathbb{R}$. Suppose $\xi \in \mathfrak{C}^{1,2}([a, b] \times \mathbb{X})$, and denote by $\xi_t, \xi_x$, and $\xi_{x x}$ respectively its first-order partial derivative in $t$, first-order partial derivative in $x$, and second-order partial derivative in $x$. Define the following generator:
$$\begin{array}{r} \mathscr{A}^{\mathbf{u}} \xi(t, x)=\xi_t(t, x)+\xi_x(t, x) \mu^{\mathbf{u}}(t, x)+\frac{1}{2} \operatorname{tr}\left(\xi_{x x}(t, x)^{\top} r^{\mathbf{u}}(t, x)\right), \ t \in[a, b], x \in \mathbb{X} \end{array}$$
For each fixed $(\tau, y) \in[0, T) \times \mathbb{X}$, denote
\begin{aligned} &f^{\tau, y}(t, x):=\mathbb{E}{t, x}\left[F\left(\tau, y, X^{\hat{\mathbf{u}}}(T)\right)\right] \ &g(t, x):=\mathbb{E}{t, x}\left[X^{\hat{\mathbf{u}}}(T)\right], t \in[0, T], x \in \mathbb{X} \end{aligned}
In addition, for fixed $(\tau, y) \in[0, T) \times \mathbb{X}$ and $s \in[0, T]$, denote
$$c^{\tau, y, s}(t, x):=\mathbb{E}_{t, x}\left[C^{\tau, y, \hat{\mathbf{u}}}\left(s, X^{\hat{\mathbf{u}}}(s)\right)\right], t \in[0, s], x \in \mathbb{X}$$

## 数学代写|随机分析代写Stochastic Calculus代考|Extended HJB

Define the continuation value of a strategy $\hat{\mathbf{u}}$, denoted as $V^{\hat{\mathbf{u}}}(t, x),(t, x) \in[0, T] \times \mathbb{X}$, to be the objective value over time and state under this strategy, i.e.,
$$V^{\hat{\mathbf{u}}}(t, x):=J(t, x ; \hat{\mathbf{u}})=H^{t, x}(t, x)+G(t, x, g(t, x)),$$
where
\begin{aligned} H^{\tau, y}(t, x): &=\mathbb{E}{t, x}\left[\int_t^T C^{\tau, y, \hat{\mathbf{u}}}\left(s, X^{\hat{\mathbf{u}}}(s)\right) d s+F\left(\tau, y, X^{\hat{\mathbf{u}}}(T)\right)\right] \ &=\int_t^T c^{\tau, y, s}(t, x) d s+f^{\tau, y}(t, x) \end{aligned} Assuming certain regularity conditions and applying the operator $\mathscr{A}^u$ to $V^{\hat{\mathbf{u}}}(t, x)$, we derive \begin{aligned} \mathscr{A}^u V^{\hat{\mathbf{u}}}(t, x) &=-C^{t, x, \hat{\mathbf{u}}}(t, x)+\int_t^T \mathscr{A}^u c^{t, x, s}(t, x) d s+\mathscr{A}^u f^{t, x}(t, x) \ &+G_z(t, x, g(t, x)) \mathscr{A}^u g(t, x)+\mathscr{A}{\tau, y}^u H^{t, x}(t, x)+\mathscr{A}{\tau, y}^u G(t, x, g(t, x)) \ &+\operatorname{tr}\left(\left(H{x y}^{t, x}(t, x)+G_{z y}(t, x, g(t, x))^{\top} g_x(t, x)\right)^{\top} Y^u(t, x)\right) \ &+\frac{1}{2} G_{z z}(t, x, g(t, x)) \operatorname{tr}\left(g_x(t, x) g_x(t, x)^{\top} \Upsilon^u(t, x)\right) \end{aligned}
where $H_{x y}^{\tau, y}(t, x)$ denotes the cross partial derivative of $H^{\tau, y}(t, x)$ in $x$ and $y, G_{z y}(\tau, y, z)$ the cross partial derivative of $G(\tau, y, z)$ in $z$ and $y$, and $G_{z z}(\tau, y, z)$ the second-order derivative of $G(\tau, y, z)$ in $z$. For each fixed $(t, x), \mathscr{A}{\tau, y}^u H^{\tau, y}(t, x)$ denotes the generator of $\mathscr{A}^u$ applied to $H^{\tau, y}(t, x)$ as a function of $(\tau, y)$, i.e., $\mathscr{A}{\tau, y}^u H^{\tau, y}(t, x):=\mathscr{A}^u \ell(\tau, y)$, where $\ell(\tau, y):=H^{\tau, y}(t, x),(\tau, y) \in[0, T) \times \mathbb{X}$, and $\mathscr{A}_{\tau, y}^u G(\tau, y, g(t, x))$ is defined similarly.

## 数学代写|随机分析代写STOCHASTIC CALCULUS代考|SUFFICIENT AND NECESSARY CONDITION

$$\mathscr{A}^{\mathrm{u}} \xi(t, x)=\xi_t(t, x)+\xi_x(t, x) \mu^{\mathrm{u}}(t, x)+\frac{1}{2} \operatorname{tr}\left(\xi_{x x}(t, x)^{\top} r^{\mathrm{u}}(t, x)\right), t \in[a, b], x \in \mathbb{X}$$

$\$ \\begin[aligned} \&f {\mid tau, y} t, x:={\operatorname{mathbb}[E}{t, x} \mid l e f t I \& Gt, x:=\mid mathbb {E}{t, x}|l| eft X^{\wedge}{{ hat:{mathbf{u}}}(T)\对 , \mathrm{t} \backslash 在 ,x \backslash in \backslash \operatorname{mathbb}{x} lend { 对齐} Inaddition, forfixed \(\tau, y) \in[0, T) \times \mathbb{X} \ and $\$ s \in[0, T] \$$, denote c^{\wedge}{\backslash tau, y, s] t, x:={\operatorname{mathbb}{E}{[t, x} \mid l left , \mathrm{t} \backslash 在 ,x \backslash in \backslash \operatorname{mathbb}{X} \ \$$

## 数学代写|随机分析代写STOCHASTIC CALCULUS代考|EXTENDED HJB

$$V^{\hat{\mathrm{u}}}(t, x):=J(t, x ; \hat{\mathbf{u}})=H^{t, x}(t, x)+G(t, x, g(t, x)),$$

$$H^{\tau, y}(t, x):=\mathbb{E} t, x\left[\int_t^T C^{\tau, y, \hat{\mathbf{u}}}\left(s, X^{\hat{\mathbf{u}}}(s)\right) d s+F\left(\tau, y, X^{\hat{\mathbf{u}}}(T)\right)\right] \quad=\int_t^T c^{\tau, y, s}(t, x) d s+f^{\tau, y}(t, x)$$

$$\mathscr{A}^u V^{\hat{\mathrm{u}}}(t, x)=-C^{t, x}, \hat{\mathrm{u}}(t, x)+\int_t^T \mathscr{A}^u e^{t, x, s}(t, x) d s+\mathscr{A}^u f^{t, x}(t, x) \quad+G_z(t, x, g(t, x)) \mathscr{A}^u g(t, x)+\mathscr{A} \tau, y^u H^{t, x}(t, x)+\mathscr{A} \tau, y^u G(t, x, g(t, x))+\operatorname{tr}((H$$

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