explicit solution processes for nonlinear jump ...

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Article

Journal of Nonlinear Mathematical Physics, Vol. 17, No. 3 (2010) 281–292 ¨ c G. Unal, H. Turkeri and C. M. Khalique DOI: 10.1142/S1402925110000908

J. Nonlinear Math. Phys. 2010.17:281-292. Downloaded from www.worldscientific.com by 103.14.233.154 on 06/19/14. For personal use only.

EXPLICIT SOLUTION PROCESSES FOR NONLINEAR JUMP-DIFFUSION EQUATIONS ∗,†,§ ¨ GAZANFER UNAL , HASRET TURKERI‡,¶ and CHAUDRY MASOOD KHALIQUE∗, ∗

International Institute for Symmetry Analysis and Mathematical Modelling Department of Mathematical Sciences North-West University, Mafikeng Campus Private Bag X2046, Mmabatho 2735 Republic of South Africa †

Department of International Finance and Department of Mathematics Yeditepe University, Istanbul, Turkey ‡

Department of Mathematics Istanbul Technical University Istanbul, Turkey § [email protected] ¶ [email protected] [email protected] Received 18 April 2009 Accepted 7 December 2009 Recent studies have shown that the nonlinear jump-diffusion models give results which are in agreement with financial data. Here we provide linearization criteria together with transformations which linearize the nonlinear jump-diffusion models with compound Poisson processes. Furthermore, we introduce the stochastic integrating factor to solve the linear jump-diffusion equations. Extended Cox–Ingersoll–Ross, Brennan–Schwartz and Epstein models are shown to be linearizable and their explicit solutions are presented. Keywords: Compound Poisson processes; linearization conditions; stochastic integrating factor; explicit solution processes; stochastic differential equations.

1. Introduction Spiky behavior observed in financial data has been accounted for jumps [8, 17]. Recent financial studies [1, 2, 8, 9, 11, 17, 18] have shown that the jumps play a prominent role in asset price and the interest rates. Financial models in these studies are given by jumpdiffusion equations with compound Poisson processes. It has been shown [8, 17] that jumps are due to surprises in macroeconomy. Moreover majority of these financial models are nonlinear jump-diffusion equations [23]. To the best of our knowledge there exists no method to provide explicit (exact analytical) solutions for these models. However, extension of normal form theory (initiated by Poincaré) to the nonlinear stochastic differential equations 281

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allows one to study the stochastic bifurcations analytically (via approxiamte linearization process) as it has been expounded in [4]. In this paper we undertake this task and provide linearization criteria for nonlinear jump-diffusion equations. We then use stochastic integrating factors for linearizable equations and obtain the explicit solutions (stochastic processes). Let (Ω, F, P ) be a probability space with filtration {Ft }t≥0 . Consider a real-valued stochastic process Xt (t ≥ 0) adapted to the filtration and satisfying the nonlinear jumpdiffusion equation of the form


dXt = f (Xt− , t)dt + g(Xt− , t)dWt + r(Xt− , t)dCt ,

X0 = x0

(1.1)

where dWt is the infinitesimal increment of the Wiener process Wt (t ≥ 0) and independently dCt is the infinitesimal increment of the compound Poisson process dCt = ZNt dNt ,

Ct =

Nt

Zi .

(1.2)

i=1

Here Nt (t ≥ 0) is a Poisson process with arrival rate λ. Zi are independent and identically distributed random variables and independent of Wt and Nt . We take Nt and Xt as right continuous. Xt− is the left limit of Xt at t− just before the jump at t. Nonlinear jumpdiffusion Eq. (1.1) could also be considered as a stochastic differential equation driven by finite time activity Lévy process since the latter can be decomposed into a drift, a Wiener and a compound Poisson process according to Lévy–Itˆ o decomposition theorem [3]. The outline of the paper is as follows. In Sec. 2, we derive the linearization criteria imposed on f (Xt− , t), g(Xt− , t) and r(Xt− , t). We also determine the invertible transformations which linearize the nonlinear jump-diffusion Eq. (1.1). Then in Sec. 3 we introduce the stochastic integrating factors to solve linear jump-diffusion equations and provide solution processes for insurance [16, 20] equations and asset price model [18]. In Sec. 4 we show that Cox–Ingersoll–Ross (CIS) model [7], Brennan and Schwartz model [5] and Epstein model [10] are linearizable and give their explicit solutions. These explicit solutions can be used to check the numerical schemes and more importantly to develop derivative pricing formulas. Finally, concluding remarks are made in Sec. 5. 2. Linearization Criteria for Nonlinear Jump-Diffusion Equations There is no general method to solve the nonlinear jump-diffusion equations which involve compound Poisson processes. Here we will look for invertible mappings to transform nonlinear jump-diffusion Eq. (1.1) to a linear one. This can only be done when f (Xt− , t), g(Xt− , t) and r(Xt− , t) satisfy certain conditions. In this section linearization criteria together with the linearizing transformations will be determined. We seek an invertible mapping h : R × R+ → R Yt = h(Xt , t)

with

∂h = 0, ∂X

∂h = 0 ∂t

(2.1)

to transform (1.1) into a linear equation of the form dYt = (a1 (t)Yt− + a2 (t))dt + (b1 (t)Yt− + b2 (t))dWt +(c1 (t, ZNt )Yt− + c2 (t, ZNt ))dNt .

(2.2)

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Itˆ o lemma [22] for h(X, t) leads to ∂h(Xt− , t) ∂h(Xt− , t) 1 2 ∂ 2 h(Xt− , t) + f (Xt− , t) + g (Xt− , t) dt dYt = 2 ∂t ∂Xt− 2 ∂Xt− ∂h(Xt− , t) dWt + g(Xt− , t) ∂Xt− + [h(Xt− + r(Xt− , t)ZNt , t) − h(Xt− , t)]dNt .

(2.3)


From (2.2) and (2.3) we obtain the following deterministic PDEs: ∂h(x, t) 1 2 ∂ 2 h(x, t) ∂h(x, t) + f (x, t) + g (x, t) = a1 (t)h(x, t) + a2 (t), ∂t ∂x 2 ∂x2 ∂h(x, t) = b1 (t)h(x, t) + b2 (t), g(x, t) ∂x h(x + r(x, t)z, t) − h(x, t) = c1 (t, z)h(x, t) + c2 (t, z).

(2.4) (2.5) (2.6)

Here Xt− = x and ZNt = z. The PDE (2.5) has two distinct solutions for b1 (t) = 0 (t ∈ [0, T ]) and for b1 (t) = 0 (t ∈ [0, T ]) and so we consider each case separately. Case 1. b1 (t) = 0 (t ∈ [0, T ]) In this case the solution of (2.5) is

h(x, t) = b2 (t)

dx , g(x, t)

(2.7)

where we have chosen the arbitrary function of integration to be zero, because it does not alter the course of linearization. Substituting (2.7) into (2.4) we obtain 1 dx ∂ f (x, t) 1 ∂ + b2 (t) dx + b2 (t) − b2 (t) g(x, t) b˙ 2 (t) g(x, t) ∂t g(x, t) g(x, t) 2 ∂x dx + a2 (t), (2.8) = a1 (t)b2 (t) g(x, t) where the overhead dot denotes differentiation with respect to t. Differentiating (2.8) and rearranging yields, 1 dx (b˙ 2 (t) − a1 (t)b2 (t)) g(x, t) 1 f (x, t) 1 ∂ ∂ dx + − g(x, t) = a2 (t). (2.9) + b2 (t) ∂t g(x, t) g(x, t) 2 ∂x Differentiation of (2.9) with respect to x leads to 1 ∂K ∂ b˙ 2 (t) − a1 (t)b2 (t) + b2 (t) + = 0, g(x, t) ∂t g(x, t) ∂x where K=

f (x, t) 1 ∂g(x, t) − . g(x, t) 2 ∂x

(2.10)

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From (2.10) we obtain b˙ 2 (t) − a1 (t)b2 (t) = −g(x, t)L b2 (t) where


∂ L= ∂t

1 g(x, t)

+

(2.11)

∂K . ∂x

Differentiation of (2.11) with respect to x leads to ∂ g(x, t)L = 0. ∂x

(2.12)

Rewriting (2.6) as h(x + r(x, t)z, t) = (1 + c1 (t, z))h(x, t) + c2 (t, z) and differentiating it with respect to x yields ∂r(x, t) ∂h(x, t) ∂h(x + r(x, t)zt , t) 1+ z = (1 + c1 (t, z)) . ∂x ∂x ∂x Substitution of (2.7) into (2.14) gives b2 (t) ∂r(x, t) b2 (t) 1+ z = (1 + c1 (t, z)) g(x + r(x, t)z, t) ∂x g(x, t) and rearranging (2.15) yields ∂r(x, t) g(x, t) 1+ z = 1 + c1 (t, z). g(x + r(x, t)z, t) ∂z Differentiation of (2.16) with respect to x leads to g(x, t) ∂r(x, t) ∂ (1 + z) = 0. ∂x g(x + r(x, t)z, t) ∂x

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

Hence, the linearization criterion (2.12) and (2.17) must be satisfied for (1.1) to be linearizable via transformation (2.7). Case 2. b1 (t) = 0 (t ∈ [0, T ]) The solution of (2.5) in this case becomes dx b2 (t) + H(t) exp b1 (t) . h(x, t) = − b1 (t) g(x, t)

(2.18)

We choose H(t) = 1 and b2 (t) = 0 because it does not affect the linearization conditions. Therefore dx . (2.19) h(x, t) = exp b1 (t) g(x, t)

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Substitution of h(x, t) into (2.5) gives, 1 f (x, t) 1 ∂ b21 ∂ dx ˙b1 + b1 dx + − g(x, t) + − a1 g(x, t) ∂t g(x, t) g(x, t) 2 ∂x 2 dX = a2 (t). (2.20) × exp b1 (t) g(X, t) After some algebra we obtain


g(x, t)

∂(g(x, t)L) ˙ + b1 + b1 (t)g(x, t)L = 0. ∂x

Differentiation of (2.21) with respect to x leads to ∂ ∂ ∂ g(x, t) (g(x, t)L) + b1 (t) (g(x, t)L) = 0 ∂x ∂x ∂x

(2.21)

(2.22)

and so b1 (t) = −

∂ ∂x

∂ (g(x, t)L) g(x, t) ∂x ∂ ∂x

(g(x, t)L)

.

Differentiation of (2.23) with respect to x gives

∂ ∂ (g(x, t)L) g(x, t) ∂ ∂x ∂x = 0. ∂ ∂x (g(x, t)L) ∂x

(2.23)

(2.24)

Substitution of (2.19) into (2.14) gives ∂r(x, t) dx b1 (t) exp b1 (t) 1+ z g(x + r(x, t)z, t) g(x, t) ∂x b1 (t) dx exp b1 (t) . = (1 + c1 (t, z)) g(x, t) g(x, t)

(2.25)

Rearranging (2.25) we obtain

dx exp b 1 g(x+r(x,t)z,t) g(x, t) ∂r(x, t)

z = 1 + c1 (t, z). 1+ dx ∂x g(x + r(x, t)z, t) exp b 1

(2.26)

g(x,t)

Differentiation of (2.26) yields ∂r(x, t) ∂r(x, t) 1 1 ∂ 1+ z Φ − 1+ z Φb1 − ∂x ∂x ∂x g(x + r(x, t)z, t) g(x, t) dx exp b1 (t) g(x+r(x,t)z,t) = 0, (2.27) × dx exp b1 (t) g(x,t)

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where Φ=

g(x, t) . g(x + r(x, t)z, t)


Substitution of the value of b1 (t) from (2.23) into (2.27) yields ∂r(x, t) ∂r(x, t) 1 1 ∂ 1+ z Φ + 1+ z − ∂x ∂x ∂x g(x + r(x, t)z, t) g(x, t)

∂ ∂ ∂x g(x, t) ∂x (g(x, t)L) × Φ = 0. (2.28) ∂ ∂x (g(x, t)L) Hence, the linearization criterion (2.24) and (2.28) must be satisfied for (1.1) to be linearizable via transformation (2.19). Note that when ∂ g(x, t) ∂r(x, t) ∂ (gL) = 0 and 1+ z =0 ∂x ∂x g(x + r(x, t)z, t) ∂x hold then the transformation

Y =

dx g(x, t)

casts (1.1) into dYt = (a1 (t)Yt− + a2 (t))dt + b2 (t)dWt + (c1 (t, ZNt )Yt− + c2 (t, ZNt ))dNt . Also when ∂ ∂x and ∂ ∂x

∂ ∂x

∂ (g(x, t)L) g(x, t) ∂x ∂ ∂x

(g(x, t)L)

=0

∂r(x, t) ∂r(x, t) 1+ z Φ + 1+ z ∂x ∂x

∂ ∂ 1 1 ∂x g(x, t) ∂x (g(x, t)L) Φ=0 − × ∂ g(x + r(x, t)z, t) g(x, t) ∂x (g(x, t)L)

are satisfied the transformation

h(x, t) = exp b1 (t)

dx g(x, t)

transforms (1.1) to dYt = (a1 (t)Yt− + a2 (t))dt + (b1 (t)Yt− + b2 (t))dWt + (c1 (t, ZNt )Yt− + c2 (t, ZNt ))dNt where b1 (t) = −

∂ ∂x

∂ (g(x, t)L) g(x, t) ∂x ∂ ∂x (g(x, t)L)

.

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We now summarize our result in the following theorem. Theorem 1. A nonlinear stochastic differential equation (1.1) is linearizable via the transformation dx h(x, t) = g(x, t) if and only if (2.12) and (2.17) are satisfied or via h(x, t) = exp b1 (t)

dx g(x, t)


if and only if (2.24) and (2.28) are satisfied where b1 (t) is given by (2.23). 3. Stochastic Integrating Factor Method for Linear Jump-Diffusion Equations To the best of our knowledge there is no work on the solution methods for the jump-diffusion equations which involve compound Poisson processes in the literature. Here we will develop the method of the stochastic integrating factors. Definition 1. The function M = M (t, Wt , Nt ; Zi ) with property d(M Yt ) = D1 (t, M )dt + D2 (t, M )dWt + D3 (t)Zt dPt is an integrating factor for the linear jump-diffusion Eq. (2.2). We now consider the chain rule [15] d(M Yt ) = M dYt + Yt dM + dYt dM.

(3.1)

Here dM is [14] dM =

1 ∂2M ∂M + ∂t 2 ∂y 2

dt +

∂M dy + (M (t, y, n + 1) − M (t, y, n)) dn ∂y

(3.2)

and dM dYt is dM dYt =

∂M (b1 Yt + b2 )dt + (M (t, y, n + 1) − M (t, y, n)) ∂y × (c1 (t, z)Yt + c2 (t, z))dn,

(3.3)

y = Wt and n = Nt and we have made use of the following multiplication rules [14] dtdNt = 0,

dNt dWt = 0,

dNtm = dNt ,

dtdWt = 0,

dWt dWt = dt,

dWtm = 0.

Equation (3.1) can now be rewritten as 1 ∂2M ∂M ∂M ∂M + + a1 (t)M dt + b1 (t)M + dy + b1 (t) d(M Yt ) = ∂t 2 ∂y 2 ∂y ∂y + [(1 + c1 (t, z)) (M (n + 1) − M (n))] dn] Yt− ∂M dt + b2 (t)M dy + c2 (t, z)M dn. + a2 (t)M + b2 (t) ∂y

(3.4)

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The right-hand side of the equation should not involve the variable Y to comply with the definition of the integrating factor. This leads to the following three equations: 1 ∂2M ∂M ∂M + + a1 (t)M = 0, + b1 (t) 2 ∂t 2 ∂y ∂y

(3.5)

∂M + b1 (t)M = 0, ∂y

(3.6)

(1 + c1 (t, z))M (t, y, n + 1) = M (t, y, n).

(3.7)


Equation (3.7) has a solution of the form M (t, Wt , Nt ; Zi ) = M1 (t, Wt )

Nt i=1

1 , 1 + c1 (t, Zi )

(3.8)

where M1 is an arbitrary function of its arguments. Substitution of this value of M into (3.6) yields M (t, Wt , Nt ) = e−

Rt

b1 (s)dWs +q(t)

Nt i=1

1 1 + c1 (t, Zi )

(3.9)

and finally substitution of (3.9) into (3.5) leads to dq b21 (t) − + a1 (t) − Ω(t, Nt ; Zi ) = 0, dt 2

(3.10)

where Ω(t, Nt ; Zi ) =

Nt

1 i=1 1+c1 (t,Zi ) . Nt 1 i=1 1+c1 (t,Zi )

d dt

Equation (3.10) has the solution q(t) =

1 2

t

b21 (s)ds −

t

a1 (s)ds +

t

Ω(s, Ns ; Zi )ds

(3.11)

and substitution of this value of q(t) into (3.9) yields 1 M (t, Wt , Nt ; Zi ) = 1 + c1 (t, Zi ) i=1 t t t 1 2 b (s) − a1 (s) ds − b1 (s)dWs + Ω(s, Ns )ds . × exp 2 1 N t

(3.12)

Invoking (3.12) in (3.4) leads to d(M Yt ) = (a2 (t) − b1 (t)b2 (t))M dt + b2 (t)M dWt + c2 (t, ZNt )M (Nt + 1)dNt .

(3.13)

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Integrating (3.13) we obtain the solution t t 1 (a2 (s) − b1 (s)b2 (s))M (s, Ws , Ps )ds + b2 (s)M (s, Ws , Ns )dWs Yt = M 0 0 t + c2 (s, ZNs )M (Ns + 1)dNs + Y0 . (3.14) 0


We now consider two linear examples. The accumulated value of aggregate claim Yt up to time t, can be modelled by compound Poisson process Ct given in (1.2). In this case Zi are the claim amounts and Nt is the number of claims up to time t. An insurance model with risk free interest rate δ is given by [16, 20] dYt = δYt− dt + dCt ,

Y0 = 0.

Notice that for this example we have a1 = δ, a2 = 0, b1 = 0, b2 = 0, c1 = 0 and c2 = 1. Integrating factor (3.12) now becomes M (t) = e−δt . Aggregate claim amount Yt is t ZNs eδ(t−s) ds, Yt = 0

Yt =

Nt

Zi eδ(t−si ) .

i=1

Jump-diffusion model for the asset price has been proposed by Merton in [19]. Merton model of asset price Yt has been extended by Kou in [18] to incorporate the asymmetric leptokurtic features in asset pricing, and the volatility smile. Kou’s model is given by dYt = a1 Yt− dt + b1 Yt− dWt + c1 Yt− dCt ,

Y0 = y 0 ,

where a1 , b1 and c1 are constant parameters which will be determined by calibration. In this case stochastic integrating factor (3.12) becomes Nt 1 2 1 b1 − a1 t − b1 Wt . M (t, Wt , Nt ; Zi ) = exp 2 1 + c1 Zi i=1

Asset price is

Yt = Y0 exp

Nt 1 2 b1 − a1 t − b1 Wt (1 + c1 Zi ) . 2 i=1

4. Applications There has been great evidence in the observations of financial data that the tails in the returns of interest rates are due to jumps [1, 2]. Impact of jumps on economy have been discussed in [17] and it is shown that surprises in macroeconomy leads to jumps in the interest rate. Following Johannes [17] we now consider some nonlinear jump diffusion models of interest rates. Explicit solution to these interest rate models are important since the pricing derivative securities relies on the interest rate processes [18]. First example is an extension of the celebrated interest rate model CIR [7] with an additional jump term and it reads (4.1) dXt = α (β − Xt− ) dt + σ Xt− dWt + Xt− dCt , X(0) = x0 .

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√ This model satisfies the linearization conditions (2.12) and (2.17) when σ = 2 αβ. Therefore, according to Theorem 1, the transformation 1 Xt (4.2) Yt = √ αβ linearizes (4.1) to


α dYt = − Yt− dt + dWt + ( 1 + Zi − 1)Yt− dNt . 2

(4.3)

Invoking formula given in (3.14) we obtain N Nt s a t 1 √ Yt = ( 1 + Zi ) exp − (t − s) dWs . 2 1 + Zi 0 i=1 i=1 Hence the interest rate process is N N 2 s t t a a √ 1 √ x0 + αβ ( 1 + Zi )e− 2 t . e 2 s dWs Xt = 1 + Z i 0 i=1 i=1 Our second example is an extension of Brennan and Schwartz interest rate model [5] with an additional jump term (log mean-reverting Ornstein–Uhlenbeck equation with an additional jump [21]) is dXt = µXt− (θ − ln Xt− ) dt + ρXt− dWt + Xt− dCt ,

X(0) = x0 .

(4.4)

This model also satisfies the linearization conditions (2.12) and (2.17) for any µ and ρ. Hence, according to Theorem 1, the transformation Yt = linearizes (4.4) to dYt =

µ ρ µθ − Yt− + + ρ 2 ρ

1 ln Xt , ρ

dt + dWt +

(4.5)

1 ln (1 + ZNt ) dNt . ρ

(4.6)

Using formula given in (3.14) we obtain t t µ µ ρ µθ + exp (s − t) ds + exp (s − t) dWs Yt = 2 ρ ρ ρ 0 0 t µ 1 ln[1 + ZNs ] exp (s − t) dNs . + ρ 0 ρ Hence the interest rate Xt is   t t µ µ ρ µθ + exp (s − t) ds + exp (s − t) dWs  ρ 2 ρ ρ ρ   0 0 Xt = x0 exp . t   µ 1 ln[1 + ZNs ] exp (s − t) dNs . + ρ 0 ρ The third example we consider here arises in several fields. For instance in population growth model in a noisy environment with a jump term [12] and in valuation of the firm [10].

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Equation Criteria (2.12), (2.17)

291

Table 1. Linearizable Equations and Solutions. p dXt = α(β − Xt− )dt + σ Xt− dWt + Xt− dCt , X(0) = x0 . Solution Process RtQ s √ Q t √ a a √ 2 s dWs )))2 √ 1 ( 1 + Zi )e− 2 t ( 0 N Xt = ( x0 + αβ( N i=1 i=1 ( 1+Z )e i




dXt = µXt− (θ − ln Xt− )dt + ρXt− dWt + Xt− dCt , X(0) = x0 . Solution Process 3 Rt [µ ρ (s−t)] ds ρ( 0 ( ρ2 + µθ ρ )e 5 Xt = x0 exp 4 R µ Rt t µ + 0 e[ ρ (s−t)] dWs + 0 ρ1 ln[1 + Zt ]e[ ρ (s−t)] dPs ) 2

dXt = ξXt− (η − Xt− )dt + δXt− dWt + Xt− dCt , X(0) = x0 . Solution Process Rt Q s Xt = X0 ( i=1 ( Zi1+1 )[ 0 ξ N i=1 (Zi + 1) QNt

2

× exp[( δ2 − ξη)(t − s) + δ(Wt − Ws )]ds])−1

It is dXt = ξXt− (η − Xt− ) dt + δXt− dW + Xt− dCt ,

X(0) = x0 .

(4.7)

This model satisfies the linearization conditions (2.24, 2.28) for any ξ, η and δ. Hence according to Theorem 1 the transformation Yt =

1 , Xt

(4.8)

linearizes (4.7) into

ZNs Yt− dNt . dYt = (δ2 − ξη)Yt− + ξ dt + δYt− dWt + − ZNs + 1

(4.9)

Using formula given in (3.14) we obtain t 2 Nt Ns δ 1 − ξη (t − s) + δ(Wt − Ws ) ds . ξ (Zi + 1) exp Yt = Zi + 1 2 0 i=1

i=1

Hence the solution process Xt is N t 2 −1 Ns t δ 1 − ξη (t − s) + δ(Wt − Ws ) ds ξ (Zi + 1) exp . Xt = x0 Zi + 1 2 0 i=1

i=1

The results are reproduced in Table 1. 5. Concluding Remarks We have shown that when certain conditions are satisfied, nonlinear jump-diffusion equations are linearizable via invertible transformations. Stochastic integrating factors were introduced to solve the linear jump-diffusion equations. Explicit solution processes have been obtained for extended CIR, Brennan–Schwartz and Epstein models of finance via linearization and stochastic integrating factor methods.

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Acknowledgments ¨ G. Unal thanks the Faculty Research Committee of FAST, North-West University, Mafikeng Campus, for the financial support and CM Khalique for his kind hospitality during his visit to IISAMM. The authors thank the anonymous reviewer for bringing to their attention the reference by L. Arnold and P. Imkeller.


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