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首页马尔科夫过程最优停止理论与高维金融衍生品定价
"1999年的文章'Optimal stopping of Markov processes',作者John N. Tsitsiklis和Benjamin Van Roy探讨了离散时间遍历马尔可夫过程的最优停止时间问题,该研究在金融领域,尤其是高维金融衍生品定价中有应用。他们利用希尔伯特空间理论来分析并提出近似算法。" 在本文中,作者深入研究了离散时间的遍历马尔可夫过程,这是一个在随机决策和优化问题中常见的数学模型。他们特别关注了具有折扣奖励的最优停止时间问题,这个问题在金融工程中至关重要,因为许多金融衍生品的价值取决于何时执行或行使权利。 与以往的工作相比,作者引入了一种新颖的方法,将每阶段和终端奖励函数视为特定希尔伯特空间中的元素。希尔伯特空间是泛函分析的一个核心概念,它提供了一个结构严谨的环境来处理无限维向量空间上的线性算子和内积,这对于处理复杂的随机过程非常有用。这种方法不仅简化了贝尔曼方程解的存在性和唯一性的证明,还为研究近似解决方案提供了一个优雅的框架。 为了处理高维度问题,作者提出了一个随机逼近算法,通过调整基函数线性组合的权重来逼近价值函数。他们证明了这个算法在概率上收敛,并且其极限具有一些理想的性质。这为解决实际计算中的复杂问题提供了实用的工具。 通过一个计算案例研究,他们展示了该近似方法的实用性,可能涉及到模拟和估计高维度金融衍生品的价格。这样的案例研究对于理解在真实世界情境下如何应用这些理论至关重要。 这篇论文为理解和求解马尔可夫过程的最优停止问题提供了新的见解和方法,特别是在处理高维金融衍生品定价问题时,这为金融领域的研究和实践提供了有价值的理论支持和算法工具。
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1842 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 10, OCTOBER 1999
2) The stopping time , defined by
is an optimal stopping time. (The minimum of an empty
set is taken to be
.)
3) The function
is equal to [in the sense of ].
B. Preliminaries
Our first lemma establishes that the operator
is a nonex-
pansion in
.
Lemma 1: Under Assumption 1, we have
Proof: The proof of the lemma involves Jensen’s in-
equality and the Tonelli–Fubini theorem. In particular, for any
, we have
The following lemma establishes that is a contraction on
.
Lemma 2: Under Assumptions 1–3, the operator
satisfies
Proof: For any scalars and
It follows that for any and
Given this fact, the result easily follows from Lemma 1.
The fact that is a contraction implies that it has a unique
fixed point in
(by unique here, we mean unique
up to the equivalence classes of
). This establishes part
1) of the theorem.
Let
denote the fixed point of . Let us define a second
operator
by
if
otherwise
(Note that is the dynamic programming operator corre-
sponding to the case of a fixed policy, namely, the policy
corresponding to the stopping time
defined in the statement
of the above theorem.) The following lemma establishes that
is also a contraction, and furthermore, the fixed point of
this contraction is equal to
(in the sense of ).
Lemma 3: Under Assumptions 1–3, the operator
satis-
fies
Furthermore, is the unique fixed point of .
Proof: We have
where the final inequality follows from Lemma 1.
Recall that
uniquely satisfies , or written
differently
This equation can also be rewritten as
if
otherwise
almost surely with respect to
. Note that for almost all (a
set
with ),
if and only if . Hence, satisfies
if
otherwise
almost surely with respect to
, or more concisely,
. Since is a contraction, it has a unique fixed point
in
, and this fixed point is .
C. Proof of Theorem 1
Part 1) of the result follows from Lemma 2. As for Part 3),
we have
if
otherwise
and since is a contraction with fixed point (Lemma 3),
it follows that
We are left with the task of proving Part 2). For any
nonnegative integer
, we have
for some scalar that is independent of , where the equality
follows from the Tonelli–Fubini theorem and stationarity. By
arguments standard to the theory of finite-horizon dynamic
programming
(This equality is simply saying that the optimal reward for an
-horizon problem is obtained by applying iterations of the
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