Optimal consumption and investment for markets with random coefficients.

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Optimal consumption and investment for markets with random coefficients.

Berdjane Belkacem, Serguei Pergamenchtchikov

To cite this version:

Berdjane Belkacem, Serguei Pergamenchtchikov. Optimal consumption and investment for markets with random coefficients.. 2011. �hal-00563577v2�

Optimal consumption and investment for markets with random coefficients.

Belkacem Berdjane^†and Serguei Pergamenshchikov^‡ December 9, 2011

Abstract

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynami- cal programming approach leads to an investigation of the Hamilton Jacobi Bellman (HJB) equation which is a highly non linear partial differential equation (PDE) of the second oder. By using the Feyn- man - Kac representation we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of the it- erative numerical schemes for both the value function and the optimal portfolio. We show, that in this case, the optimal convergence rate is super geometrical, i.e. is more rapid than any geometrical one. We apply our results to a stochastic volatility financial market.

Key words : Black-Scholes market, Stochastic volatility, Opti- mal consumption and Investment, Hamilton-Jacobi-Bellman equation, Feynman - Kac formula, Fixed point solution.

Mathematical Subject Classification (2000) 91B28, 93E20

∗The paper is supported by the RFBR-Grant 09-01-00172-a.

†Laboratoire L2CSP, de l’universit´e de Tizi-Ouzou (Alg´erie) & Laboratoire LMRS, de l’universit´e de Rouen (France); email: berdjane [email protected]

‡Laboratoire de Math´ematiques Raphael Salem, Avenue de l’Universit´e, BP. 12, Uni- versit´e de Rouen, F76801, Saint Etienne du Rouvray, Cedex France and Department of Mathematics and Mechanics,Tomsk State University, Lenin str. 36, 634041 Tomsk, Russia, e-mail: [email protected]

1 Introduction

One of the principal questions in mathematical finance is the optimal consumption-investment problem for continuous time market models.

This paper deals with an investment problem aiming at optimal con- sumption during a fixed investment interval [0, T] in addition to an optimal terminal wealth at maturity T. Such problems are of prime interest for the institutional investor, selling asset funds to their cus- tomers, who are entitled to certain payment during the duration of an investment contract and expect a high return at maturity. The classical approach to this problem goes back to Merton [15] and in- volves utility functions, more precisely, the expected utility serves as the functional which has to be optimized. By applying results from the stochastic control, explicit solutions have been obtained for fi- nancial markets with nonrandom coefficients (see, e.g. [9], [11] and references therein). Since then, there has been a growing interest in consumption and investment problems and the Merton problem has been extended in many directions. One of the generalisations con- siders financial models with random coefficients such as stochastic volatility markets (see, e.g., [5]). In this paper for the CRRA (Con- stant Relative Risk Aversion) utility functions we consider the opti- mal consumption-investment problem for a Black-Scholes type model with coefficients depending on a diffusion process which is referred as the external stochastic factor. The pure investment problem for such models is considered in [17] and [16]. In these papers the au- thors use the dynamic programming approach and they show that the nonlinear HJB equation can be transformed in a quasilinear PDE.

The similar approach has been used in [12] for optimal consumption- investment problems with the default risk for financial markets with non random coefficients. Furthermore, in [3], by making use of the Girsanov measure transformation the authors study a pure optimal consumption problem for stochastic volatility financial markets. In [1]

and [6] the authors use dual methods. Usually, the classical existence and uniqueness theorem for the HJB equation is shown by the lin- ear PDE methods (see, for example, chapter VI.6 and appendix E in [4]). In this paper we use the approach proposed in [2] for the optimal consumption-investment problem for financial markets with random coefficients depending on pure jumps processes. Unfortunately, we can not apply directly this method for our case since in [2] the HJB equa- tion is the integro-differential equation of the first oder. In our case

it is a highly non linear PDE of the second oder. Similarly to [2] we study the HJB equation through the Feynman - Kac representation.

We introduce a special metric space in which the Feynman - Kac map- ping is contracted. Taking this into account we show the fixed-point theorem for this mapping and we show that the fixed-point solution is the classical unique solution for HJB equation in our case. Moreover, using the verification theorem we provide the explicite expressions for the optimal investment and consumption which depend on the HJB solution. Therefore, to calculate the optimal strategies one needs to study numerical schemes for HJB equation. To this end we find an explicite upper bound for the approximation accuracy. Then, we min- imize this bound and we get the optimal convergence rate for both the value function and the optimal financial strategies. It turns out that in this case this rate is super geometrical.

The rest of the paper is organized as follows. In section 2 we intro- duce the financial market, we state the main conditions on the market parameters and we write the HJB equation. In section 3 we state the main results of the paper. Section 4 presents a stochastic volatility model as an example of applications of our results. In Section 5 we study the properties of the Feynman - Kac mapping. In Section 6 the corresponding verification theorem is stated. The proofs of the main results are given in Section 7. In Section 8 we consider a numerical example. In Appendix some auxiliary results are given.

2 Market model

Let (Ω,FT,(Ft)_0≤t≤T,P) be a standard filtered probability space with two standard independent (Ft)_0≤t≤T adapted Wiener processes (W_t)_0≤t≤T and (V_t)_0≤t≤T taking their values inR^d andR^m respectively, i.e.

W_t= (W_t¹, . . . , W_t^d)^′ and V_t= (V_t¹, . . . , V_t^m)^′.

The prime^′denotes the transposition. Our financial market consists of oneriskless bond(S₀(t))_0≤t≤T anddrisky stocks(S_i(t))_0≤t≤T governed by the following equations:







dS₀(t) =r(t, Y_t)S₀(t)dt ,

dS_i(t) =S_i(t)µ_i(t, Y_t)dt+S_i(t) Pd

j=1 σ_ij(t, Y_t) dW_t^j,

(2.1)

with S₀(0) = 1 and S_i(0) =s_i for 1≤i≤ d. In this model r(t, y) ∈ R+ is the riskless interest rate,µ(t, y) = (µ₁(t, y), . . . , µ_d(t, y))^′ is the vector ofstock-appreciation ratesand σ(t, y) = (σ_ij(t, y))_1≤i,j≤dis the matrix of stock-volatilities. For all y ∈ R^m the coefficients r(·, y), µ(·, y) ∈R^d and σ(·, y) ∈ M

d are nonrandom c`adl`ag functions. Here Md denotes the set of quadratic matrix of order d. Moreover, in just the same way as in [17] we assume, that the stochastic factorY valued inR^m has a dynamics governed by the following stochastic differential equation:

dY_t=F(t, Y_t) dt+βdU_t, (2.2) where F is a [0, T]×R^m → R^m nonrandom function and β is fixed positive parameter. The process U is the standard Brownian motion defined as

U_t=ρV_t+p

1−ρ²σ_∗W_t, (2.3) where 0≤ρ≤1 andσ_∗ is a fixedm×dmatrix for which σ_∗σ_∗^′ =I_m. Here I_m is the identity matrix of order m.

Moreover, we setK= [0, T]×R^m and we note, that for the model (2.1) the risk premium is the K →R^dfunction defined as

θ(t, y) =σ⁻¹(t, y)(µ(t, y)−r(t, y)1_d), (2.4) where 1_d = (1, . . . ,1)^′ ∈ R^d. Similarly to [10] we consider the frac- tional portfolio process

ϕ(t) = (ϕ₁(t), . . . , ϕ_d(t))^′ ∈R^d,

i.e. ϕ_i(t) represent the fraction of the wealth process X_t invested in thei-th stock at the timet. The fractions for consumptions we denote byc= (c_t)_0≤t≤T. In this case the wealth process satisfies the following stochastic equation

dX_t=X_t(r(t, Y_t) +π^′_tθ(t, Y_t)−c_t)dt+X_tπ_t^′dW_t, (2.5) where π_t = σ(t, Y_t)ϕ_t and the initial endowment X₀ = x. Now we describe the set of all admissible strategies. A portfolio control (finan- cial strategy) ϑ= (ϑ_t)_t≥0 = ((π_t, c_t))_t≥0 is said to beadmissibleif it is (Ft)_0≤t≤T - progressively measurable with values inR^d×[0,∞), such that

kπkT :=

Z T

0

|π_t|²dt <∞ and Z T

0

c_tdt <∞ a.s. (2.6)

and the equation (2.5) has a unique strong a.s. positive continuous so- lution (X_t^ϑ)_0≤t≤T on [0, T]. We denote the set ofadmissible portfolios controls byV.

In this paper we consider an agent using the CRRA utility function x^γ for 0< γ <1. The goal is to maximize the expected utilities from the consumption on the time interval [0, T] and from the terminal wealth. Then for any x ∈ R^d, y ∈R^m and ϑ∈ V the value function of agent is

J(x, y, ϑ) :=Ex,y

Z T 0

c^γ_t (X_t^ϑ)^γdt + (X_T^ϑ)^γ

,

were E_x,y is the conditional expectation forX₀^ϑ=x and Y₀=y. Our goal is to maximize this function, i.e.

sup

ϑ∈V

J(x, y, ϑ). (2.7)

Remark 2.1. Note, that (2.7) is the classical Merton optimization problem for the market (2.1), (see, e.g. [15], [2]). Pure investment cases of this problem are studied in [17] and [16].

2.1 Conditions

To list the conditions for the model (2.1) we set

α(t, y) =F(t, y) + β_∗σ_∗θ(t, y), (2.8) where β_∗ =γp

1−ρ²β/(1−γ). Moreover, we denote byC^i,k(K) the space of the functions f(t, y₁, . . . , y_m) which are itimes differentiable with respect to t ∈ [0, T] and j times differentiable with respect to (y_j)_1≤j≤m.

We assume that the market parameters satisfy the following condi- tions:

A₁) The functionsr :K →R

+, µ:K →R^d

+ and σ :K →M

d belong to C^1,1(K) and have bounded derivatives. Moreover, for all (t, y)∈ K the matrix σ(t, y)is non-degenerated and

sup

(t,y)∈K |r(t, y)|+|µ(t, y)|+|σ⁻¹(t, y)|

<∞.

A₂) The K →R^m functionF(·,·) belongs to C^1,1(K) and its partial derivatives are bounded.

This condition provides the existance of a unique strong solution for the equation (2.2). On the interval [t, T] we denote this solution by

Y^t,y = (Y_v^t,y)_t≤v≤T withY_t^t,y =y.

A₃) There exist [0, T] ×R → R continuously differentiable func- tions (a_i,j(·,·))_1≤i,j≤m such that for any 0 ≤ t ≤ T and for any x = (x₁, . . . , x_m)^′ from R^m the ith component [α(·,·)]_i of the vector α(t, x) has the form

[α(t, x)]_i = Xm

l=1

a_i,l(t, x_j) (2.9) and, moreover,

α_∗= max

i,j max

t,z max

|a_i,j(t, z)|,

∂a_i,j(t, z)

∂t ,

∂a_i,j(t, z)

∂z

<∞.

In the sequel, for any vector x from Rⁿ we denote theith compo- nent by [x]_i.

Remark 2.2. In Section 4 we check the conditionA₃) for a two asset stochastic volatility financial market.

2.2 The HJB equation

Now we introduce the HJB equation for the problem (2.7). To this end, for any differentiable K → Rfunction f we denote by D_yf(t, y) its gradient with respect toy ∈R^m, i.e.

D_yf(t, y) = ∂

∂y₁f(t, y), . . . , ∂

∂y_mf(t, y) ′

.

Let now z(t, ς) be any [0, T]×R×R^m → Rtwo times differentiable function. Hereς = (x, y)^′,x∈Rand y∈R^m. We set

D_ςz(t, ς) = z_x(t, ς), D_yz(t, ς)′

and

D_ς,ςz(t, ς) =





∂²z(t, ς)

∂²x ; (D_x,yz(t, ς))^′ D_x,yz(t, ς) ; D_y,yz(t, ς)



 ,

where

D_x,yz(t, ς) =

∂²z(t, ς)

∂x∂y₁ , . . . ,∂²z(t, ς)

∂x∂y_m ′

and

D_y,yz(t, ς) = ∂²z(t, ς)

∂y_j∂y_i

!

1≤i,j≤m

. Let nowq∈R^m+1 andM∈M

m+1 be fixed parameters of the follow- ing form

q= (q₁,eq)^′ and M= µ; eµ^′ e µ; M₀

!

, (2.10)

where q₁, µ ∈ R, eq,µe ∈ R^m and M₀ ∈ M

m. For these parameters withq₁ >0 we define the Hamilton function as

H(t, ς,q,M) =x r(t, y)q₁+ (F(t, y))^′qe+ 1 γ₁

γ q₁

γ₁−1

+|θ(t, y)q₁+βp

1−ρ²σ^′_∗µe|² 2|µ| +β²

2 trM₀, (2.11) where γ₁ = (1−γ)⁻¹. In this case the HJB equation is

( z_t(t, ς) +H(t, ς,D_ςz(t, ς),D_ς,ςz(t, ς)) = 0, z(T, ς) =x^γ.

(2.12) To study this equation we make use of the distortion power transfor- mation introduced in [17], i.e. for a positive function h we represent z(t, y) as

z(t, ς) =x^γh^ε(t, y), ε= 1−γ

1−ρ²γ . (2.13) It is easy to deduce that the function h satisfies the following quasi- linear PDE:

















h_t(t, y) +Q(t, y)h(t, y) + (α(t, y))^′D_yh(t, y) +β²

2 trD_y,yh(t, y) + 1 q_∗

1 h(t, y)

q_∗−1

= 0 ; h(T, y) = 1,

(2.14)

where

Q(t, y) = γ(1−ρ²γ) 1−γ

r(t, y) + |θ(t, y)|² 2 (1−γ)

. (2.15) Note that, by the condition A₁) this function and its y-gradient are bounded. Therefore, we can set

Q_∗ = sup

(t,y)∈K

Q(t, y) and D_∗ = sup

(t,y)∈K |D_yQ(t, y)|. (2.16) Our goal is to study the equation (2.14). By making use of the probabilistic representation for the linear PDE (the Feynman-Kac for- mula) we show in Proposition 5.4, that the solution of this equation is the fixed-point solution for a special mapping of the integral type which will be introduced in the next section.

3 Main results

First, to study the equation (2.14) we introduce a special functional space. Let X be the set of K → [1,∞) functions from C^0,1(K) such that

kfk0,1 = sup

(t,y)∈K

(|f(t, y)|+|D_xf(t, y)|) ≤r^∗, (3.1) where

r^∗=r₀+m H^∗

q_∗ (2√

T +T) +D_∗T e^{(α∗+Q∗)T}

, (3.2) r₀=e^Q∗T+ (e^Q∗T−1)/Q_∗q_∗. The parameterH^∗ will be defined below in Section 3.1. To define a metrics in X we set

κ=Q_∗+ζ+ 1 +mL_∗ and L_∗ = (1 +r^∗q_∗+T D_∗)e^α∗T, (3.3) where the positive parameterζwill be specified later,q_∗= (1−ρ²γ)⁻¹. Now for anyf andg from X we define the metrics as

̺_∗(f, g) =kf−gk∗, (3.4) where

kfk∗ = sup

(t,y)∈K

e^−κ(T−t) (|f(t, y)|+|D_xf(t, y)|) .

Moreover, to write the Feynman-Kac representation for the equation (2.14) we need to use the stochastic process (η_v)_0≤v≤T governed by the following stochastic differential equation

dη_t=α(t, η_t)dt+β dU_t. (3.5) It should be noted that the condition A₃) provides the existence and the uniqueness of a strong solution on any time interval [t, T] and initial condition y from R^m. We denote this solution by (η_v^t,y)_t≤v≤T. Using this process we define the X → X Feynman-Kac mappingL:

Lf(t, y) =EG(t, T, y) + 1 q_∗

Z T

t Hf(t, s, y) ds , (3.6) where G(t, s, y) = expRs

t Q(u, ηu^t,y)du and Hf(t, s, y) =E f(s, η^t,y_s )1−q_∗

G(t, s, y). (3.7) To solve the equation (2.14) we need to find the fixed-point solution for the mapping LinX, i.e.

Lh =h . (3.8)

To this end we construct the following iterated scheme. We seth₀ ≡1 and

h_n(t, y) =Lh_n−1(t, y) for n≥1. (3.9) First, we study the behavior of the deviation

∆_n=h(t, y)−h_n(t, y).

In the following theorem we show that the contraction coefficient for the operator (3.6) is given by

λ= 1 +mL_∗

ζ+ 1 +mL_∗ (3.10)

and an appropriate choice ofζgives the super-geometrical convergence rate for the sequence (h_n)_n≥1.

Theorem 3.1. Under the conditions A₁)–A₃) the equation (3.8) has a unique solution h in X such that for any n≥1 and ζ >0

sup

(t,y)∈K |∆_n(t, y)|+D_y∆_n(t, y)≤B^∗λⁿ, (3.11) where B^∗=e^κT(1 +r^∗)/(1−λ) andκ is given in (3.3).

The proof of this theorem is given in Section 7.

Now we can minimize the upper bound (3.11) over ζ >0. Indeed, setting ζe=ζ/(1 +mL_∗) and Te= (1 +mL_∗)T, we obtain

B^∗λⁿ=C^∗ exp{g_n(eζ)}, where C^∗ = (1 +r^∗)e^{(Q∗+1+mL∗)T} and

g_n(x) =xTe−lnx−(n−1) ln(1 +x). Now we minimize this function over x >0, i.e.

minx>0g_n(x) =x^∗_nTe−lnx^∗_n−(n−1) ln(1 +x^∗_n), where

x^∗_n= q

(Te−n)²+ 4Te+n−Te

2Te .

Therefore, for

ζ =ζ_n^∗= (1 +mL_∗)x^∗_n we obtain the optimal upper bound (3.11).

Corollary 3.2. Under the conditions A₁)–A₃) the equation (3.8) has a unique solution h in X such that for any n≥1

sup

(t,y)∈K |∆_n(t, y)|+D_y∆_n(t, y)≤U^∗_n, (3.12) where U^∗_n=C^∗ exp{g^∗_n}.

Remark 3.1. One can check directly that for some δ >0 U^∗_n=O(n^−δn) as n→ ∞.

This means that the convergence rate is more rapid than any geomet- rical one, i.e. it is super geometrical.

Theorem 3.3. Assume, that the conditions A₁)–A₃) hold. Then the optimal value of J(x, y, ϑ) for optimization problem (2.7) is given by

maxϑ∈V J(x, y, ϑ) = J(x, y, ϑ^∗) =x^γ(h(0, y))^ε,

where h(t, y) is the unique solution of the equation (2.14). Moreover, for all 0 ≤t≤T an optimal financial strategy ϑ^∗ = (π^∗, c^∗) is of the form









π_t^∗=π^∗(t, Y_t) = θ(t, Y_t) 1−γ +εp

1−ρ²βσ_∗D_yh(t, Y_t) (1−γ)h(t, Y_t) ; c^∗_t =c^∗(t, Y_t) = (h(t, Y_t))^−q∗ .

(3.13)

The optimal wealth process (X_t^∗)_0≤t≤T satisfies the following stochastic equation

dX_t^∗ =a^∗(t, Y_t)X_t^∗dt+X_t^∗(b^∗(t, Y_t))^′dW_t, X₀^∗ = x , (3.14) where

a^∗(t, y) = |θ(t, y)|²

1−γ + εp

1−ρ²β

(1−γ)h(t, y)(σ_∗D_yh(t, y))^′θ(t, y) +r(t, y)−(h(t, y))^−q∗ ;

b^∗(t, y) = θ(t, y)

1−γ + εp

1−ρ²β

(1−γ)h(t, y)σ_∗D_yh(t, y). The proof of this theorem is given in Section 7.

Remark 3.2. Note that the optimal strategy (3.13) coincides with the well-known Merton strategy in the case ρ = 1, i.e. when the process Y is independent of the brownian motion W generating the financial market (2.1). Note also that, in the case 0 ≤ ρ < 1, the optimal investment strategy in (3.13) depends on the external factorY through the derivative of the function h. The first term in this expression is the well-known Merton strategy and the second term is the impact of the external factor.

To calculate the optimal strategy in (3.13) we use the sequence (h_n)_n≥1, i.e. we set

π^∗_n(t, y) = θ(t, y)

1−γ + εp

1−ρ²β

(1−γ)h_n(t, y)σ_∗D_yh_n(t, y) and

c^∗_n(t, y) = (h_n(t, y))^−q∗ . Theorem 3.1–Theorem 3.3 imply the following result

Theorem 3.4. Assume, that the conditions A₁)–A₃) hold. Then for any n≥1

sup

(t,y)∈K |π^∗(t, y)−π_n^∗(t, y)|+|c^∗(t, y)−c^∗_n(t, y)|

≤B^∗₁U^∗_n,

where B^∗₁=γ₁βεp

1−ρ²|σ_∗|r^∗+q_∗.

Remark 3.3. Note that in this paper we use only the power utility functionx^γ with0< γ <1. It seems that the question about the upper bound (3.11) is open for general utility functions of the CRRA type.

The method to obtain the bound (3.11) is based on the explicite form of the utility function. The heart of this method is to show that the operator (3.6)is contracted in a suitable functional space. To this end one needs to make use of the power uitility.

3.1 Formula for H^∗

To write the upper bound for the partial derivatives of the function Hf(t, s, y) (see Lemma A.2) we define for any q >0

φ_q=√

2e^Tφeq and ι_q =qα_∗(β⁻²+T), (3.15) where φe_q = q Q_∗+β²α_∗/2

+ (2m²+ 1)q²β⁻²α²_∗ . Using these pa- rameters we set

H^∗ = 2 max

2φ₁(ι₁+α_∗) + Ψ_∗(φ₂)^1/2, 1 β√

2π

e^e2ι , (3.16) where Ψ_∗ =

q

T α_∗+ (T D_∗+T α_∗(α_∗+ 1 +β²/2) +β⁻²α_∗)² and e

ι=β²T (max (ι₁, ι₂) +α_∗)².

4 Two-asset stochastic volatility model

In this section we consider an important example of the model (2.1) used in financial markets with the stochastic volatilities (see, e.g., [16]). This model is defined as follows:

dS₀(t) =r S₀(t)dt

and

dS₁(t) =µ₁S₁(t)dt+S₁(t)σ₁(t, Y_t¹) dW_t¹+b₁dW_t²

; dS₂(t) =µ₂S₂(t)dt+S₂(t)σ₂(t, Y_t²) dW_t¹+b₂dW_t²

,

whereb₁ 6=b₂, the processY_t= (Y_t¹, Y_t²)^′ is gouverned by the follow- ing stochastic differential equations

dY_t¹=F₁(t, Y_t¹) dt+βdU¹_t, dY_t²=F₂(t, Y_t²) dt+βdU²_t. Here Uⁱ_t=ρV_tⁱ+p

1−ρ²W_tⁱ for some 0≤ρ ≤1, i.e. the parameter σ_∗ = 1 in (2.3). Assume, that the [0, T]×R → R functions F_i are bounded and have bounded derivatives, i.e.

F_max = max

1≤i≤2 sup

(t,y)

max

|F_i(t, y)|,

∂F_i(t, y)

∂t ,

∂F_i(t, y)

∂y

<∞.

We assume also, that the [0, T]×R→Rfunctionsσ_i are such, that σ_max= max

1≤i≤2 sup

(t,y)

max

∂σ_i(t, y)

∂t ,

∂σ_i(t, y)

∂y

<∞ and

σ_min= inf

(t,y) min (|σ₁(t, y)|,|σ₂(t, y)|)>0.

Obviously, that in this case the matrix σ(t, y₁, y₂) is non-degenerated and

σ⁻¹(t, y₁, y₂) =







b^∗₁

σ₁(t, y₁) ;− b^∗₁ σ₂(t, y₂)

− b^∗₂

σ₁(t, y₁) ; b^∗₂ σ₂(t, y₂)





 ,

where b^∗₁ = b₂(b₂ −b₁)⁻¹ and b^∗₂ = (b₂ −b₁)⁻¹. Therefore, the components of the function (2.4) are

[θ(t, y₁, y₂)]_i= eb_1,i

σ₁(t, y₁)+ eb_2,i

σ₂(t, y₂), 1≤i≤2,

wherebe_1,1=b^∗₁(µ₁−r),be_2,1 =−b^∗₁(µ₂−r),be_1,2 =−b^∗₂(µ₁−r) and be_2,2 = b^∗₂(µ₂ −r). From this we obtain, that in this case function

(2.8) has the form (2.9), i.e.

[α(t, y₁, y₂)]_i =F_i(t, y_i) +β_∗ be_1,i

σ₁(t, y₁) + be_2,i σ₂(t, y₂)

!

, 1≤i≤2. It is easy to see, that for this model the conditionsA₁)–A₃) hold with

α_∗ ≤F_max+eb_∗ 1 +σ_max

σ_min and be_∗ =β_∗ max

1≤i,j≤2(|be_1,i|).

5 Properties of the mapping L

Proposition 5.1. Assume, that the conditions A₁)–A₃) hold. Then Lf ∈ X for any f ∈ X.

Proof. Obviously, that for anyf ∈ X the mappingLf ≥1. Moreover, setting

fe_s=f(s, η^t,y_s ), (5.1) we represent Lf(t, y) as

Lf(t, y) =EG(t, T, y) + 1 q_∗

Z T

t

E

fe_s1−q_∗

G(t, s, y)ds . (5.2) Therefore, taking into account that fe_s≥1 andq_∗≥1 we get

Lf(t, y) ≤ e^Q∗(T−t)+ Z T

t

1

q_∗ e^Q∗(s−t)ds ≤ r₀, (5.3) where the upper boundr₀is defined in (3.2). Moreover, Lemmas A.4–

A.5 yield

∂

∂y_iLf(t, y) =E ∂

∂y_iG(t, T, y) + 1 q_∗

Z _T

t

∂

∂y_iHf(t, s, y) ds . Therefore, applying here (5.3), (A.16) and Lemmas A.2 we get

Lf(t, y) +D_yLf(t, y) ≤r^∗.