upper bounds for ruin probabilities under stochastic interest rate and optimal investment strategies

Acta Mathematica Sinica, English Series

Jul., 2012, Vol. 28, No. 7, pp. 1421–1430

Published online: February 21, 2012

DOI: 10.1007/s10114-012-0153-9

Http://www.ActaMath.com

Acta Mathematica Sinica, English Series© Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2012

Upper Bounds for Ruin Probabilities under Stochastic Interest Rate

and Optimal Investment Strategies

Jin Zhu LI Rong WUSchool of Mathematical Sciences and LPMC, Nankai University,

Tianjin 300071, P. R. China

E-mail : [email protected] [email protected]

Abstract In this paper, we study the upper bounds for ruin probabilities of an insurance company

which invests its wealth in a stock and a bond. We assume that the interest rate of the bond is stochastic

and it is described by a Cox–Ingersoll–Ross (CIR) model. For the stock price process, we consider both

the case of constant volatility (driven by an O–U process) and the case of stochastic volatility (driven

by a CIR model). In each case, under certain conditions, we obtain the minimal upper bound for ruin

probability as well as the corresponding optimal investment strategy by a pure probabilistic method.

Keywords Cox–Ingersoll–Ross model, jump-diffusion model, optimal investment, Ornstein–Uhlen-

beck (O–U) process, ruin probability, stochastic interest rate

MR(2000) Subject Classification 91B30, 60H30

1 Introduction

In recent years, there has been an increasing attention on the topic about ruin probability of aninsurance company which has the possibility to invest in a risk asset. One of the seminal papersis [1], in which an optimal investment strategy minimizing the ruin probability is obtained.In that paper, it is assumed that no money is invested in the bond and the risk process isdescribed by its diffusion approximation. Hipp and Plum [2] considered the jump-diffusionwealth model and obtained the investment strategy which minimizes the ruin probability bysolving a Hamilton–Jacobi–Bellman (HJB) equation. Under the same hypotheses as in [2],Gaier et al. [3] derived an exponential bound that improves the classical Lundberg result. Theinvestment strategy used in their paper is investing a constant amount of money in the stock,and such strategy was proved to be asymptotically optimal by Hipp and Schmidli [4]. Utilizingdynamic programming principle, Yang and Zhang [5] studied the optimal policy for a generalobjective function which contains minimizing the ruin probability as a special case. Assumingthat the stock price is an O–U process, Baev and Bondarev [6] obtained an estimate of theupper bound for ruin probability and the optimal investment strategy minimizing the estimate.Recently, Liang and Guo [7] considered the optimal investment and reinsurance to minimizethe upper bound for ruin probability from a different point of view to maximize the adjustment

Received March 19, 2010, revised December 21, 2010, accepted March 23, 2011

Supported by National Basic Research Program of China (973 Program, Grant No. 2007CB814905) and National

Natural Science Foundation of China (Grant No. 10871102)

1422 Li J. Z. and Wu R.

coefficient. They obtained the explicit results in the diffusion approximation case as well as inthe jump-diffusion case.

It is worth saying that all the works mentioned above assumed that the interest rate isa constant and the stock price follows a geometric Brownian motion or an O–U process withconstant volatility. The main method they used is solving the HJB equations according to thestochastic control theory.

In this paper, more realistically, we study the jump-diffusion wealth models with a stochasticinterest rate described by a CIR model. For the stock price process, we consider both the caseof constant volatility (driven by an O–U process) and the case of stochastic volatility (drivenby a CIR model). In each case, by a pure probabilistic method, we obtain the minimal upperbound for ruin probability and the corresponding optimal investment strategy.

The remainder of this paper is organized as follows. In Section 2, we introduce the wealthmodels and some necessary preliminaries. In Section 3, we obtain the minimal upper boundfor ruin probability and the corresponding optimal investment strategy when the stock price isdriven by an O–U process with constant volatility. In Section 4, we study the problem whenthe stock price is driven by a CIR model with stochastic volatility. At last, in Section 5, weconclude this paper by giving a comprehensive theorem.

2 Models and Preliminaries

Let (Ω, F , P ) be a probability space with filtration {Ft} containing all objects defined in thefollowing.

Without loss of generality, we assume that there are only two assets available for an insurancecompany in the financial market: a bond and a stock. The price of the bond at time t is denotedby Bt which satisfies

dBt = rtBtdt. (2.1)

Here rt is the stochastic interest rate described by the CIR model

drt = (β − αrt)dt + σ0√

rtdW(0)t , r0 = 0, (2.2)

where {W (0)t } is a standard Brownian motion, β, α and σ0 are positive constants satisfying

2β > σ20 which implies that rt ≥ 0 for all t ≥ 0 (see [8]).

For the stock price process St, there are various stochastic models that can be used todescribe it. The commonly-used one is the geometric Brownian motion with the form

St = S0 exp{

δt − σ2

2t + σWt

}, (2.3)

where S0, δ and σ are positive constants, and {Wt} is a standard Brownian motion. This modelhas been widely used in describing the stock price because of its simple expression.

In this paper, we propose different stock price models which generalize the model (2.3). Forthe case of constant volatility, we assume that

S(1)t = S0 exp

{δ1t − σ2

1

2t + ξt

}, (2.4)

where {ξt} is an O–U process satisfying the stochastic differential equation (SDE)

dξt = −γξtdt + σ1dW(1)t , ξ0 = 0. (2.5)

Upper Bounds for Ruin Probabilities 1423

Here γ, σ1 are positive constants, and {W (1)t } is a standard Brownian motion independent of

{W (0)t }. Applying Ito’s formula on (2.4) leads to

dS(1)t = S

(1)t [(δ1 − γξt)dt + σ1dW

(1)t ]. (2.6)

For the case of stochastic volatility, let

S(2)t = S0 exp {δ2t − bt + ηt} ,

where {ηt} follows a CIR model satisfying the SDE

dηt = (b − aηt)dt + σ2√

ηtdW(2)t , η0 = 0. (2.7)

Here b, a, σ2 are positive constants with 2b > σ22 , and {W (2)

t } is a standard Brownian motionindependent of {W (0)

t }. We can also derive the differential form of (2.7) as

dS(2)t = S

(2)t

{[δ2 +

(σ2

2

2− a

)ηt

]dt + σ2

√ηtdW

(2)t

}. (2.8)

In what follows, we denote by c > 0 the premium rate of the insurance company. Theaccumulated claim process is modeled by a compound Poisson process Rt, that is,

Rt =Nt∑i=1

Yi,

where {Nt} is a standard Poisson process with intensity λ > 0, and the claim sizes {Yi} arepositive i.i.d. random variables with the common distribution function F and with finite meanμ. We also assume that {Yi} are independent of {Nt}, and that the safety loading c > λμ

holds. It is well known that Rt can be represented as the stochastic integral with respect tothe Poisson measure, that is

Rt =∫ t

0

∫yv(dy, ds), (2.9)

where v(·, ·) is the Poisson measure satisfying Ev(A, t) = λt∫

AF (dy) for every measurable set

A and t > 0.Let the positive constant πi be the proportion of the total wealth invested in the stock

{S(i)t }, i = 1, 2. We suppose that {Rt} is independent of {Bt}, and {S(i)

t }, i = 1, 2. If wedenote by X

(i)t the wealth process of the company at time t with X

(i)0 = ui, then we have

dX(i)t = πiX

(i)t

dS(i)t

S(i)t

+ (1 − πi)X(i)t

dBt

Bt+ cdt − dRt, i = 1, 2. (2.10)

Substituting (2.1), (2.6), (2.8) and (2.9) into (2.10) shows that

dX(1)t = X

(1)t [δ1π1 + (1 − π1)rt − γπ1ξt] dt

+ X(1)t σ1π1dW

(1)t + cdt −

∫yv(dy, dt) (2.11)

and

dX(2)t = X

(2)t

[δ2π2 + (1 − π2)rt +

(σ2

2

2− a

)π2ηt

]dt

+ X(2)t σ2π2

√ηtdW

(2)t + cdt −

∫yv(dy, dt). (2.12)


Meanwhile, we introduce two auxiliary processes:

X(1)t = exp

{−δ1π1t − (1 − π1)

∫ t

0

rsds + γπ1

∫ t

0

ξsds +σ2

1π21

2t − σ1π1W

(1)t

}(2.13)

and

X(2)t = exp

{− δ2π2t − (1 − π2)

∫ t

0

rsds +[σ2

2π22

2−

(σ2

2

2− a

)π2

] ∫ t

0

ηsds

− σ2π2

∫ t

0

√ηsdW (2)

s

}. (2.14)

Easy calculation by Ito’s formula leads to

d(X(i)t X

(i)t ) = X

(i)t

(cdt −

∫yv(dy, dt)

), i = 1, 2.

Noting X(i)0 = 1, we have

X(i)t = (X(i)

t )−1

[ui +

∫ t

0

X(i)s (c − λμ)ds −

∫ t

0

∫X(i)

s yv(dy, ds)]

, i = 1, 2, (2.15)

where v(A, t) = v(A, t) − Ev(A, t).The central task of this paper is to find the sharper upper bounds for ruin probabilities

defined asΦ(i)(ui) = P

{inf

0<t<∞X

(i)t < 0

∣∣∣X(i)0 = ui

}, i = 1, 2. (2.16)

3 The Case of Constant Volatility

In this section, we consider wealth model (2.11) in which the stock price is driven by an O–Uprocess with constant volatility. Additionally, we assume 0 < π1 ≤ 1. By (2.15) and notingthat X

(i)t > 0 , we get, for a fix T > 0,

P{

inf0<t≤T

X(i)t < 0

∣∣∣X(i)0 = ui

}

= P

{inf

0<t≤T(X(i)

t )−1

[ui +

∫ t

0


∫ t

0

∫X(i)

s yv(dy, ds)]

< 0}

= P

{inf

0<t≤T

[ui +

∫ t

0


∫ t

0

∫X(i)

s yv(dy, ds)]

< 0}

≤ P

{sup

0<t≤T

[ ∫ t

0

∫X(i)

s yv(dy, ds)]

> ui

}

≤ E( ∫ T

0

∫X

(i)s yv(dy, ds)

)2

u2i

, i = 1, 2, (3.1)

where the last inequality follows from the Kolmogorov inequality for martingales. The remainingproblem is to estimate

sup0≤t<∞

E

(∫ t

0

∫X(i)

s yv(dy, ds))2

, i = 1, 2. (3.2)

For this purpose, when i = 1, we need the following results.

Proposition 3.1 Consider the O–U processes {ξt} and the CIR process {rt} defined as (2.5)and (2.2) respectively. We have


(i) for any positive constant θ,

E(e−θξt) = exp{

θ2σ21

4γ(1 − e−2γt)

}; (3.3)

(ii) for any positive constants ι and κ,

E(e−ιrt−κ

∫ t0 rsds)

=

[2√

α2 + 2σ20κ exp{α−

√α2+2σ2

0κ

2t}

(ισ20 + α)

(1 − exp{−√

α2 + 2σ20κ · t}) +

√α2 + 2σ2

0κ(1 + exp{−√

α2 + 2σ20κ · t})

] 2β

σ20 . (3.4)

Proof (i) It is well known that (see [9]) {ξt} is a stationary normal Markov process withEξt = 0 and

Var ξt =σ2

1

2γ(1 − e−2γt).

Hence, for every t > 0, the random variable

ξt√σ21

2γ (1 − e−2γt)

has the standard normal distribution, and (3.3) follows according to the form of the Laplacetransform for the standard normal distribution.

(ii) Let Ψt =∫ t

0rsds. Recalling (2.2), the generator of the process (t, Ψt, rt) acting on a

function f(t, Ψ, r) belonging to its domain is given by

Af(t, Ψ, r) =∂f

∂t+ r

∂f

∂Ψ+ (β − αr)

∂f

∂r+

12σ2

0r∂2f

∂r2. (3.5)

We aim to find a function f such that Af = 0, and then we get that {f(t, Ψt, rt)} is a martingale.Set

f(t, Ψ, r) = exp{−A(t)r − κΨ + B(t)}, (3.6)

where κ is a positive constant, A(t) and B(t) are deterministic functions. Inserting (3.6)into (3.5) leads to the differential equation(

12σ2

0A2(t) + αA(t) − κ − A′(t)

)r + B′(t) − βA(t) = 0. (3.7)

Solving (3.7), we obtain

A(t) =C exp{√α2 + 2σ2

0κ · t}(α +√

α2 + 2σ20κ) + (α − √

α2 + 2σ20κ)

σ20(1 − C exp{

√α2 + 2σ2

0κ · t}) (3.8)

and

B(t) = β

∫ t

0

A(s)ds

=β

σ20

[− (

α +√

α2 + 2σ20κ

)t − 2 ln

(C − exp{−√

α2 + 2σ20κ · t}

C − 1

)], (3.9)

where C is a constant. Therefore, we obtain a martingale

f(t, Ψt, rt) = exp{−A(t)rt − κΨt + B(t)}, t ≥ 0.


Noting that r0 = Ψ0 = 0, we have, for every t ≥ 0,

E(e−A(t)rt−κΨt

)= e−B(t). (3.10)

Let A(t) = ι. Using (3.8), we have

C = exp{−

√α2 + 2σ2

0κ · t} ισ2

0 + α − √α2 + 2σ2

0κ

ισ20 + α +

√α2 + 2σ2

0κ. (3.11)

Substituting (3.11) and (3.9) into (3.10) leads to (3.4). �Based on Proposition 3.1, we can give an estimate of (3.2).

Proposition 3.2 Let Z(1)t denote

∫ t

0

∫X

(1)s yv(dy, ds). We assume that σ0 ≤ α ≤ √

2σ0, andthe claim sizes {Yi} have the finite second-order moment EY 2

i = �. Then,(i) if δ1 − β

α < 3σ21, sup0≤t<∞ E(Z(1)

t )2 attains the minimal upper bound at

π∗11 =

δ1 + 2βσ20

α3 − βα

3σ21 + 2βσ2

0α3

, (3.12)

and correspondingly we have

sup0≤t<∞

E(Z(1)t )2 ≤ 2

2β

σ20 λ�

D∗11

,

where

D∗11 =

(δ1 + 2βσ2

0α3 − β

α

)2

3σ21 + 2βσ2

0α3

+2β

α

(1 − σ2

0

α2

); (3.13)

(ii) if δ1 − βα ≥ 3σ2

1, sup0≤t<∞ E(Z(1)t )2 attains the minimal upper bound at π∗2

1 = 1, andcorrespondingly we have

sup0≤t<∞

E(Z(1)t )2 ≤ 2

2β

σ20 λ�

D∗21

,

whereD∗2

1 = 2δ1 − 3σ21 . (3.14)

Proof Clearly,

Z(1)t =

∫ t

0

∫X(1)

s yv(dy, ds)

is a martingale. Therefore, we have (see [10])

E(Z(1)t )2 = E

(⟨Z(1)

⟩t

)= λ

∫ t

0

∫y2 · E(

X(1)s

)2F (dy)ds = λ�

∫ t

0

E(X(1)

s

)2ds, (3.15)

where⟨Z(1)

⟩denotes the quadratic variational process of Z(1).

Recalling (2.13) and (2.5), we have

E(X

(1)t

)2 = E

(exp

{− 2δ1π1t − 2(1 − π1)

∫ t

0

rsds + σ21π

21t

− 2π1

(− γ

∫ t

0

ξsds + σ1W(1)t

)})

= exp{−2δ1π1t + σ2

1π21t

}E

(exp

{−2(1 − π1)

∫ t

0

rsds

})E(exp {−2π1ξt}). (3.16)


Using Proposition 3.1 (i) with θ = 2π1, we get

E(exp {−2π1ξt}) = exp{

σ21π

21

γ(1 − e−2γt)

}≤ exp

{2σ2

1π21t

}, (3.17)

and using Proposition 3.1 (ii) with ι = 0, κ = 2(1 − π1) leads to

E

(exp

{− 2(1 − π1)

∫ t

0

rsds

})≤ 2

2β

σ20 exp

{αβ

σ20

(1 −

√1 +

4σ20

α2(1 − π1)

)t

}. (3.18)

Substituting (3.17) and (3.18) into (3.16), and noting that√

1 + x ≥ 1 + 12x − 1

8x2 for everyx ≥ 0, we obtain

E(X

(1)t

)2 ≤ 22β

σ20 exp

{[− 2δ1π1 + 3σ2

1π21 +

αβ

σ20

− αβ

σ20

(1 +

2σ20

α2(1 − π1) − 2σ4

0

α4(1 − π1)2

)]t

}

= 22β

σ20 exp

{[(3σ2

1 +2βσ2

0

α3

)π2

1 − 2(

δ1 +2βσ2

0

α3− β

α

)π1

− 2β

α

(1 − σ2

0

α2

)]t

}. (3.19)

If δ1 − βα < 3σ2

1 , the right-hand side of (3.19) attains the minimal value at

π∗11 =

δ1 + 2βσ20

α3 − βα

3σ21 + 2βσ2

0α3

,

and we have

E(X

(1)t

)2 ≤ 22β

σ20 e−D

∗11 t, (3.20)

where

D∗11 =

(δ1 + 2βσ2

0α3 − β

α

)2

3σ21 + 2βσ2

0α3

+2β

α

(1 − σ2

0

α2

).

Substituting (3.20) into (3.15) leads to the first proposition.On the other hand, if δ1 − β

α ≥ 3σ21 , the right-hand side of (3.19) attains the minimal value

at π∗21 = 1 and we have

E(X(1)t )2 ≤ 2

2β

σ20 e−D

∗21 t, (3.21)

where

D∗21 = 2δ1 − 3σ2

1 .

Substituting (3.21) into (3.15), we complete the proof of the second proposition. �Applying Proposition 3.2 and recalling (2.16) and (3.1), we can obtain the following relations

by letting T → ∞ in (3.1):

Φ(1)(u1) ≤

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

22β

σ20 λ�

D∗11 u2

1

, if δ1 − β

α< 3σ2

1 , π1 = π∗11 ,

22β

σ20 λ�

D∗21 u2

1

, if δ1 − β

α≥ 3σ2

1 , π1 = π∗21 .


4 The Case of Stochastic Volatility

In this section, we study wealth model (2.12) in which the stock price is driven by a CIR modelwith stochastic volatility. We assume 0 < π2 ≤ 1. Similar to Proposition 3.2, we have thefollowing results:

Proposition 4.1 Let Z(2)t denote

∫ t

0

∫X

(2)s yv(dy, ds). We assume that all the assumptions

in Proposition 3.2 hold, and additionally suppose δ2 > b. Then,(i) if δ2 − β

α < b, sup0≤t<∞ E(Z(2)t )2 attains the minimal upper bound at

π∗12 =

δ2 − b + 2βσ20

α3 − βα

2βσ20

α3

, (4.1)

and correspondingly we have

sup0≤t<∞

E(Z(2)t )2 ≤ 2

2β

σ20+ 2b

σ22 · λ�

D∗12

,

where

D∗12 =

(δ2 − b + 2βσ2

0α3 − β

α

)2

2βσ20

α3

+2β

α

(1 − σ2

0

α2

); (4.2)

(ii) if δ2 − βα ≥ b, sup0≤t<∞ E(Z(2)

t )2 attains the minimal upper bound at π∗22 = 1, and

correspondingly we have

sup0≤t<∞

E(Z(2)t )2 ≤ 2

2β

σ20+ 2b

σ22 · λ�

D∗22

,

whereD∗2

2 = 2(δ2 − b). (4.3)

Proof Using the same argument as in Proposition 3.2, we have

E(Z(2)t ) = λ�

∫ t

0

E(X(2)

s

)2ds. (4.4)

Recalling (2.14) and (2.7), after some simplifications, we have

E(X(2)t )2 = e−2(δ2−b)π2t · E

(exp

{−2(1 − π2)

∫ t

0

rsds

})

· E(

exp{−2π2ηt − σ2

2π2(1 − π2)∫ t

0

ηsds

}).

It is obvious that Proposition 3.1 (ii) also holds for the CIR process {ηt} if we substitute b, a, σ2

for β, α, σ0 respectively. Therefore, by (3.4), we get

E

(exp

{−2π2ηt − σ2

2π2(1 − π2)∫ t

0

ηsds

})

≤ 22b

σ22 exp

{ab

σ22

(1 −

√1 +

2σ42

a2π2(1 − π2)

)t

}≤ 2

2b

σ22 . (4.5)

Substituting π2 for π1 in (3.18) and considering (4.5), we obtain

E(X(2)t )2


≤ 22β

σ20+ 2b

σ22 · exp

{[− 2(δ2 − b)π2 +

αβ

σ20

− αβ

σ20

(1 +

2σ20

α2(1 − π2) − 2σ4

0

α4(1 − π2)2

)]t

}

= 22β

σ20+ 2b

σ22 · exp

{[2βσ2

0

α3π2

2 − 2(

δ2 − b +2βσ2

0

α3− β

α

)π2 − 2β

α

(1 − σ2

0

α2

)]t

}. (4.6)

If δ2 − βα < b, the right-hand side of (4.6) attains the minimal value at

π∗12 =

δ2 − b + 2βσ20

α3 − βα

2βσ20

α3

,

and we have

E(X(2)t )2 ≤ 2

2β

σ20+ 2b

σ22 · e−D

∗12 t, (4.7)

where

D∗12 =

(δ2 − b + 2βσ2

0α3 − β

α

)2

2βσ20

α3

+2β

α

(1 − σ2

0

α2

).

Substituting (4.7) into (4.4) leads to the first proposition.On the other hand, if δ2 − β

α ≥ b, the right-hand side of (4.6) attains the minimal value atπ∗2

2 = 1 and we have

E(X(2)t )2 ≤ 2

2β

σ20+ 2b

σ22 · e−D

∗22 t, (4.8)

where

D∗22 = 2(δ2 − b).

Substituting (4.8) into (4.4), we complete the proof of the second proposition. �Applying Proposition 4.1 and recalling (2.16) and (3.1), we can obtain the following relations

by letting T → ∞ in (3.1):

Φ(2)(u2) ≤

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

22β

σ20+ 2b

σ22 · λ�

D∗12 u2

2

, if δ2 − β

α< b, π2 = π∗1

2 ,

22β

σ20+ 2b

σ22 · λ�

D∗22 u2

2

, if δ2 − β

α≥ b, π2 = π∗2

2 .

5 Conclusion

In this paper, we consider the upper bounds for ruin probabilities of two types of wealth modeldefined as (2.11) and (2.12). In both models, we describe the stochastic interest rate by CIRmodel (2.2) and assume the risk process is the compound Poisson process written as (2.9).For the stock price process, we let it be driven by O–U process (2.5) with constant volatilityin (2.11), and by CIR model (2.7) with stochastic volatility in (2.12). Under some restrictedconditions on the parameters, we obtain the minimal upper bounds for ruin probabilities andthe corresponding optimal investment strategies, respectively, for these two wealth models bya pure probabilistic method. Now, we conclude the key results of this paper by the followingcomprehensive theorem.

Theorem 5.1 Consider wealth models defined as (2.11) and (2.12). We assume that σ0 ≤α ≤ √

2σ0 and that the claim sizes {Yi} have the finite second-order moment EY 2i = �.


(i) (The case of constant volatility) For wealth model (2.11), we have

Φ(1)(u1) ≤

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

22β

σ20 λ�

D∗11 u2

1

, if δ1 − β

α< 3σ2

1 , π1 = π∗11 ,

22β

σ20 λ�

D∗21 u2

1

, if δ1 − β

α≥ 3σ2

1 , π1 = π∗21 ,

where D∗11 is given by (3.13), π∗1

1 is given by (3.12), D∗21 is given by (3.14) and π∗2

1 = 1.(ii) (The case of stochastic volatility) Additionally assume that δ2 > b. For wealth model

(2.12), we have

Φ(2)(u2) ≤

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

22β

σ20+ 2b

σ22 · λ�

D∗12 u2

2

, if δ2 − β

α< b, π2 = π∗1

2 ,

22β

σ20+ 2b

σ22 · λ�

D∗22 u2

2

, if δ2 − β

α≥ b, π2 = π∗2

2 ,

where D∗12 is given by (4.2), π∗1

2 is given by (4.1), D∗22 is given by (4.3) and π∗2

2 = 1. �

Acknowledgements The authors are most grateful to the anonymous referees for their verythorough reading of the paper and valuable suggestions.

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upper bounds for ruin probabilities under stochastic interest rate and optimal investment strategies

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