Volume 2010, Article ID 347105,21pages doi:10.1155/2010/347105
Research Article
Diffusion Approximations of
the Geometric Markov Renewal Processes and Option Price Formulas
Anatoliy Swishchuk
^{1}and M. Shafiqul Islam
^{2}1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive, NW, Calgary, Alberta, Canada T2N 1N4
2Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PE, Canada C1A 4P3
Correspondence should be addressed to M. Shafiqul Islam,sislam@upei.ca Received 3 August 2010; Accepted 8 November 2010
Academic Editor: Aihua Xia
Copyrightq2010 A. Swishchuk and M. S. Islam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the geometric Markov renewal processes as a model for a security market and study this processes in a diﬀusion approximation scheme. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes in diﬀusion scheme are presented.
We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diﬀusion geometric Markov renewal processes.
1. Introduction
Let Nt be a standard Poisson process and Yk_{k∈Z}_{} be i.i.d. random variables which are independent ofNtandS^{∗}_{0} >0. The geometric compound Poisson processes
S^{∗}_{t} S^{∗}_{0}
Nt
k1
1Yk, t >0, 1.1
is a trading model in many financial applications with pure jumps1, page 214. Motivated by the geometric compound Poisson processes 1.1, Swishchuk and Islam 2 studied the Geometric Markov renewal processes 2.5 see Section 2 for a security market in a series scheme. The geometric Markov renewal processes 2.5 are also known as a switched-switching process. Averaging and diﬀusion approximation methods are important
approximation methods for a switched-switching system. Averaging schemes of the geometric Markov renewal processes2.5were studied in2.
The singular perturbation technique of a reducible invertible-operator is one of the techniques for the construction of averaging and diﬀusion schemes for a switched- switching process. Strong ergodicity assumption for the switching process means that the singular perturbation problem has a solution with some additional nonrestrictive conditions.
Averaging and diﬀusion approximation schemes for switched-switching processes in the form of random evolutions were studied in3, page 157and1, page 41. In this paper, we introduce diﬀusion approximation of the geometric Markov renewal processes. We study a discrete Markov-modulatedB, S-security market described by a geometric Markov renewal processGMRP. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes in diﬀusion scheme are presented. We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diﬀusion geometric Markov renewal processes.
The paper is organized as follows. In Section 2 we review the definition of the geometric Markov renewal processesGMRPfrom2. Moreover we present notation and summarize results such as random evolution of GMRP, Markov renewal equation for GMRP, infinitesimal operator of GMRP, and martingale property of GMRP. InSection 3we present diﬀusion approximation of GMRP in ergodic, merged, and double-averaging schemes. In Section 4 we present proofs of the above-mentioned results.Section 4contains solution of martingale problem, weak convergence, rates of convergence for GMRP, and characterization of the limit measure. In Section 5 we present merged diﬀusion GMRP in the case of two ergodic classes. European call option pricing formula for ergodic, merged, and diﬀusion GMRP are presented inSection 6.
2. The Geometric Markov Renewal Processes (GMRP)
In this section we present the Geometric Markov renewal processes. We closely follow2.
Let Ω,B,Ft,P be a standard probability space with complete filtrationFt and let xk_{k∈Z}_{} be a Markov chain in the phase space X,Xwith transition probability Px, A, wherex∈X, A ∈ X. Letθk_{k∈Z}_{}be a renewal process which is a sequence of independent and identically distributed i.i.d. random variables with a common distribution function Fx : P{w : θ_{k}w ≤ x}. The random variables θk_{k∈Z}_{} can be interpreted as lifetimes operating periods, holding times, renewal periods of a certain system in a random environment. From the renewal processθk_{k∈Z}_{}we can construct another renewal process τk_{k∈Z}_{}defined by
τk:^{k}
n0
θn. 2.1
The random variablesτ_{k}are called renewal timesor jump times. The process
vt:sup{k:τk≤t} 2.2
is called the counting process.
Definition 2.1see1,4. A homogeneous two-dimensional Markov chainxn, θn_{n∈Z}_{}on the phase spaceX×R_{} is called a Markov renewal processMRPif its transition probabilities are given by the semi-Markov kernel
Qx, A, t P{x_{n1}∈A, θ_{n1}≤t|x_{n} x}, ∀x∈X, A∈ X, t∈R_{}. 2.3 Definition 2.2. The process
xt:x_{vt} 2.4
is called a semi-Markov process.
The ergodic theorem for a Markov renewal process and a semi-Markov process respectively can be found in3, page 195,1, page 66, and4, page 113.
Letxn, θ_{n}_{n∈Z}_{}be a Markov renewal process on the phase spaceX×R_{}with the semi- Markov kernelQx, A, tdefined in2.3, and letxt:x_{vt}be a semi-Markov process where the counting processvtis defined in2.2. Letρxbe a bounded continuous function onX such thatρx>−1. We define the geometric Markov renewal processGMRP{St}_{t∈R}_{}as a stochastic functionalS_{t}defined by
S_{t}:S_{0} vt
k1
1ρxk
, t∈R_{}, 2.5
whereS0 >0 is the initial value ofSt. We call this processSt_{t∈R}_{}a geometric Markov renewal process by analogy with the geometric compound Poisson processes
S^{∗}_{t} S^{∗}_{0}
Nt
k1
1Yk, 2.6
whereS^{∗}_{0} > 0,Ntis a standard Poisson process,Yk_{k∈Z}_{} are i.i.d. random variables. The geometric compound Poisson processes{S^{∗}_{t}}_{t∈R}_{}in2.6is a trading model in many financial applications as a pure jump model5,6. The geometric Markov renewal processes{St}_{t∈R}_{} in2.5will be our main trading model in further analysis.
Jump semi-Markov random evolutions, infinitesimal operators, and Martingale property of the GMRP were presented in2. For the convenience of readers we repeat them again in the following.
2.1. Jump Semi-Markov Random Evolutions
LetC0Rbe the space of continuous functions onRvanishing at infinity, and let us define a family of bounded contracting operatorsDxonC0R_{}as follows:
Dxfs:f s
1ρx
, x∈X, s∈R. 2.7
With these contraction operatorsDxwe define the following jump semi-Markov random evolutionJSMREVtof the geometric Markov renewal processes{St}_{t∈R}_{}in2.5:
Vt ^{vt}
k1
Dxk:D x_{vt}
◦D x_{vt−1}
◦ · · · ◦Dx1. 2.8
Using2.7we obtain from2.8
Vtfs ^{vt}
k1
Dxkfs f
s vt
k1
1ρxk
fSt, 2.9
whereS_{t}is defined in2.5andS_{0} s. LetQx, A, tbe a semi-Markov kernel for Markov renewal processxn;θn_{n∈Z}_{}, that is,Qx, A, t Px, AGxt, wherePx, Ais the transition probability of the Markov chainxn_{n∈Z}_{}andG_{x}tis defined byG_{x}t:Pθ_{n1}≤t|x_{n}x.
Let
ut, x:E_{x}
Vtgxt :E
Vtgxt|x0 x 2.10
be the mean value of the semi-Markov random evolutionVtin2.9.
The following theorem is proved in1, page 60and4, page 38.
Theorem 2.3. The mean valueut, xin2.10of the semi-Markov random evolutionVtgiven by the solution of the following Markov renewal equation (MRE):
ut, x− _{t}
0
X
Q
x, dy, ds D
y u
t−s, y
G_{x}tgx, 2.11
whereG_{x}t 1−G_{x}t, Gxt:Pθ_{n1} ≤t|x_{n}x, gxis a bounded and continuous function onX.
2.2. Infinitesimal Operators of the GMRP
Let1ρ_{T}x: ρx
T , T >0, 2.12
S^{T}_{t} :S_{0}
vtT k1
1ρ_{T}xk S_{0}
vtT
k1
1T^{−1}ρxk
. 2.13
A detailed information aboutρTxandS^{T}_{t} can be found in Section 4 of2. It can be easily shown that
lnS^{T}_{t} S0 ^{vtT}^{}
k1
ln
1 ρxk T
. 2.14
To describe martingale properties of the GMRP St_{t∈R}_{} in 2.5 we need to find an infinitesimal operator of the process
ηt:^{vt}
k1
ln
1ρxk
. 2.15
Letγt:t−τ_{vt}and consider the processxt, γtonX×R_{}. It is a Markov process with infinitesimal operator
Qfx, t : df
dt g_{x}t Gxt
X
P x, dy
f y,0
−fx, t , 2.16
wheregxt : dGxt/dt,Gxt 1−Gxt, wherefx, t ∈ CX×R_{}. The infinitesimal operator for the process lnSthas the form:
Afz, x g_{x}t Gxt
X
P x, dy
f zln
1ρ y
, x
−fz, x , 2.17
wherez:lnS0. The processlnSt, xt, γtis a Markov process onR_{}×X×R_{}with the infinitesimal operator
Lf z, x, t Af z, x, t Qfz, x, t, 2.18 where the operatorsA andQ are defined in2.17and2.18, respectively. Thus we obtain that the process
mt:f
lnSt, xt, γt
−fz, x,0− _{t}
0
AQ f
lnSu, xu, γu
du 2.19
is anFt-martingale, whereFt : σxs, γs; 0≤s ≤t. Ifxt : x_{vt} is a Markov process with kernel
Qx, A, t Px, A
1−e^{−λxt}
, 2.20
namely,Gxt 1−e^{−λxt}, thengxt λxe^{−λxt},Gxt e^{−λxt},gxt/Gxt λx, and the operatorAin2.17has the form:
Afz λx
X
P x, dy
f zln
1ρ
y
−fz . 2.21
The processlnSt, xtonR_{}×Xis a Markov process with infinitesimal operator
Lfz, x Afz, x Qfz, x, 2.22
where
Qfz, x λx
X
P x, dy
f y
−fx
. 2.23
It follows that the process
mt:flnSt, xt−fz, x− _{t}
0
AQ
flnSu, xudu 2.24
is anFt-martingale, whereFt:σxu; 0≤u≤t.
2.3. Martingale Property of the GMRP
Consider the geometric Markov renewal processesSt_{t∈R}_{}
StS0
vt k1
1ρxk
. 2.25
Fort∈0, Tlet us define
Lt:L0
vt k1
hxk, EL01, 2.26
wherehxis a bounded continuous function such that
X
h y
P x, dy
1,
X
h y
P x, dy
p y
0. 2.27
If EL_{T} 1, then geometric Markov renewal processS_{t} in2.25 is an Ft, P^{∗}-martingale, where measureP^{∗}is defined as follows:
dP^{∗} dP LT, Ft:σxs; 0≤s≤t.
2.28
In the discrete case we have
SnS0
n k1
1ρxk
. 2.29
LetLn : L0_{n}
k1hxk,EL0 1, wherehxis defined in2.27. IfELN 1, thenSn is an Ft, P^{∗}-martingale, wheredP^{∗}/dP L_{N}, andFn:σxk; 0≤k≤n.
3. Diffusion Approximation of the Geometric Markov Renewal Process (GMRP)
Under an additional balance condition, averaging eﬀect leads to diﬀusion approximation of the geometric Markov renewal processGMRP. In fact, we consider the counting process vtin2.5in the new accelerated scale of timetT^{2}, that is,v ≡ vtT^{2}. Due to more rapid changes of states of the system under the balance condition, the fluctuations are described by a diﬀusion processes.
3.1. Ergodic Diffusion Approximation
Let us suppose that balance condition is fulfilled for functionalS^{T}_{t} S0_{vtT}
k1 1ρTxk:
ρ
Xpdx
XP x, dy
ρ y
m 0, 3.1
wherepxis ergodic distribution of Markov chainxk_{k∈Z}_{}. ThenSt S_{0}, for allt ∈R^{}. ConsiderS^{T}_{t} in the new scale of timetT^{2}:
S_{T}t:S^{T}_{tT}2S_{0}
v^{tT}^{2}
k1
1T^{−1}ρxk
. 3.2
Due to more rapid jumps of vtT^{2}the process S_{T}t will be fluctuated near the pointS_{0} as T → ∞. By similar arguments similar to 4.3–4.5 in 2, we obtain the following expression:
lnSTt S_{0} T^{−1}
v^{tT}^{2}
k1
ρxk−1 2T^{−2}
v^{tT}^{2}
k1
ρ^{2}xk T^{−2}
v^{tT}^{2}
k1
r
T^{−1}ρxk
ρ^{2}xk. 3.3
Algorithms of ergodic averaging give the limit result for the second term in3.3 see1, page 43and4, page 88:
Tlim→∞
1 2T^{−2}
v^{tT}^{2}
k1
ρ^{2}xk
1
2tρ_{2}, 3.4
whereρ_{2} :
Xpdx
XPx, dyρ^{2}y/m. Using algorithms of diﬀusion approximation with respect to the first term in3.3we obtain4, page 88:
Tlim→∞T^{−1}
vtT^{2} k1
ρxk σρwt, 3.5
where σ_{ρ}^{2} :
Xpdx1/2
XPx, dyρ^{2}y
XPx, dyρyR0Px, dyρy/m,R0 is a potential3, page 68, ofxn_{n∈Z}_{},wtis a standard Wiener process. The last term in3.3
goes to zero asT → ∞. LetSt be the limiting process forSTtin3.3asT → ∞. Taking limit on both sides of3.3we obtain
Tlim→∞lnSTt
S0 lnSt
S0 σρwt−1
2tρ2, 3.6
whereσ_{ρ}^{2}andρ2are defined in3.4and3.5, respectively. From3.6we obtain
St S_{0}e^{σ}^{ρ}^{wt−1/2t}^{ρ}^{2}S_{0}e^{−1/2t}^{ρ}^{2}e^{σ}^{ρ}^{wt}. 3.7
Thus,St satisfies the following stochastic diﬀerential equationSDE:
dSt St 1
2
σ_{ρ}^{2}−ρ_{2}
dtσ_{ρ}dwt
. 3.8
In this way we have the following corollary.
Corollary 3.1. The ergodic diﬀusion GMRP has the form
St S_{0}e^{−1/2t}^{ρ}^{2}e^{σ}^{ρ}^{wt}, 3.9
and it satisfies the following SDE:
dSt St 1
2
σ_{ρ}−ρ_{2}
dtσ_{ρ}dwt. 3.10
3.2. Merged Diffusion Approximation
Let us suppose that the balance condition satisfies the following:
ρk
Xkp_{k}dx
XkP x, dy
ρ y
mk 0, 3.11
for all k 1,2, . . . , r where xn_{n∈Z}_{} is the supporting embedded Markov chain,p_{k} is the stationary density for the ergodic componentXk,mkis defined in2, and conditions of reducibility ofX are fulfilled. Using the algorithms of merged averaging1,3,4we obtain from the second part of the right hand side in3.3:
Tlim→∞
1 2T^{−1}
v^{tT}^{2}
k1
ρ^{2}xk
1 2
_{t}
0
ρ_{2}xsds, 3.12
where
ρ_{2}k:
Xkpkdx
XkP x, dy
ρ^{2} y
mk 3.13
using the algorithm of merged diﬀusion approximation that1,3,4obtain from the first part of the right hand side in3.3:
Tlim→∞T^{−1}
v^{tT}^{2}
k1
ρxk
_{t}
0
σρxsdws, 3.14
where σ_{ρ}^{2}k:
Xk
p_{k}dx
Xk
P x, dy
ρ^{2} y
Xk
P x, dy
ρ y
R_{0}
Xk
P x, dy
ρ y
mk . 3.15
The third term in3.3goes to 0 asT → ∞. In this way, from3.3we obtain:
Tlim→∞lnSTt
S_{0} lnSt S_{0}
_{t}
0
σρxsdws − 1 2
_{t}
0
ρ2xsds, 3.16
whereSt is the limitSTtasT → ∞. From3.16we obtain
St S0e^{−1/2}^{}^{t}^{0}^{ρ}^{}^{2}^{}^{xsds}^{}^{t}^{0}^{σ}^{}^{ρ}^{}^{xsdws}. 3.17
Stochastic diﬀerential equationSDEfor ˇSthas the following form:
dSt St 1
2
σ_{ρ}^{2}xt −ρ_{2}xt
dtσ_{ρ}xtdwt, 3.18
wherext is a merged Markov process.
In this way we have the following corollary.
Corollary 3.2. Merged diﬀusion GMRP has the form3.17and satisfies the SDE3.18.
3.3. Diffusion Approximation under Double Averaging
Let us suppose that the phase space X {1,2, . . . , r} of the merged Markov process xt consists of one ergodic class with stationary distributionspk; k {1,2, . . . r}. Let us also suppose that the balance condition is fulfilled:
r k1
pkρk 0. 3.19
Then using the algorithms of diﬀusion approximation under double averagingsee3, page 188,1, page 49and4, page 93we obtain:
Tlim→∞lnS_{T}t
S0 lnStˇ
S0 σˇ_{ρ}wt−1
2pˇ_{2}t, 3.20
where
ˇ σ_{ρ}^{2}:^{r}
k1
pkσ_{ρ}^{2}k, pˇ2:^{r}
k1
pkρ2k, 3.21
andρ_{2}kandσ_{ρ}^{2}kare defined in3.13and3.15, respectively. Thus, we obtain from3.20:
St ˇ S0e^{−1/2ˇ}^{p}^{2}^{t}^{σ}^{ˇ}^{ρ}^{wt}. 3.22 Corollary 3.3. The diﬀusion GMRP under double averaging has the form
St ˇ S_{0}e^{−1/2}^{ρ}^{ˇ}^{2}^{t}^{σ}^{ˇ}^{ρ}^{wt}, 3.23 and satisfies the SDE
dStˇ Stˇ 1
2
ˇ σ_{ρ}^{2}−ρˇ2
dtσˇρdwt. 3.24
4. Proofs
In this section we present proofs of results inSection 3. All the above-mentioned results are obtained from the general results for semi-Markov random evolutions3,4in series scheme.
The main steps of proof are1weak convergence ofS^{T}_{t} in Skorokhod spaceD_{R}0,∞ 7, page 148;2solution of martingale problem for the limit processSt; 3characterization of the limit measure for the limit processSt; 4uniqueness of solution of martingale problem.
We also give here the rate of convergence in the diﬀusion approximation scheme.
4.1. Diffusion Approximation (DA)
LetG^{T}_{t} :T^{−1}
vtT^{2} k0
ρxk, G^{T}_{n} :G^{T}_{τ}
nT^{−1}, G^{T}_{0} lns, 4.1
and the balance condition is satisfied:
ρ:
X
pdx
X
P x, dy
ρ y
0. 4.2
Let us define the functions
φ^{T}s, x:fs T^{−1}φ^{1}_{f}s, x T^{−2}φ_{f}^{2}s, x, 4.3
whereφ^{1}_{f} andφ_{f}^{2}are defined as follows:
P−Iφ_{f}^{1}s, x ρxfs, P−Iφ^{2}_{f}s, x
−Ax A
fs, 4.4
where
A:
X
pdxAx, 4.5
andAx: ρ^{2}x/2ρxR0−Iρxd^{2}/ds^{2}. From the balance condition4.2and equality ΠA−Ax 0 it follows that both equations in 4.3 simultaneously solvable and the solutionsφ^{i}_{f}s, xare bounded functions,i1,2.
We note that
f S^{T}_{n1}
−f G^{T}_{n}
1
Tρxndfxn
ds 4.6
and define
φ^{T}s, x:fs T^{−1}φ^{1}_{f}s, x T^{−2}φ_{f}^{2}s, x, 4.7 whereφ^{1}_{f}s, xandφ^{2}_{f}s, xare defined in4.4and4.5, respectively. We note, thatG^{T}_{n1}− G^{T}_{n} T^{−1}ρxn.
4.2. Martingale Problem for the Limiting Problem
G0tin DA
Let us introduce the family of functions:ψ^{T}s, t:φ^{T}
G^{T}^{tT}^{2}, x_{tT}^{2}_{}
−φ^{T}
G^{T}^{st}^{2}, x_{sT}^{2}_{}
−^{tT}^{2}^{−1}
j^{sT}^{2} E
φ^{T}
G^{T}_{j}, x_{j1}
−φ^{T} G^{T}_{j}, x_{j}
| Fj
,
4.8
whereφ^{T} are defined in4.7andG^{T}_{j} is defined by
G^{T}_{τ}
n/T 1
T n k0
ρxk. 4.9
Functions ψ^{T}s, t are FtT^{2}-martingale by t. Taking into account the expression4.6and 4.7, we find the following expression:
ψ^{T}s, t f
G^{T}^{tT}^{2}
−f
G^{T}^{sT}^{2}
φ^{1}_{f}
G^{T}^{tT}^{2}, x_{tT}^{2}_{}
−φ^{1}_{f}
G^{T}^{st}^{2}, x_{sT}^{2}_{}
^{2}
φ^{2}_{f}
G^{T}^{tT}^{2}, x_{tT}^{2}_{}
−φ_{f}^{2}
G^{T}^{sT}^{2}, x_{sT}^{2}_{}
−T^{−1}
tT^{2}−1 jsT^{2}
⎧⎨
⎩ρ xj
df G^{T}_{j} dg E
φ^{1}_{f}
G^{T}_{j}, x_{j1}
−φ_{f}^{2} G^{T}_{j}, xj
| Fj
⎫
⎬
⎭
−T^{−2}
tT^{2}−1 jsT^{2}
⎧⎪
⎨
⎪⎩2^{−1}ρ^{2} x_{j}df
G^{T}_{j} dg ρ
x_{j} E
⎛
⎜⎝dφ^{1}_{f}
G^{T}_{j}, x_{j1} dg | Fj
⎞
⎟⎠
E φ_{f}^{2}
G^{T}_{j}, x_{j1}
−φ^{2}_{f} G^{T}_{j}, x_{j}
| Fj
⎫
⎬
⎭o T^{−2}
f
G^{T}^{tT}^{2}
−f
G^{T}^{sT}^{2}
φ_{f}^{1}
G^{T}^{tT}^{2}, x_{tT}^{2}_{}
−φ^{1}_{f}
G^{T}^{sT}^{2}, x_{sT}^{2}_{}
T^{−2}
φ^{2}_{f}
G^{T}^{tT}^{2}, x_{tT}^{2}_{}
−φ^{2}_{f}
G^{T}^{sT}^{2}, x_{sT}^{2}_{}
−T^{−2} ^{tT}^{2}^{−1}
j^{sT}^{2} Af
G^{T}_{j} O
T^{−2} , 4.10
whereOT^{−2}is the sum of terms withT^{−2}nd order. Sinceψ^{T}0, tisFtT^{2}-martingale with respect to measure Q_{T}, generated by process G_{T}t in 4.1, then for every scalar linear continuous functionalη^{s}_{0}we have from4.8-4.10:
0E^{T}
ψ^{T}s, tη^{s}_{0} E^{T}
⎡
⎣
⎛
⎝f
G^{T}^{tT}^{2}
−f
G^{T}^{sT}^{2}
−T^{−2} ^{tT}^{2}^{−1}
j^{sT}^{2} Af
G^{T}_{j}⎞
⎠η^{s}_{0}
⎤
⎦
−T^{−1}E^{T}
φ^{1}_{f}
G^{T}^{tT}^{2}, x_{tT}^{2}_{}
−φ_{f}^{1}
G^{T}^{sT}^{2}, x_{sT}^{2}_{}
η_{0}^{s}
−T^{−2}E^{T}
φ^{2}_{f}
G^{T}^{tT}^{2}, x_{tT}^{2}_{}
−φ_{f}^{2}
G^{T}^{sT}^{2}, x_{sT}^{2}_{}
η_{0}^{s}
−O T^{−2}
,
4.11
whereE^{T} is a mean value by measureQT. If the processG^{T}_{tT}2 converges weakly to some processG_{0}tasT → ∞, then from4.11we obtain
0E^{T} (
fG0t−fG0s− _{t}
s
AfG 0udu )
, 4.12
that is, the process
fG0t−fG0s− _{t}
s
AfG 0udu 4.13
is a continuous QT-martingale. Since A is the second order diﬀerential operator and coeﬃcientσ_{1}^{2}is positively defined, where
σ_{1}^{2}:
X
πdx (ρ^{2}x
2 ρxR0ρx )
, 4.14
then the processG0tis a Wiener process with varianceσ_{1}^{2}in4.14:G0t σwt. Taking into account the renewal theorem forvt, namely,T^{−1}vtT^{2}→T→∞t/m, and the following representation
GTt T^{−1}
v^{tT}^{2}
k0
ρxk T^{−1} ^{tT}^{2}
k0
ρxk T^{−1}
vtT^{2}
k^{tT}^{2}^{1}ρxk 4.15
we obtain, replacingtT^{2}byvtT^{2}, that processGTtconverges weakly to the processG0t asT → ∞, which is the solution of such martingale problem:
f G0t
−f G0s
− _{t}
s
A0f G0u
du 4.16
is a continuousQ_{T}-martingale, whereA_{0}:A/m, and Ais defined in4.5-4.5.
4.3. Weak Convergence of the Processes
GTtin DA
From the representation of the processGTtit follows thatΔTs, t:|GTt−G_{T}s|
****
**T^{−1}
v^{tT}^{2}
kv^{sT}^{2}^{1} ρxk
****
**
≤T^{−1}sup
x ρx***v tT^{2}
−v sT^{2}
−1***.
4.17
This representation gives the following estimation:
|ΔTt1, t_{2}||ΔTt2, t_{3}| ≤T^{−2}
sup
x
ρx _{2}***v
t_{3}T^{2}
−v
t_{1}T^{2}***^{2}. 4.18
Taking into account the same reasonings as in2we obtain the weak convergence of the processesGTtin DA.
4.4. Characterization of the Limiting Measure
Qfor
QTas
T → ∞in DA
From Section 4.3 see also Section 4.1.4 of 2 it follows that there exists a sequence Tnsuch that measures QTn converge weakly to some measure Q onDR0,∞ asT → ∞, whereD_{R}0,∞is the Skorokhod space7, page 148. This measure is the solution of such martingale problem: the following process
ms, t:f G_{0}t
−f G_{0}s
− _{t}
s
A_{0}f G_{0}u
du 4.19
is aQ-martingale for allfg∈C^{2}Rand
Ems, tη_{0}^{s}0, 4.20
for scalar continuous bounded functionalη^{s}_{0},Eis a mean value by measureQ. From4.19it follows thatE^{T}m^{T}s, tη^{s}_{0}0, and it is necessary to show that the limiting passing in4.1goes to the process in3.12asT → ∞. From equality4.11we find that limTn→∞E^{T}^{n}ms, tη^{s}_{0} Ems, tη^{s}_{0}. Moreover, from the following expression
***E^{T}ms, tη^{s}_{0}−Ems, tη^{s}_{0}***≤***
E^{T}−E
ms, tη^{s}_{0}***E^{T}***ms, t−m^{T}s, t******η^{s}_{0}***−→T→∞0, 4.21 we obtain that there exists the measureQonD_{R}0,∞which solves the martingale problem for the operatorA_{0}or, equivalently, for the processG_{0}tin the form4.12. Uniqueness of the solution of the martingale problem follows from the fact that operatorA0generates the unique semigroup with respects to the Wiener process with varianceσ_{1}^{2}in4.14. As long as the semigroup is unique then the limit processG0tis unique. See3, Chapter 1.
4.5. Calculation of the Quadratic Variation for GMRP
IfG^{T}_{n} G^{T}_{T}_{−1}_{τ}n, the sequence
m^{T}_{n}:G^{T}_{n}−G^{T}_{0} −^{n−1}
k0
E
G^{T}_{k1}−G^{T}_{k} | Fk
, G^{T}_{0} g, 4.22
isFn-martingale, where Fn : σ{xk, θk; 0 ≤ k ≤ n}. From the definition it follows that the characteristicm^{T}_{n}of the martingalem^{T}_{n}has the form
+ m^{T}_{n},
^{n−1}
k0
E
m^{T}_{k1}−m^{T}_{k}_{2}
| Fk
. 4.23
To calculatem^{T}_{n}let us representm^{T}_{n}in4.22in the form of martingale-diﬀerence:
m^{T}_{n} ^{n−1}
k0
G^{T}_{k1}−E
G^{T}_{k1}| Fk
. 4.24
From representation
G^{T}_{n1}−G^{T}_{n} 1
Tρxn 4.25
it follows thatEG^{T}_{k1}| Fk G^{T}_{k}T^{−1}ρxk, that is why G^{T}_{k1}−E
G^{T}_{k1}| Fk
T^{−1}
ρxk−P ρxk
. 4.26
Since from4.22it follows that
m^{T}_{k1}−m^{T}_{k} G^{T}_{k1}−E
G^{T}_{k1}| Fk
T^{−1}
ρxk−P ρxk
, 4.27
then substituting4.27in4.23we obtain +
m^{T}_{n}, T^{−2}
n−1 k0
I−Pρxk ^{2}. 4.28
In an averaging schemesee2for GMRP in the scale of timetTwe obtain thatm^{T}_{tT}goes to zero asT → ∞in probability, which follows from4.27:
+ m^{T}_{tT}_{},
T^{−2}
tT−1
k0
I−Pρxk ^{2}−→0 asT −→∞ 4.29
for allt ∈ R_{}. In the diﬀusion approximation scheme for GMRP in scale of timetT^{2} from 4.27we obtain that characteristicm^{T}_{tT}2does not go to zero asT → ∞since
- m^{T}^{tT}^{2}
. T^{−2}
^{tT}^{2}^{−1}
k0
I−Pρxk ^{2}−→tσ_{1}^{2}, 4.30
whereσ_{1}^{2}:
XπdxI−Pρx^{2}.
4.6. Rates of Convergence for GMRP
Consider the representation4.22for martingalem^{T}_{n}. It follows that
G^{T}_{n} gm^{T}_{n}^{n−1}
k0
E
G^{T}_{k1}−G^{T}_{k}| Fk
. 4.31
In diﬀusion approximation scheme for GMRP the limit for the processG^{T}_{tT}2asT → ∞will be diﬀusion processSt see 3.10. Ifm0tis the limiting martingale form^{T}_{tT}2in4.22as T → ∞, then from4.31and3.10we obtain
E
G^{T}^{tT}^{2}−St
E
m^{T}^{tT}^{2}−m0t
T^{−1} ^{tT}^{2}^{−1}
k0
ρxk−St. 4.32
SinceEm^{T}_{tT}2−m_{0}t 0,becausem^{T}_{tT}_{2}_{} andm_{0}tare zero-mean martingalesthen from 4.32we obtain:
****E
G^{T}^{tT}^{2}−St **
**≤T^{−1}
****
**
^{tT}^{2}^{−1}
k0
ρxk−StT
****
**. 4.33 Taking into account the balance condition
Xπdxρx 0 and the central limit theorem for a Markov chain4, page 98, we obtain
****
**
^{tT}^{2}^{−1}
k0
ρxk−StT
****
**C1t0, 4.34 whereC1t0is a constant depending ont0,t∈0, t0. From4.33,4.2, and4.32we obtain:
****E
G^{T}^{tT}^{2}−St **
**≤T^{−1}C_{1}t0. 4.35 Thus, the rates of convergence in diﬀusion scheme has the orderT^{−1}.
5. Merged Diffusion Geometric Markov Renewal Process in the Case of Two Ergodic Classes
5.1. Two Ergodic Classes
LetPx, A: P{x_{n1} ∈ A|x_{n} x}be the transition probabilities of supporting embedded reducible Markov chain{xn}_{n≥0}in the phase spaceX. Let us have two ergodic classesX_{0}and X1of the phase space such that:
XX0∪X1, X0∩X1∅. 5.1
Let{X {0,1},V} be the measurable merged phase space. A stochastic kernelP0x, Ais consistent with the splitting5.1in the following way:
P_{0}x, Xk 1_{k}:
⎧⎨
⎩
1, x∈X_{k}, 0, x /∈Xk,
k0,1. 5.2
Let the supporting embedded Markov chain xn_{n∈Z}_{} with the transition probabilities P_{0}x, Abe uniformly ergodic in each classX_{k}, k 0,1 and have a stationary distribution π_{k}dxin the classesX_{k},k0,1:
πkA
Xk
πkdxP0x, A, A⊂Xk, k0,1. 5.3
Let the stationary escape probabilities of the embedded Markov chainxn_{n∈Z}_{}with transition probabilitiesPx, A:P{xn1 ∈A|xnx}be positive and suﬃciently small, that is,
qkA
Xk
πkdxPx, X\Xk>0, k0,1. 5.4
Let the stationary sojourn time in the classes of states be uniformly bounded, namely,
0≤C1≤mk:
Xk
πkdxmx≤C2, k0,1, 5.5
where
mx: _{∞}
0
G_{x}tdt. 5.6
5.2. Algorithms of Phase Averaging with Two Ergodic Classes
The merged Markov chainx_{n}_{n∈Z}_{}in merged phase spaceXis given by matrix of transition probabilities
P p_{kr}
k,r0,1;
p_{01} 1−p_{11}
X1
π_{1}dxPx, X0 1−
X1
π_{1}dxPx, X1;
p01 1−p00
X0
π0dxPx, X1 1−
X0
π0dxPx, X0.
5.7
Aspkr/0,k0,1, thenxnhas virtual transitions. IntensitiesΛkof sojourn timesθk,k0,1, of the merged MRP are calculated as follows:
Λk 1
m_{k}, m_{k}
Xk
π_{k}dxmx, k0,1. 5.8
And, finally, the merged MRP xn,θ_{n∈Z}_{} in the merged phase space X is given by the stochastic matrix
Qt Q_{kr}
k,r0,1:p_{kr}
1−e^{−}^{Λ}^{}^{k}^{t}
, k, r0,1. 5.9
Hence, the initial semi-Markov system is merged to a Markov system with two classes.
5.3. Merged Diffusion Approximation in the Case of Two Ergodic Classes
The merged diﬀusion GMRP in the case of two ergodic classes has the form:St S_{0}e^{−1/2}^{}^{0}^{t}^{ρ}^{}^{2}^{}^{xsds}^{}^{0}^{t}^{σ}^{}^{ρ}^{}^{xsdws} 5.10
which satisfies the stochastic diﬀerential equationSDE:
dSt St 1
2
σ_{ρ}^{2}xt−ρ^{2}xt
dtσ_{ρ}xtdwt, 5.11
where
ρ^{2}1:
X1
p_{1}dx
X1
P x, dy
ρ^{2} y
m1 ,
ρ^{2}0:
X0
p0dx
X0
P x, dy
ρ^{2} y
m0 ,
σ^{2}_{ρ}1:
X1
p1dx
X1
P x, dy
ρ^{2} y
X1
P x, dy
ρ y
R0
X1
P x, dy
ρ y
m1 ,
σ^{2}_{ρ}0:
X0
p_{0}dx
X0
P x, dy
ρ^{2} y
X0
P x, dy
ρ y
R_{0}
X0
P x, dy
ρ y
m0 ,
5.12
xtis a merged Markov process inX {0,1}with stochastic matrixQt in5.9.
6. European Call Option Pricing Formulas for Diffusion GMRP 6.1. Ergodic Geometric Markov Renewal Process
As we have seen inSection 3, an ergodic diﬀusion GMRPSt satisfies the following SDE see3.10:
dSt St 1
2
σρ−ρ2
dtσρdwt, 6.1
where ρ_{2}
X
pdx
X
P x, dy
ρ^{2} y
m , 6.2
σ_{ρ}^{2}
X
pdx (1
2
X
P x, dy
ρ^{2} y
X
P x, dy
ρ y
R_{0}P x, dy
ρ y
m . 6.3
The risk-neutral measureP^{∗}for the process in6.1is:
dP^{∗} P exp
/
−θt−1 2θ^{2}wt
0
, 6.4
where
θ
1/2
σ_{ρ}−ρ_{2}
−r σρ
. 6.5
UnderP^{∗}, the processe^{−rt}S_{t}is a martingale and the processw^{∗}t wt θtis a Brownian motion. In this way, in the risk-neutral world, the processS_{t}has the following form
dSt
St rdtσ_{ρ}dw^{∗}t. 6.6
Using Black-Scholes formulasee8we obtain the European call option pricing formula for our model6.6:
CS0Φd_{}−Ke^{−rT}Φd_{−}, 6.7
where
d_{} lnS0/K
r 1/2σρt σ_{ρ}√
t ,
d_{−} lnS0/K
r−1/2σρt σρ
√t ,
6.8
Φxis a normal distribution andσ_{ρ}is defined in6.3.