測度值隨機過程與財務應用(II)

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行政院國家科學委員會專題研究計畫期中進度報告

測度值隨機過程與財務應用(2/3)

計畫類別：個別型計畫

計畫編號： NSC92-2115-M-009-005-

執行期間： 92 年 08 月 01 日至 93 年 07 月 31 日

執行單位：國立交通大學應用數學系

計畫主持人：許元春

報告類型：精簡報告

處理方式：本計畫可公開查詢

中華民國 93 年 5 月 27 日

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中文摘要：

Affine 隨機過程是取值在

R

₊m

×

R

n

上的一種馬可夫過程。這種隨機過程具有

多樣的隨機性質(如平均迴歸、跳躍、隨機波動)及良好的可分析性，所以在財務

領域裡有非常廣泛的應用。最近 Duffie, Filipovic 和 Schachermayer[2003]建構所

有平滑的 affine 隨機過程。同時，他們也建立平滑 affine 隨機過程和超過程的密

切關聯。應用這個關係，我們建構更一般的 affine 隨機過程及研究他們的路徑性

質及其在財務上的應用。

關鍵詞：affine 過程、超過程、有限基空間、利率結構、債券選擇權定價、違約

風險

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YUAN-CHUNG SHEU

Abstract. Affine processes is a class of Markov processes taking values in Rm

+× Rn. The rich variety of alternative types of random behavior(e.g., mean

reversion, stochastic volatility, and jumps) and analytically tractable for affine processes make them ideal models for financial applications. Duffie, Filipovic and Schachermayer[DFS03] characterized all regular affine processes. Connec-tions between regular affine processes and superprocesses with a finite base space were also established. Based on this observation, we construct more general affine processes and investigate sample path properties and financial applications of these processes.

1. Affine Processes

A Markov transition function in a measurable space (E, B) is a function p(r, x; t, B), r <

t ∈ R, x ∈ E, B ∈ B which is B-measurable in x and which is a measure in B subject

to the conditions:

(A) R_Ep(r, x; t, dy)p(t, y; u, B) = p(r, x; u, B) for all r < t < u, x ∈ E and all B ∈ B.

(B) p(r, x; t, E) ≤ 1 for all r, x, t.

To every Markov transition function p there corresponds a family of linear operator

Tr

t acting on functions by the formula

Tr tf (x) =

Z

E

p(r, x; t, dy)f (y).

It follows from (B) that Tr

sTts= Ttr for all r < s < t ∈ R. We call T the Markov

semigroup corresponding to the transition function p.

We assume that (E, B) is a measurable Luzin space. To every Markov transition function p there corresponds a Markov process ξ = (ξt, F(I), Πr,x) such that

Πr,x{ξt∈ B} = p(r, x; t, B), Πr,x{ξt1 ∈ B1, · · · , ξtn∈ Bn} = Z B1×···×Bn p(r, x; t1, dy1)p(t1, y1; t2, dy2) · · · p(tn−1, yn−1; tn, dyn) for n ≥ 2, t1< t2< · · · < tn.

If the transition function p(r, x; t, dy) satisfies a condition

p(r, x; t, B) = p(r + s, x; t, B)

for all r, s, t, B, then ξ = (ξt, F(I), Πr,x) is time homogeneous. In this case we

consider only process ξ = (ξt, Ft, Πx) where Πx = Π0,x, Ft = F[0, t], and ξt is

defined for all t ≥ 0.

Key words and phrases. affine process, superprocess, finite base space, term structure of in-terest rates, bond option pricing, default risk.

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2 YUAN-CHUNG SHEU

From this point on we consider E = Rm

+ × Rn and write d = m + n. We say

that ξ = (ξt, F(I), Πr,x) is affine if for every r < t ∈ R and λ ∈ Cd, there exists

v(r, t, λ) ∈ C and u(r, t, λ) ∈ Cd _{such that}

Πr,x[exp{i < λ, ξt>}] = exp{v(r, t, λ) + u(r, t, λ) · x}

for all x ∈ E. Clearly if ξ is time homogeneous, then we have u(r, t, λ) = u(t − r, λ) and v(r, t, λ) = v(t − r, λ).

Example (Ornstein-Uhlenbeck process) Consider an Ornstein-Uhlenbeck process

dξt= α(l − ξt)dt + σdwt

where w is a standard Brownian and α, l and σ are positive constants. Through an application of Ito’s formula, we get

ξt= e−αt[ξ0+ l(eαt− 1) + σ

Z t 0

eαs_dw s].

This implies that ξtis normally distributed with mean

Πxξt= e−αt[x + l(eαt− 1)] and variance V arξt= σ 2 2α(1 − e −2αt_).

This implies that

Πxeiλξt = exp{v(t, λ) + u(t, λ)x}

with v(t, λ) = iλe−αt_l(eαt_{− 1) −} λ2α2 4α (1 − e −2αt₎ and u(t, λ) = iλe−αt_.

Example (Feller’s diffusions)

Feller considered a class of processes that includes the square-rot diffusions

dξt= α(l − ξt)dt + σ

p

ξtdwt

where w is a standard Brownian. We consider the case that α, l and σ are pos-itive constants. Based on results of Feller[Fe51], Cox, Ingersoll and Ross[CIR85] noted that the distribution of ξtgiven ξu for some u < t, is distributed as σ2(1 −

e−α(t−u)_{)/4α times a noncentral chi-square distribution χ}2

ν(λ) with degree of

free-dom ν = 4lα σ2 and noncentrality λ = 4αe −α(t−u) σ2_{(1 − e}−α(t−u)₎ξu.

Therefore the Laplace transform of ξtis given by

Πxe−λξt= 1

(2λc + 1)2lα/σ2exp{−

λcf

2λc + 1} with c = σ2_{/4α(1 − e}−αt_{) and f = 4xα/(σ}2_(eαt_{− 1)).}

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2. Characterizations and Applications of affine processes When m = 1 and n = 0, the affine process ξ takes values in R+ and is also

called a continuous- state process with immigration. It was first studied by Kawazu and Watanabe[KW71] as a continuous limit of Galton-Watson branchinh processes with immigration. Kawazu and Watanabe[KW71] showed that if ξ is a stocastically continuous affine process in R+, then for every λ > 0, t > 0 and x ∈ R+, we have

Πxe−λξt = exp{−utx −

Z _t

0

φ(us)ds}

where ut= u(t, λ) satisfies

du

dt = −ϕ(u) , u(0) = λ

with

ϕ(u) = αu2− βu − γ +

Z

R+

[e−uy− 1 + u(1 ∧ y)]µ(dy)

and

φ(u) = c + bu +

Z

R+

(1 − e−uy_)ν(dy).

( Here we assume that

α ≥ 0, γ ≥ 0, b ≥ 0, c ≥ 0, β ∈ R

and µ, ν are two measures on (0, ∞0 satisfying Z _∞ 0 (1 ∧ y)µ(dy) < ∞ and Z _∞ 0 (1 ∧ y)ν(dy) < ∞.)

For general m and n, Duffie, Filipovic and Schachermayer[DFS03] characterized all regular affine processes. In particular they obtained that if ξ is regular, then

Πxe<λ,xt> = exp{< u(t, λ), x > +

Z _t

0

φ(u(s, λ))ds}

where u satisfies some generalized Riccati equations and φ is in a class of nonlinear functions. It is worth noting that if n = 0, then u solves the differential log-Laplace equation corresponging to some superprocess with a finite base space. Based on this observation, we can construct more general affine processes through general superprocesses. We also study sample path properties for general affine processes. The rich variety of alternative types of random behavior(e.g., mean reversion, stochastic volatility, and jumps) and analytically tractable for affine processes make them ideal models for financial applications(see, e.g., Duffie, Filipovic and Schacher-mayer[DFS03] and references therein.)

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4 YUAN-CHUNG SHEU

References

[1] J.C. Cox, J.E. Ingersoll and S.A.Ross, A theory of the term structure of interest rates, Econo-metrica, 53 (1985), 129–151.

[2] D.Duffie, D.Filipovic and W. Schachermayer, Affine processes and applications in finance, Anna. Applied Probab., 13 (2003), 984–1053.

[3] E.B.Dynkin, An introduction to branching measure-valued processes, American Mathematical Society, 1994.

[4] W. Feller, Two singular diffusion problems, Annals of Mathematics, 54 (1951), 173–182. [5] K. Kawazu and S.Watanabe, Branching processes with immigration and related limit theorems,

Theory Probab. Appl., 16 (1971), 36–54.

Department of Applied Mathematics, National Chiao-Tung University, Hsinchu, Tai-wan