the Geometric Markov Renewal Processes and Option Price Formulas

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

and M. Shafiqul Islam

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∈Zbe 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_kw ≤ 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∈Zwe can construct another renewal process τk_k∈Zdefined by

τk:^k

n0

θn. 2.1

The random variablesτ_kare 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∈Zon 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_nn∈Zbe a Markov renewal process on the phase spaceX×Rwith the semi- Markov kernelQx, A, tdefined in2.3, and letxt:x_vtbe 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∈Ras a stochastic functionalS_tdefined by

S_t:S₀ vt

k1

1ρxk

, t∈R, 2.5

whereS0 >0 is the initial value ofSt. We call this processSt_t∈Ra 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∈Rin2.6is 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 operatorsDxonC0Ras 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∈Rin2.5:

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_tis defined in2.5andS₀ 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∈ZandG_xtis defined byG_xt:Pθ_n1≤t|x_nx.

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_xtgx, 2.11

whereG_xt 1−G_xt, Gxt:Pθ_n1 ≤t|x_nx, gxis a bounded and continuous function onX.

2.2. Infinitesimal Operators of the GMRP

1ρ_Tx: ρx

T , T >0, 2.12

S^T_t :S₀

vtT k1

1ρ_Txk S₀

vtT

k1

1T⁻¹ρxk

. 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_vtand consider the processxt, γtonX×R. It is a Markov process with infinitesimal operator

Qfx, t : df

dt g_xt 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_xt 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×Rwith 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 timetT², that is,v ≡ vtT². 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 S₀, for allt ∈R. ConsiderS^T_t in the new scale of timetT²:

S_Tt:S^T_tT2S₀

v^tT2

k1

1T⁻¹ρxk

. 3.2

Due to more rapid jumps of vtT²the process S_Tt will be fluctuated near the pointS₀ as T → ∞. By similar arguments similar to 4.3–4.5 in 2, we obtain the following expression:

lnSTt S₀ T⁻¹

v^tT2

k1

ρxk−1 2T⁻²

v^tT2

k1

ρ²xk T⁻²

v^tT2

k1

r

T⁻¹ρxk

ρ²xk. 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^tT2

k1

ρ²xk

1

2tρ₂, 3.4

whereρ₂ :

Xpdx

XPx, dyρ²y/m. Using algorithms of diffusion approximation with respect to the first term in3.3we obtain4, page 88:

Tlim→∞T⁻¹

vtT² k1

ρxk σρwt, 3.5

where σ_ρ² :

Xpdx1/2

XPx, dyρ²y

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_ρ²andρ2are defined in3.4and3.5, respectively. From3.6we obtain

St S₀e^{σρwt−1/2tρ2}S₀e^−1/2tρ2e^σρwt. 3.7

Thus,St satisfies the following stochastic differential equationSDE:

dSt St 1

2

σ_ρ²−ρ₂

dtσ_ρdwt

. 3.8

In this way we have the following corollary.

Corollary 3.1. The ergodic diffusion GMRP has the form

St S₀e^−1/2tρ2e^σρwt, 3.9

and it satisfies the following SDE:

dSt St 1

2

σ_ρ−ρ₂

dtσ_ρdwt. 3.10

3.2. Merged Diffusion Approximation

Let us suppose that the balance condition satisfies the following:

ρk

Xkp_kdx

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^tT2

k1

ρ²xk

1 2

_t

0

ρ₂xsds, 3.12

where

ρ₂k:

Xkpkdx

XkP x, dy

ρ² 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^tT2

k1

ρxk

_t

0

σρxsdws, 3.14

where σ_ρ²k:

Xk

p_kdx

Xk

P x, dy

ρ² y

Xk

P x, dy

ρ y

R₀

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

S₀ lnSt S₀

_t

0

σρxsdws − 1 2

_t

0

ρ2xsds, 3.16

whereSt is the limitSTtasT → ∞. From3.16we obtain

St S0e^{−1/2t0ρ2xsdst0σρxsdws}. 3.17

Stochastic differential equationSDEfor ˇSthas the following form:

dSt St 1

2

σ_ρ²xt −ρ₂xt

dtσ_ρxtdwt, 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_Tt

S0 lnStˇ

S0 σˇ_ρwt−1

2pˇ₂t, 3.20

where

ˇ σ_ρ²:^r

k1

pkσ_ρ²k, pˇ2:^r

k1

pkρ2k, 3.21

andρ₂kandσ_ρ²kare defined in3.13and3.15, respectively. Thus, we obtain from3.20:

St ˇ S0e^{−1/2ˇp2tσˇρwt}. 3.22 Corollary 3.3. The diffusion GMRP under double averaging has the form

St ˇ S₀e^{−1/2ρˇ2tσˇρwt}, 3.23 and satisfies the SDE

dStˇ Stˇ 1

2

ˇ σ_ρ²−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_R0,∞ 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)

G^T_t :T⁻¹

vtT² k0

ρxk, G^T_n :G^T_τ

nT⁻¹, G^T₀ 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^Ts, x:fs T⁻¹φ¹_fs, x T⁻²φ_f²s, x, 4.3

whereφ¹_f andφ_f²are defined as follows:

P−Iφ_f¹s, x ρxfs, P−Iφ²_fs, x

−Ax A

fs, 4.4

where

A:

X

pdxAx, 4.5

andAx: ρ²x/2ρxR0−Iρxd²/ds². From the balance condition4.2and equality ΠA−Ax 0 it follows that both equations in 4.3 simultaneously solvable and the solutionsφⁱ_fs, xare bounded functions,i1,2.

We note that

f S^T_n1

−f G^T_n

1

Tρxndfxn

ds 4.6

and define

φ^Ts, x:fs T⁻¹φ¹_fs, x T⁻²φ_f²s, x, 4.7 whereφ¹_fs, xandφ²_fs, xare defined in4.4and4.5, respectively. We note, thatG^T_n1− G^T_n T⁻¹ρxn.

4.2. Martingale Problem for the Limiting Problem

in DA

ψ^Ts, t:φ^T

G^TtT2, x_tT²

−φ^T

G^Tst2, x_sT²

−^tT2−1

j^sT2 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^Ts, t are FtT²-martingale by t. Taking into account the expression4.6and 4.7, we find the following expression:

ψ^Ts, t f

G^TtT2

−f

G^TsT2

φ¹_f

G^TtT2, x_tT²

−φ¹_f

G^Tst2, x_sT²

²

φ²_f

G^TtT2, x_tT²

−φ_f²

G^TsT2, x_sT²

−T⁻¹

tT²−1 jsT²

⎧⎨

⎩ρ xj

df G^T_j dg E

φ¹_f

G^T_j, x_j1

−φ_f² G^T_j, xj

| Fj

⎫

⎬

⎭

−T⁻²

tT²−1 jsT²

⎧⎪

⎨

⎪⎩2⁻¹ρ² x_jdf

G^T_j dg ρ

x_j E

⎛

⎜⎝dφ¹_f

G^T_j, x_j1 dg | Fj

⎞

⎟⎠

E φ_f²

G^T_j, x_j1

−φ²_f G^T_j, x_j

| Fj

⎫

⎬

⎭o T⁻²

f

G^TtT2

−f

G^TsT2

φ_f¹

G^TtT2, x_tT²

−φ¹_f

G^TsT2, x_sT²

T⁻²

φ²_f

G^TtT2, x_tT²

−φ²_f

G^TsT2, x_sT²

−T^{−2 tT2−1}

j^sT2 Af

G^T_j O

T⁻² , 4.10

whereOT⁻²is the sum of terms withT⁻²nd order. Sinceψ^T0, tisFtT²-martingale with respect to measure Q_T, generated by process G_Tt in 4.1, then for every scalar linear continuous functionalη^s₀we have from4.8-4.10:

0E^T

ψ^Ts, tη^s₀ E^T

⎡

⎣

⎛

⎝f

G^TtT2

−f

G^TsT2

−T^{−2 tT2−1}

j^sT2 Af

G^T_j⎞

⎠η^s₀

⎤

⎦

−T⁻¹E^T

φ¹_f

G^TtT2, x_tT²

−φ_f¹

G^TsT2, x_sT²

η₀^s

−T⁻²E^T

φ²_f

G^TtT2, x_tT²

−φ_f²

G^TsT2, x_sT²

η₀^s

−O T⁻²

,

4.11

whereE^T is a mean value by measureQT. If the processG^T_tT2 converges weakly to some processG₀tasT → ∞, 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 coefficientσ₁²is positively defined, where

σ₁²:

X

πdx (ρ²x

2 ρxR0ρx )

, 4.14

then the processG0tis a Wiener process with varianceσ₁²in4.14:G0t σwt. Taking into account the renewal theorem forvt, namely,T⁻¹vtT²→T→∞t/m, and the following representation

GTt T⁻¹

v^tT2

k0

ρxk T^{−1 tT2}

k0

ρxk T⁻¹

vtT²

k^tT21ρxk 4.15

we obtain, replacingtT²byvtT², 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, whereA₀:A/m, and Ais defined in4.5-4.5.

4.3. Weak Convergence of the Processes

in DA

ΔTs, t:|GTt−G_Ts|

****

**T⁻¹

v^tT2

kv^sT21 ρxk

****

**

≤T⁻¹sup

x ρx***v tT²

−v sT²

−1***.

4.17

This representation gives the following estimation:

|ΔTt1, t₂||ΔTt2, t₃| ≤T⁻²

sup

x

ρx ₂***v

t₃T²

−v

t₁T²***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

for

as

in DA

such that measures QTn converge weakly to some measure Q onDR0,∞ asT → ∞, whereD_R0,∞is the Skorokhod space7, page 148. This measure is the solution of such martingale problem: the following process

ms, t:f G₀t

−f G₀s

− _t

s

A₀f G₀u

du 4.19

is aQ-martingale for allfg∈C²Rand

Ems, tη₀^s0, 4.20

for scalar continuous bounded functionalη^s₀,Eis a mean value by measureQ. From4.19it follows thatE^Tm^Ts, tη^s₀0, and it is necessary to show that the limiting passing in4.1goes to the process in3.12asT → ∞. From equality4.11we find that limTn→∞E^Tnms, tη^s₀ Ems, tη^s₀. Moreover, from the following expression

***E^Tms, tη^s₀−Ems, tη^s₀***≤***

E^T−E

ms, tη^s₀***E^T***ms, t−m^Ts, t******η^s₀***−→T→∞0, 4.21 we obtain that there exists the measureQonD_R0,∞which solves the martingale problem for the operatorA₀or, equivalently, for the processG₀tin 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 varianceσ₁²in4.14. As long as the semigroup is unique then the limit processG0tis unique. See3, Chapter 1.

4.5. Calculation of the Quadratic Variation for GMRP

n, the sequence

m^T_n:G^T_n−G^T₀ −ⁿ⁻¹

k0

E

G^T_k1−G^T_k | Fk

, G^T₀ g, 4.22

isFn-martingale, where Fn : σ{xk, θk; 0 ≤ k ≤ n}. From the definition it follows that the characteristicm^T_nof the martingalem^T_nhas the form

+ m^T_n,

ⁿ⁻¹

k0

E

m^T_k1−m^T_k2

| Fk

. 4.23

To calculatem^T_nlet us representm^T_nin4.22in 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_kT⁻¹ρxk, 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_tTgoes 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 timetT² from 4.27we obtain that characteristicm^T_tT2does not go to zero asT → ∞since

- m^TtT2

. T⁻²

^tT2−1

k0

I−Pρxk ²−→tσ₁², 4.30

whereσ₁²:

XπdxI−Pρx².

4.6. Rates of Convergence for GMRP

Consider the representation4.22for martingalem^T_n. It follows that

G^T_n gm^T_nⁿ⁻¹

k0

E

G^T_k1−G^T_k| Fk

. 4.31

In diffusion approximation scheme for GMRP the limit for the processG^T_tT2asT → ∞will be diffusion processSt see 3.10. Ifm0tis the limiting martingale form^T_tT2in4.22as T → ∞, then from4.31and3.10we obtain

E

G^TtT2−St

E

m^TtT2−m0t

T^{−1 tT2−1}

k0

ρxk−St. 4.32

SinceEm^T_tT2−m₀t 0,becausem^T_tT2 andm₀tare zero-mean martingalesthen from 4.32we obtain:

****E

G^TtT2−St **

**≤T⁻¹

****

**

^tT2−1

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

****

**

^tT2−1

k0

ρxk−StT

****

**C1t0, 4.34 whereC1t0is a constant depending ont0,t∈0, t0. From4.33,4.2, and4.32we obtain:

****E

G^TtT2−St **

**≤T⁻¹C₁t0. 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≥0in the phase spaceX. Let us have two ergodic classesX₀and 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:

P₀x, 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 P₀x, Abe uniformly ergodic in each classX_k, k 0,1 and have a stationary distribution π_kdxin 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∈Zwith 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_xtdt. 5.6

5.2. Algorithms of Phase Averaging with Two Ergodic Classes

The merged Markov chainx_nn∈Zin merged phase spaceXis given by matrix of transition probabilities

P p_kr

k,r0,1;

p₀₁ 1−p₁₁

X1

π₁dxPx, X0 1−

X1

π₁dxPx, 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_kdxmx, 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^−Λkt

, 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

St S₀e^{−1/20tρ2xsds0tσρxsdws} 5.10

which satisfies the stochastic differential equationSDE:

dSt St 1

2

σ_ρ²xt−ρ²xt

dtσ_ρxtdwt, 5.11

where

ρ²1:

X1

p₁dx

X1

P x, dy

ρ² y

m1 ,

ρ²0:

X0

p0dx

X0

P x, dy

ρ² y

m0 ,

σ²_ρ1:

X1

p1dx

X1

P x, dy

ρ² y

X1

P x, dy

ρ y

R0

X1

P x, dy

ρ y

m1 ,

σ²_ρ0:

X0

p₀dx

X0

P x, dy

ρ² y

X0

P x, dy

ρ y

R₀

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 ρ₂

X

pdx

X

P x, dy

ρ² y

m , 6.2

σ_ρ²

X

pdx (1

2

X

P x, dy

ρ² y

X

P x, dy

ρ y

R₀P x, dy

ρ y

m . 6.3

The risk-neutral measureP^∗for the process in6.1is:

dP^∗ P exp

/

−θt−1 2θ²wt

0

, 6.4

where

θ

1/2

σ_ρ−ρ₂

−r σρ

. 6.5

UnderP^∗, the processe^−rtS_tis a martingale and the processw^∗t wt θtis a Brownian motion. In this way, in the risk-neutral world, the processS_thas the following form

dSt

St rdtσ_ρdw^∗t. 6.6

Using Black-Scholes formulasee8we obtain the European call option pricing formula for our model6.6:

CS0Φd−Ke^−rTΦd₋, 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_ρis defined in6.3.