# 统计代写|随机控制代写Stochastic Control代考|MATH8170

#### Doug I. Jones

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## 统计代写|随机控制代写Stochastic Control代考|Mean-Variance Portfolio Selection

The following example is from [FØS3].
Consider a financial market with two investment possibilities, one risk free (e.g.. a bond or bank account) and one risky (e.g., a stock), whose prices $S_{0}(t), S_{1}(t)$ at time $t \in[0, T]$ are given by
(bond) $\mathrm{d} S_{0}(t)=\rho_{t} S_{0}(t) \mathrm{d} t, \quad S_{0}(0)=1$
(stock) $\mathrm{d} S_{1}(t)=S_{1}\left(t^{-}\right)\left[\mu_{t} \mathrm{~d} t+\sigma_{t} \mathrm{~d} B(t)+\int_{\mathbb{R}} \gamma(t, z) \tilde{N}(\mathrm{~d} t, \mathrm{~d} z)\right], \quad S_{1}(0)>0$,
$(5.2 .42)$
where $\rho_{t}>0, \mu_{t}, \sigma_{t}$, and $\gamma(t, z) \geq-1$ are given bounded deterministic functions. We assume that the function
$$t \rightarrow \int_{\mathbb{R}} \gamma^{2}(t, z) \nu(\mathrm{d} z) \text { is locally bounded. }$$

We may regard this market as a jump diffusion extension of the classical BlackScholes market (see e.g. [Ø1]).

A portfolio in this market is a two-dimensional càdlàg and adapted process $\theta(t)=$ $\left(\theta_{0}(t), \theta_{1}(t)\right)$ giving the number of units of bonds and stocks, respectively, held at time $t$ by an agent.
The corresponding wealth process $X(t)=X^{(\theta)}(t)$ is defined by
$$X(t)=\theta_{0}(t) S_{0}(t)+\theta_{1}(t) S_{1}(t), \quad t \in[0, T]$$

## 统计代写|随机控制代写Stochastic Control代考|The Maximum Principle with Infinite Horizon

In the previous section we have presented maximum principles for optimal control of an Itô-Lévy process with a performance functional defined within a finite time interval $[0, T]$, and we saw that these principles involved BSDEs in the adjoint processes $(p(t), q(t), r(t))$ with given terminal value $p(T)$ at time $T$. If the performance functional is over the infinite horizon $[0, \infty)$, however, it is not clear how to formulate the terminal value condition in the corresponding infinite horizon BSDE.

In this section we will study this situation. Our system is a controlled Itô-Lévy process of the form
\left{\begin{aligned} d X(t)=& b(t, X(t), u(t), \omega) d t+\sigma(t, X(t), u(t), \omega) d B(t) \ &+\int_{\mathbb{R}{0}} \theta(t, X(t), u(t), \zeta, \omega) \tilde{N}(d t, d \zeta) ; t \in[0, \infty) \ X(0)=& x ; \end{aligned}\right. where $u(t)$ is the control process and \begin{aligned} &b:[0, \infty) \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathcal{U} \times \Omega \rightarrow \mathbb{R} \ &\sigma:[0, \infty) \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathcal{U} \times \Omega \rightarrow \mathbb{R} \ &\theta:[0, \infty) \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathcal{U} \times \mathbb{R}{0} \times \Omega \rightarrow \mathbb{R} \end{aligned}
are given functions such that for all $t, b(t, x, u), \sigma(t, x, u)$ and $\theta(t, x, u, \zeta)$ are $\mathcal{J}{t^{-}}$ predictable processes for all $x \in \mathbb{R}$ and $u \in \mathcal{U}$ and $\zeta \in \mathbb{R}{0}$. We assume that $b, \sigma, \theta$ are $C^{1}$ (i.e. continuously differentiable) with respect to $x, u$, for all $t, \zeta$ and a.a. $\omega$. Let $\mathbb{G}=\left{\mathcal{G}{t}\right}{t \geq 0}$ be a given subfiltration of $\mathbb{F}$, in the sense that $\mathcal{G}{t} \subset \mathcal{F}{t}$ for all $t$. We assume that $\mathbb{G}$ satisfies the usual conditions. The sigma-algebra $\mathcal{G}{t}$ represents the information available to the controller at time $t$. Let $\mathcal{U}$ be a non-empty subset of $\mathbb{R}$. We let $\mathcal{A}{\mathrm{G}}$ denote a given family of admissible $\mathbb{G}$-predictable control processes $u$ such that there exists a unique associated solution $X(t)=X^{(u)}(t)$ of (5.3.1) with $E\left[\int_{0}^{T} X^{2}(t) d t\right]<\infty$, for all $T<\infty$.

## 统计代写|随机控制代写Stochastic Control代考|Mean-Variance Portfolio Selection

(bond)给出 $\mathrm{d} S_{0}(t)=\rho_{t} S_{0}(t) \mathrm{d} t, \quad S_{0}(0)=1$
(股票)
$\mathrm{d} S_{1}(t)=S_{1}\left(t^{-}\right)\left[\mu_{t} \mathrm{~d} t+\sigma_{t} \mathrm{~d} B(t)+\int_{\mathbb{R}} \gamma(t, z) \tilde{N}(\mathrm{~d} t, \mathrm{~d} z)\right], \quad S_{1}(0)>0$
$(5.2 .42)$

$t \rightarrow \int_{\mathbb{R}} \gamma^{2}(t, z) \nu(\mathrm{d} z)$ is locally bounded.

$$X(t)=\theta_{0}(t) S_{0}(t)+\theta_{1}(t) S_{1}(t), \quad t \in[0, T]$$

## 统计代写|随机控制代写Stochastic Control代考|The Maximum Principle with Infinite Horizon

$\$ \$$Veft { 的受控 Itô-Lévy 过程$$
d X(t)=b(t, X(t), u(t), \omega) d t+\sigma(t, X(t), u(t), \omega) d B(t)+\int_{\mathbb{R} 0} \theta(t, X(t), u(t)
$$【正确的。 where \ u(t) \ i s t h e c o n t r o l p r o c e s s a n d$$
b:[0, \infty) \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathcal{U} \times \Omega \rightarrow \mathbb{R} \quad \sigma:[0, \infty) \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathcal{U} \times \Omega \rightarrow \mathbb{R}
$$\ \$$

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