隨機過程應用在財務與人口模型的指數複雜度

Exponential Complexity for Stochastic Differential Equations with Applications in Finance and Population Models

吳承憲(Chen-Hsien Wu)

指導教授 ： 呂育道

國立臺灣大學/電機資訊學院/資訊工程學研究所/碩士(2012年)

https://doi.org/10.6342/NTU.2012.02939

摘要

在這份論文中，我們先複習了伊藤定理(Ito’s lemma)和一些常見的隨機微分方程，例如：線性(linear)、對數常態(lognormal)、方根(square-root)和常數彈性變異數(constant elasticity of variance)隨機微分方程。為了建立樹狀(lattice)結構複雜度計算的基礎，我們設定樹狀結構需滿足的7條公理。
線性隨機微分方程是本論文中極重要的一種隨機過程，一則它的複雜度下界相對容易計算，二則許多隨機微分方程皆可轉換成線性微分方程來估計複雜度。對此，證明Nelson和Ramaswamy提出的轉換方法(NR轉換)具唯一性 可作為複雜度計算的依據。可在建立樹狀結構之前先應用NR轉換來評估是否存在非指數成長的樹狀結構。
我們證明四個隨機過程有指數複雜度： Ornstein–Uhlenbeck過程、CKLS模型、純量(scalar)隨機微分方程和Lewis的3/2隨機波動模型。

關鍵字

隨機過程 ； 樹狀模型 ； 指數複雜度 ； NR轉換

並列摘要

In this thesis, we first review Ito processes and some common stochastic differential equations (SDE): linear SDEs, lognormal linear SDEs, square-root SDEs, and the constant elasticity of variance (CEV) process. Then we lay out the axioms for lattices, which give us the foundations to talk about the lattice complexity.
The linear SDE plays a key role because, on one hand, it is easier to derive a lower bound for it, and on the other hand, many SDEs can be reduced to linear SDEs in deriving their lower bounds. The uniqueness of the Nelson and Ramaswamy transformation (NR transformation) that transforms an SDE into one with a constant diffusion term is proved. Its application is necessary before a lattice is built if there is any hope for the lattice to avoid exponential complexity.
Exponential lower bounds are given for the following four SDEs: the Ornstein–Uhlenbeck process, the CKLS model, the scalar SDE, and Lewis’s 3/2 model.

並列關鍵字

linear SDE ； Axioms of lattices ； exponential complexity ； NR transformation

參考文獻

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