ON DECOMPOSITION OF THE LAST PASSAGE TIME OF DIFFUSIONS MASAHIKO EGAMI1AND RUSUDAN KEVKHISHVILI2 12Graduate School of Economics Kyoto University Sakyo-ku Kyoto 606-8501 Japan

2025-05-02 0 0 629.27KB 27 页 10玖币

侵权投诉

ON DECOMPOSITION OF THE LAST PASSAGE TIME OF DIFFUSIONS

MASAHIKO EGAMI1AND RUSUDAN KEVKHISHVILI2

1,2Graduate School of Economics, Kyoto University, Sakyo-ku, Kyoto, 606-8501, Japan

ABSTRACT. For a regular transient diffusion, we provide a decomposition of its last passage time to a certain state α.

This is accomplished by transforming the original diffusion into two diffusions using the occupation time of the area

above and below α. Based on these two processes, both having a reﬂecting boundary at α, we derive the decomposition

formula of the Laplace transform of the last passage time explicitly in a simple form in terms of Green functions. This

equation also leads to the Green function’s decomposition formula. We demonstrate an application of these formulas

to a diffusion with two-valued parameters.

Keywords: diffusion; last passage time; decomposition; occupation time; Green function

Mathematics Subject Classiﬁcation (2010): 60J60

1. INTRODUCTION

This paper provides a decomposition of the last passage time’s Laplace transform and the Green function for a

general one-dimensional regular transient diffusion. Considering the last passage time to a certain state α, the proof

of the main result in Proposition 1 is based on the transformation of the original diffusion into two diffusions using

the occupation time of the area above and below α. To the best of our knowledge, the related Lemmas 2.1-2.4,

which are the foundations of Proposition 1, are fully original. They also provide new insights on the occupation

and local times of these two diffusions since we handle two local times together in analyzing a killing time and

a last passage time. An immediate and important application of this result is Theorem 1, the decomposition

of the Green function, the latter being one of the fundamental objects in applied mathematics (e.g. differential

equations (Duffy, 2015) and potential theory including its probabilistic approach (Doob, 1984; Chung and Zhao,

1995; Pinsky, 1995)). The decomposition can be done easily as demonstrated in Section 4.1 where we handle the

Ornstein-Uhlenbeck (OU) process: its Green function involves non-elementary hard-to-treat functions.

The decomposition formulas in Proposition 1 and Theorem 1 are new results. With these formulas, the behavior

of diffusions above and below a certain point αcan be analyzed separately from the original diffusion. One example

is to apply this decomposition to a diffusion whose parameters are different above and below α. We demonstrate

this point in Section 5: our results allow us to bypass the need of knowing the explicit transition density of such

diffusions by reducing the original problem to the case of two non-switching diffusions. This feature is particularly

E-mail address:egami@econ.kyoto-u.ac.jp, kevkhishvili.rusudan.2x@kyoto-u.ac.jp.

1Phone: +81-75-753-3430. 2Phone: +81-75-753-3429.

This version: June 21, 2024. The ﬁrst author is in part supported by Grant-in-Aid for Scientiﬁc Research (C) No.23K01467, Japan Society

for the Promotion of Science. The second author was in part supported by JSPS KAKENHI Grant-in-Aid for Early-Career Scientists

No.21K13324 and No.23K12501.

arXiv:2210.01321v4 [math.PR] 19 Jun 2024

2 ON DECOMPOSITION OF THE LAST PASSAGE TIME OF DIFFUSIONS

important because the transition density (and its Laplace transform) in the case of switching parameters is often

unavailable. Let us point out that there is no general established method in the literature for explicitly obtaining the

Green function of diffusions with switching parameters. We provide this method. In the special case of a Brownian

motion with two-valued drift, Beneˇ

s et al. (1980) derives its Green function using the symmetry of the Brownian

motion, the forward Kolmogorov equation (satisﬁed by the transition density), and a linear system of equations

based on various conditions satisﬁed by the density’s Laplace transform. Section 5.1 shows that the decomposition

formula saves these computations. Moreover, a diffusion with switching parameters is useful in modeling real-life

problems. For example, in Section 5.2, we show the last passage time distribution of such a process, quantifying

the leverage effect of high volatility stock. In addition, Proposition 2 derives the killing rate for the diffusion above

level αexplicitly. This is also a new ﬁnding that uncovers a connection between the component diffusions in the

decomposition formula.

The literature for the last passage time (or the last exit time) includes Doob (1957), Nagasawa (1964), Kunita

and Watanabe (1966), Salminen (1984), Rogers and Williams (1994), Chung and Walsh (2004), Revuz and Yor

(2005) as well as the studies referred therein. This object is closely related to the concepts of transience/recurrence,

Doob’s h-transform, time-reversed process, and the Martin boundary theory, and has been an important subject in

the probability literature. Salminen (1984) derives the distribution of the last passage time using the transition

density of the original diffusion, which leads to its Laplace transform in terms of the original Green function

(Borodin and Salminen, 2002, Chapter II.3.20). See also Egami and Kevkhishvili (2020). In contrast to the

existing literature, the study of this paper is the ﬁrst one to investigate the distribution of the last passage time

to αby focusing on the regions above and below αseparately. Propositions 1-3 and Theorem 1 characterize the

behavior of the original process in these two regions, and the decomposition formulas represent a new tool for

further investigation of diffusions.

A wide range of applications of last passage times in ﬁnancial modeling are discussed in Nikeghbali and Platen

(2013). These applications cover the analysis of default risk, insider trading, and option valuation, which we

summarize below. Elliott et al. (2000) and Jeanblanc and Rutkowski (2000) discuss the valuation of defaultable

claims with payoff depending on the last passage time of a ﬁrm’s value to a certain level. See also Coculescu

and Nikeghbali (2012) and Chapters 4 and 5 in Jeanblanc et al. (2009). Egami and Kevkhishvili (2020) develops

a new risk management framework for companies based on the last passage time of a leverage ratio to some

alarming level. They derive the distribution of the time interval between the last passage time and the default time.

Their analysis of company data demonstrates that the information regarding this time interval together with the

distribution of the last passage time is useful for credit risk management. To distinguish the information available

to a regular trader versus an insider, Imkeller (2002) uses the last passage time of a Brownian motion driving a

stock price process. The last passage time, which is not a stopping time to a regular trader, becomes a stopping time

to an insider by utilizing progressive enlargement of ﬁltration. This study illustrates how additional information

provided by the last passage time can create arbitrage opportunities. Last passage times have also been used in the

European put and call option pricing. The related studies are presented in Profeta et al. (2010). These studies show

that option prices can be expressed in terms of probability distributions of last passage times. See also Cheridito

et al. (2012).

The structure of the paper is the following. In the rest of this section, we summarize some mathematical facts of

one-dimensional diffusion. Section 2 is devoted to the proof of Proposition 1 and the identiﬁcation of the associated

killing rate. Section 3.2 is an example of last passage time decomposition. We present extensions and applications

ON DECOMPOSITION OF THE LAST PASSAGE TIME OF DIFFUSIONS 3

in Sections 4 and 5 where the decomposition for the Green function is established in a general setting (Section 4)

and diffusions with switching parameters are studied in Sections 5.1 and 5.2, the latter being a ﬁnancial application.

1.1. Mathematical Setup. We refer to Borodin and Salminen (2002, Chapter II), Karlin and Taylor (1981, Chap-

ter 15), Karatzas and Shreve (1998, Chapter 5), Itˆ

o and McKean (1974, Chapter 4), and Rogers and Williams

(1994, Chapter III) for diffusion processes. The main reference is the ﬁrst one. Except for the proof of (10), the

facts regarding diffusions mentioned in this subsection can be found in the references above. We cite speciﬁc

references for the facts that are not listed in Borodin and Salminen (2002, Chapter II).

Let us consider a complete probability space (Ω,F,P)with a ﬁltration F= (Ft)t≥0satisfying the usual con-

ditions. Let Xbe a regular diffusion process adapted to Fwith the state space I= (ℓ, r)⊂R. We assume X

is not killed in the interior of I, which is a standard grand assumption for a general study of regular diffusions

(e.g., see Salminen (1984) and Dayanik and Karatzas (2003)). On the other hand, if Xhits ℓor r, it is killed and

immediately transferred to the cemetery ∆/∈ I. The lifetime of Xis given by

ξ= inf{t:X(t−) = ℓor r}.

Following Rogers and Williams (1994, Chapter III), we write

X= (Ω,{Ft:t≥0},{Xt:t≥0},{Pt:t≥0},{Px:x∈ I})

where Pxdenotes the probability law of the process when it starts at x∈ I. For every t≥0, the transition function

is given by Pt:I × B(I)7→ [0,1] such that for all t, s ≥0and every Borel set A∈ B(I)

Px(Xt+s∈A| Fs) = Pt(Xs, A),Px-a.s.

The dynamics of a one-dimensional diffusion are characterized by scale function, speed measure, and killing

measure (see Appendix A.1 for deﬁnitions). The scale function and the speed measure of Xare given by s(·)and

m(·), respectively. The killing measure is given by k(·). The assumption we made above that killing does not

occur in the interior of the state space is expressed by k(dx) = 0 for x∈ I.

We assume that Xis transient. The transience is equivalent to one or both of the boundaries being attracting; that

is, s(ℓ)>−∞ and/or s(r)<+∞. See Proposition 5.22 in Karatzas and Shreve (1998, Chapter 5) and Salminen

(1984). Note that s(ℓ) := s(ℓ+) and s(r) := s(r−). Transient diffusion can also be obtained from originally

recurrent diffusion (such as Brownian motion and Ornstein-Uhlenbeck process) by including a killing boundary

in its state space. Such setup is often used in engineering, economics, ﬁnance, and other scientiﬁc ﬁelds when

dealing with real-life problems. For example, we refer the reader to Linetsky (2007) for ﬁnancial engineering

applications such as derivative pricing. Applications in neuroscience are discussed in Bibbona and Ditlevsen

(2013). For optimal stopping problems, refer to Alvarez and Matom¨

aki (2014). Hence transient diffusions are

useful in modeling.

To obtain concrete results, we set a speciﬁc assumption:

Assumption 1.

s(ℓ)>−∞ and s(r)=+∞.

Then, it holds that

Pxlim

t→ξXt=ℓ= 1,∀x∈ I

4 ON DECOMPOSITION OF THE LAST PASSAGE TIME OF DIFFUSIONS

(see Proposition 5.22 in Karatzas and Shreve (1998, Chapter 5)). That is, killing occurs at ℓ. For the later reference,

we state the deﬁnition of the killing rate of a diffusion: the inﬁnitesimal killing rate γ(x)at x∈ I is

γ(x) := lim

s↓0

s(1 −Px(ξ > s)) .(1)

Assumption 1 is necessary to ﬁx a method to prove Proposition 1. But we shall remove this assumption in Propo-

sition 3.

For every t > 0and x∈ I,Pt(x, ·) : A7→ Pt(x, A)is absolutely continuous with respect to the speed measure

Pt(x, A) = ZA

p(t;x, y)m(dy), A ∈ B(I).

As discussed in Itˆ

o and McKean (1974, Chapter 4.11), the transition density pmay be constructed to be positive

and jointly continuous in all variables as well as symmetric satisfying p(t;x, y) = p(t;y, x).

We use superscripts +and −to denote the right and left derivatives of some function fwith respect to the scale

function:

f+(x) := lim

h↓0

f(x+h)−f(x)

s(x+h)−s(x), f−(x) := lim

h↓0

f(x)−f(x−h)

s(x)−s(x−h).(2)

The inﬁnitesimal generator Gis deﬁned by

Gf:= lim

t↓0

Ptf−f

t(3)

applied to bounded continuous functions fdeﬁned in Ifor which the limit exists pointwise, is a bounded con-

tinuous function in I, and supt>0||Ptf−f

t|| <∞with the sup norm || · ||. We assume sand mare absolutely

continuous with respect to the Lebesgue measure and have smooth derivatives. With this assumption together with

a continuous second derivative of s, the generator Gcoincides with the second-order differential operator given by

Gf(x) = 1

2σ2(x)f′′(x) + µ(x)f′(x), x ∈ I (4)

where µ(·)and σ(·)denote inﬁnitesimal drift and diffusion parameters, respectively. We assume σ2(x)>0for all

x∈ I. To ensure that dXt=µ(Xt)dt+σ(Xt)dWt(with a standard Brownian motion W) has a weak solution,

we impose a standard condition on µand σ:

∀x∈ I,∃ε > 0such that Zx+ε

x−ε

1 + |µ(y)|

σ2(y)dy < ∞.

See Karatzas and Shreve (1998, Chapter 5, Theorem 5.15). Consider the equation Gu=qu for q > 0. Under the

original deﬁnition of Gin (3), it should read as follows: uis a function which satisﬁes

qZ[a,b)

u(x)m(dx) = u−(b)−u−(a)

for all a, b such that ℓ<a<b<r. But in the absolute continuous case, uis the solution to Gu=qu for Gin (4),

so that the existence of uis part of the deﬁnition of the generator. From the generator equation we have

s(x) = Zx

e−Ry2µ(u)

σ2(u)dudy, m(dx) = 2eRx2µ(u)

σ2(u)du

σ2(x)dx. (5)

Note that such deﬁnitions and assumptions for the scale function and speed measure are used in Karatzas and

Shreve (1998, Chapter 5) and Karlin and Taylor (1981, Chapter 15) and that s(x)satisﬁes Gs= 0 on I.

ON DECOMPOSITION OF THE LAST PASSAGE TIME OF DIFFUSIONS 5

The Laplace transform of the hitting time Hz:= inf{t≥0 : Xt=z}for z∈ I is given by

Exe−qHz=





ϕq(x)

ϕq(z), x ≥z,

ψq(x)

ψq(z), x ≤z, (6)

where the continuous positive functions ψqand ϕqdenote linearly independent solutions of the ODE Gf=qf

with q > 0. Here ψqis increasing while ϕqis decreasing. They are unique up to a multiplicative constant, once the

boundary conditions at ℓand rare speciﬁed. Finally, the Green function is deﬁned as

Gq(x, y) := 





ψq(y)ϕq(x)

wq, x ≥y,

ψq(x)ϕq(y)

wq, x ≤y, (7)

with the Wronskian wq:= ψ+

q(x)ϕq(x)−ψq(x)ϕ+

q(x) = ψ−

q(x)ϕq(x)−ψq(x)ϕ−

q(x). It holds that Gq(x, y) =

R∞

0e−qtp(t;x, y)dtfor x, y ∈ I.

Under Assumption 1, the killing boundary ℓis attracting and limx↓ℓExe−qHz=ψq(ℓ+)

ψq(z)= 0 for z∈ I. Hence

ψq(ℓ+) = 0. As the right boundary ris not attracting, limz↑rExe−qHz=ψq(x)

ψq(r−)= 0 for x∈ I and we obtain

ψq(r−)=+∞.

Next, due to the transience of X, we deﬁne

G0(x, y) := lim

q↓0Gq(x, y) = Z∞

p(t;x, y)dt < +∞.(8)

Following Itˆ

o and McKean (1974, Section 4.11), this quantity is represented by

G0(x, y) = 





ψ0(y)ϕ0(x)

w0, x ≥y,

ψ0(x)ϕ0(y)

w0, x ≤y, (9)

where the continuous positive functions ψ0and ϕ0denote (linearly independent) solutions of the ODE Gf= 0 and

w0:= ψ+

0(x)ϕ0(x)−ψ0(x)ϕ+

0(x) = ψ−

0(x)ϕ0(x)−ψ0(x)ϕ−

0(x).

Here ψ0is increasing while ϕ0is decreasing. These functions are uniquely determined based on the boundary

conditions and satisfy

ϕ0≡1, ψ0(ℓ+) = 0, ψ0(r−) = +∞(10)

under Assumption 1. We show a proof of (10) in Appendix A.2 since to our knowledge it is not shown in the

existing literature. The ﬁrst equation ϕ0≡1should be understood such that the solution ϕ0can be taken as unity.

Since ψ0solves Gf= 0 and is increasing, we can set ψ0(x) = w0(s(x) + constant). Then, the boundary

condition at ℓdetermines the constant, i.e.,

ψ0(x) = w0(s(x)−s(ℓ)), x ∈ I,

which in turn leads to

G0(x, y) = (s(x)−s(ℓ)) ∧(s(y)−s(ℓ)),(11)

since ϕ0≡1by (10). Note that w0is forced to be ψ−

Our analysis focuses on a decomposition of the last passage time of some ﬁxed level α∈ I which is denoted by

λα:= sup{t:Xt=α}(12)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ONDECOMPOSITIONOFTHELASTPASSAGETIMEOFDIFFUSIONSMASAHIKOEGAMI1ANDRUSUDANKEVKHISHVILI21,2GraduateSchoolofEconomics,KyotoUniversity,Sakyo-ku,Kyoto,606-8501,JapanABSTRACT.Foraregulartransientdiffusion,weprovideadecompositionofitslastpassagetimetoacertainstateα.Thisisaccomplishedbytransformingtheoriginal...

展开>> 收起<<

ON DECOMPOSITION OF THE LAST PASSAGE TIME OF DIFFUSIONS MASAHIKO EGAMI1AND RUSUDAN KEVKHISHVILI2 12Graduate School of Economics Kyoto University Sakyo-ku Kyoto 606-8501 Japan.pdf

共27页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

ON DECOMPOSITION OF THE LAST PASSAGE TIME OF DIFFUSIONS MASAHIKO EGAMI1AND RUSUDAN KEVKHISHVILI2 12Graduate School of Economics Kyoto University Sakyo-ku Kyoto 606-8501 Japan

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: