Intelligent Computational Methods for Financial Engineering

PARABOLIC PROBLEM

ABDERRAHMANE EL HACHIMI AND MOULAY RCHID SIDI AMMI Received 22 February 2004 and in revised form 16 February 2005

A time discretization technique by Euler forward scheme is proposed to deal with a nonlocal parabolic problem. Existence and uniqueness of the approximate solution are proved.

1. Introduction

In this work, we study the time discretization by Euler forward scheme of the nonlocal initial boundary value problem

∂u

∂t ⁻ u=λ f(u)

Ωf(u)dx² inΩ×]0;T[, u=0 on∂Ω×]0;T[,

u(0)=u0 inΩ,

(1.1)

withΩ⊂R^d(d≥1) a bounded regular domain andλa positive parameter. The hypothe- ses we will assume on f are the same as in [6]. We recall first that (1.1) arises by reducing the following system of two equations modeling the thermistor problem:

ut= ∇·

k(u)∇u+σ(u)∇ϕ²,

∇

σ(u)∇ϕ=0, (1.2)

whereurepresents the temperature generated by the electric current flowing through a conductor,ϕthe electric potential,σ(u) andk(u) are, respectively, the electric and ther- mal conductivities. For more description, we refer to [5,6,7,8,11] among others.

We recall also that the Euler forward method was used by several authors to treat semidiscretization of nonlinear parabolic problems, see [3,4]. Concerning problem (1.1), results of existence and uniqueness of solutions are known under particular forms of f, we refer to [2] and the references therein. On the other hand, little is known about

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:10 (2005) 1655–1664 DOI:10.1155/IJMMS.2005.1655

the solutions to the discrete problem

Uⁿ−τUⁿ=Uⁿ⁻¹+λτ fUⁿ

ΩfUⁿdx² inΩ, Uⁿ=0 on∂Ω,

U⁰=u0 inΩ.

(1.3)

Whereas, semidiscretization has been involved for the equations of the thermistor problem in [1,9]. Our aim here is to continue the study of problem (1.1) initiated in [6], where an a prioriL^∞-estimate is derived. In addition to habitual existence and uniqueness questions concerning the solutions of (1.3), we will prove some results of stability and proceed to error estimates analysis. In [1], the authors derived anL²andH¹-norm error by requiring more regularity on the solutionu, for instanceu,u_tinH²(Ω)∩W^1,∞(Ω).

Unfortunately, such smoothness is not always possible since the function f is nonlinear.

2. The semidiscrete problem

2.1. Existence and uniqueness. We consider the Euler scheme (1.3), withNτ=T,T >0 fixed, and 1≤n≤N, under the following hypotheses.

(H1) f :R→Ris a locally Lipschitzian function.

(H2) There exist positive constantsσ,c1,c2, andαsuch thatα <4/(d−2) and for all ξ∈R,

σ≤f(ξ)≤c1ξ^α+1+c2. (2.1) In the sequel, we will denote the norms in the spacesL^∞(Ω),L^k(Ω) by| · |L^∞(Ω)and| · |k, respectively, (·,·) will denote the associated inner product inL²(Ω) or the duality product betweenH₀¹(Ω) and its dualH⁻¹(Ω).

Theorem2.1. Let(H1)-(H2)be satisfied. Then, for eachn, there exists a unique solution Uⁿof (1.3) inH0¹(Ω)∩L^∞(Ω)provided thatτis small enough.

Proof. For simplicity, we writeU=Uⁿ,h(x)=Uⁿ⁻¹. Then (1.3) becomes U−τU=h(x) +λ f(U)

Ωf(U)dx² inΩ, U=0 on∂Ω.

(2.2)

Existence. Define the mapS(µ,·) byU=S(µ,v),µ∈[0, 1] if and only if U−τU=µg(x,v) inΩ,

U=0 on∂Ω, U⁰=µu0,

(2.3)

whereg(x,v)=h(x) +λ(f(v)/(_Ωf(v)dx)²). For a fixedv∈H₀¹(Ω), (2.3) has a unique solutionU∈H₀¹(Ω). Then, for eachµ∈[0, 1], the operatorS(µ,·) is well defined. More- over,S(µ,·) is compact fromH0¹(Ω) into it self. Indeed, using (H2), we have the estimate

U²₂+τ∇U²₂≤c3. (2.4)

We can easily see thatµ→S(µ,v) is continuous and thatS(0,v)=U, for anyv, if and only ifU=0. From Leray-Schauder fixed point theorem, there exists therefore a fixed pointU

ofS(µ,·).

Now, we derive an a priori estimate.

Lemma2.2. Ifu0∈L^∞(Ω), then for alln∈ {1,. . .,N},Uⁿ∈L^∞(Ω).

Proof. The proof is similar to the one used by De Th´elin in [10] concerning a very differ- ent problem and we will give here only a sketch. Suppose thatd≥2 and define

δ=





 2d

d−2 if 2< d, 2(α+ 2) ifd=2.

(2.5)

For eachk∈N∗, we consider the number qk= δ

2 k−1

(δ−γ)−(2−γ) δ

δ−2, k≥2, q1=δ,

(2.6)

we have

q_k+1=

q_k+ 2−γδ

2 withγ=α+ 2,∀k∈N^∗. (2.7) Lemma2.3. For allk∈N^∗,Uⁿ∈L^qk(Ω), and moreover

Uⁿ_∞=limUⁿ_q

k<+∞. (2.8)

Proof. We prove by recurrence thatU∈L^qk. The property is true fork=1, sinceH₀¹(Ω)⊂ L^δ(Ω). We show now thatU∈L^qk+1. Letm∈N, 1≤m≤k. Multiplying (2.2) by|U|^qm−γU, using (H2), and Young’s inequality, we get

q_m−γ+ 1

Ω|∇U|²|U|^qm−γdx≤c4|U|^qq^mm+c5. (2.9) On the other hand, we have

|U|^qq^mm+1^+2−γ≤c6

1 +qm−γ 2

2

Ω|∇U|²|U|^qm−γdx. (2.10)

Therefore, we obtain

|U|^qq^mm+1^+2−γ≤

c7+c8|U|^qq^mm

q_m+ 2−γ. (2.11)

Thus,

|U|^qq^k+1k+1

2/δ

≤

c7+c8|U|^qq^kk

qk+ 2−γ. (2.12)

The rest of the proof follows the same lines as in [10, pages 383-384].

Uniqueness. ConsiderU andV two different solutions of (2.2) and definew=U−V. Then, we have

w−τw= λτ

Ωf(U)dx²

f(U)−f(V)

+λτ

Ωf(U)−f(V)dx_Ωf(V) +f(U)dx

Ωf(U)dx²_Ωf(V)dx² f(V).

(2.13)

Multiplying (2.13) byw, integrating onΩ, and using theL^∞-estimate obtained inLemma 2.2, we get

|w|²2+τ|∇w|²2≤c9τ|w|²2. (2.14)

Therefore,w=0 ifτ≤1/c9.

We address now the question of stability.

3. Stability

Theorem3.1. Assume(H1)-(H2)hold. Then, there existsc(T,u0)>0depending on data but not onNsuch that for anyn∈ {1,. . .,N},

(a)|Uⁿ|L^∞(Ω)≤c(T,u0);

(b)|Uⁿ|²2+τⁿ_k=1|∇U^k|²2≤c(T,u0);

(c)ⁿ_k=1|U^k−U^k−1|²2≤c(T,u0).

Proof. (i) Multiplying (1.3) by|U^k|^mU^kfor some integerm≥1, usingLemma 2.2, and H¨older’s inequality, we obtain after simplification

U^k_m+2≤U^k−1_m+2+c10τ. (3.1) By induction and taking the limit in the resulting inequality asm→+∞, we get

U^k_L∞(Ω)≤cT,u0

. (3.2)

(ii) Multiplying the first equation of (1.3) byU^k and using the hypotheses on f, one easily has

U^k−U^k−1,U^k+τ∇U^k2₂≤c11τU^k₁. (3.3)

Using the elementary identity 2a(a−b)=a²−b²+ (a−b)²and summing fromk=1 to n, we obtain

Uⁿ²₂+ n k=1

U^k−U^k−12₂+τ n k=1

∇U^k2₂≤u0²

2+τc12

n k=1

U^k₁. (3.4)

Then, the inequalities (b)-(c) hold by using the uniform bound ofUⁿ inL^∞ which is

established in part (a).

4. Error estimates for solutions

We will adopt the following notations concerning the time discretization for problem (1.1). We denote the time stepτ=T/N,tⁿ=nτ, andI_n=(tⁿ,tⁿ⁻¹) forn=1,. . .,N. Ifz is a continuous function (resp., summable), defined in (0,T) with values inH⁻¹(Ω) or L²(Ω) orH0¹(Ω), we definezⁿ=z(tⁿ,·),zⁿ=(1/τ)_Inz(t,·)dt,z⁰=z⁰=z(0,·); the error e_n=u(t)₋Uⁿfor all t_∈I_n and the local errorse_uⁿandeⁿdefined bye_uⁿ₌uⁿ(t)₋Uⁿ, eⁿ=uⁿ−Uⁿ.

We have the following theorem.

Theorem4.1. Let(H1)-(H2)hold. Then, the following error bounds are satisfied:

(1)e_n²_L∞(0,T,H⁻¹(Ω))+₀^T|e_n|²dt≤c13τ, (2)e^mH⁻¹(Ω)≤c14τ^1/2,

(3)|∇_T

0 en(t)dt|2≤c15τ^1/4.

Proof. For the proof, we consider the following variational formulation of discrete prob- lem (1.3):

Uⁿ−Uⁿ⁻¹,ϕ+τ∇Uⁿ,∇ϕ= λτ

ΩfUⁿdx²

fUⁿ,ϕ, ∀ϕ∈H₀¹(Ω). (4.1)

Integrating the continuous problem (1.1) overI_n, we get uⁿ−uⁿ⁻¹,ϕ+τ∇uⁿ,∇ϕ=λτ

fuⁿ,ϕ

Ωfuⁿdx², ∀ϕ∈H₀¹(Ω). (4.2) Substracting (4.2) from (4.1) and adding fromn=1 tomwithm≤N, we obtain

m n=1

eⁿ−eⁿ⁻¹,ϕ+τ m n=1

∇e_uⁿ,∇ϕ

≤c16τ m n=1

f(u)ⁿ−fUⁿ,ϕ+c17τ m n=1

f(Uⁿ,ϕ.

(4.3)

Let (−)⁻¹be the green operator satisfying

∇(−)⁻¹ψ,∇ϕ=(ψ,ϕ)_H−1(Ω),H¹0(Ω) (4.4) for allψ∈H⁻¹(Ω),ϕ∈H₀¹(Ω). Choosingϕ=(−)⁻¹(eⁿ) as test function, we then ob- tain

I1+I2≤I3+I4, (4.5)

where

I1= m n=1

eⁿ−eⁿ⁻¹, (−)⁻¹eⁿ,

I2=τ m n=1

eⁿ_u,eⁿ,

I3≤c16τ m n=1

f(u)ⁿ−fUⁿ, (−)⁻¹eⁿ,

I4=c17τ

m n=1

fUⁿ, (−)⁻¹eⁿ .

(4.6)

With the aid of the elementary identity 2a(a−b)=a²−b²+ (a−b)²and the property of (−)⁻¹,I1reduces after straightforward calculations to

I1=1

2e^m2_H−1(Ω)+1 2

m n=1

eⁿ−eⁿ⁻¹²_H−1(Ω). (4.7)

On the other hand, I2=τ

m n=1

eⁿ_u,eⁿ

= m n=1

In

u(t)−Uⁿ,u(t)−Uⁿdt+ m n=1

In

u(t)−Uⁿ,uⁿ−u(t)dt

=I21+I22,

(4.8)

where

I21= m n=1

In

u(t)−Uⁿ,u(t)−Uⁿdt= m n=1

In

en²

2dt,

I22= m n=1

In

u(t),uⁿ−u(t)dt− m n=1

In

Uⁿ,uⁿ−u(t)dt

=I₂₂¹ +I₂₂².

(4.9)

We now estimateI₂₂¹.Using the boundedness of∂u/∂s(see [6]), we have I₂₂¹=

m n=1

In

u(t),

_tⁿ

t

∂u

∂sds

dt

≤ m n=1

In

_tⁿ

t

∂u

∂s

H⁻¹(Ω)ds

u(t)_H0¹_(Ω)dt

≤τ∂u

∂s

L²(0,t^m,H⁻¹(Ω))uL²(0,t^m,H¹0(Ω))

≤c18τ.

(4.10)

In the same manner, we have I₂₂²≤τ∂u

∂s

L²(0,t^m,H⁻¹(Ω))

τ

m n=1

Uⁿ²_H0¹_(Ω) 1/2

≤c18τ.

(4.11)

Next, we estimate the first term on the right-hand side of (4.5) by using H¨older’s and Young’s inequalities and (H1),

I3≤

m n=1

In

f(u)−fUⁿdt, (−)⁻¹eⁿ

≤c20τ^1/2 m n=1

In

f(u)−fUⁿ²₂dt 1/2

eⁿ_H−1(Ω)

≤η m n=1

In

f(u)−fUⁿ²₂dt

+c21

η τ m n=1

eⁿ²_H−1(Ω)

≤c22η m n=1

In

en²

2dt

+c21

η τ m n=1

eⁿ²_H−1(Ω).

(4.12)

Moreover, we have

I4≤c23τ+c24τ m n=1

eⁿ²_H−1(Ω). (4.13)

Choosing suitableη, we conclude that e^m2_H−1(Ω)+

m n=1

eⁿ−eⁿ⁻¹²_H−1(Ω)+ m n=1

In

e_n²₂dt

≤c25τ+c26τ m n=1

eⁿ²_H−1(Ω).

(4.14)

On the other hand, settingy^m=_m

n=1eⁿ²_H−1(Ω), then from (4.14), we get

y^m−y^m−1≤c25τ+c26τ y^m. (4.15)

By applying the discrete Gronwall inequality, we deduce that y^m≤c(T). Therefore, we have

e^m_H−1(Ω)≤c27τ^1/2. (4.16)

On the other hand, we have sup

t∈(0,tm)

en(t)_H−1(Ω)−c27τ^1/2≤ max

1_≤n≤m

en

tⁿ_H−1(Ω)= max

1_≤n≤m

eⁿ_H−1(Ω). (4.17)

Thus, we get

e_nL∞(0,T,H⁻¹(Ω))−c27τ^1/2≤ max

1≤n≤m

eⁿ_H−1(Ω). (4.18)

From the last inequality, we obtain

e_n²_L∞(0,T,H⁻¹(Ω))+ _T

0

e_n²₂dt≤c29τ, m

n=1

eⁿ−eⁿ⁻¹²_H−1(Ω)≤c29τ.

(4.19)

Choosingϕ=τ^m_n=1(uⁿ−Uⁿ) in (4.3), we get

τ

Ω

u^m−U^m _m

n=1

uⁿ−Uⁿdx

+τ² m n=1

∇

uⁿ−Uⁿ

2

2

≤c30τ²

Ω

m n=1

f(u)ⁿ−fUⁿ _m

n=1

uⁿ−Uⁿ

dx +c31τ²

m n=1

fUⁿ,

m n=1

uⁿ−Uⁿ.

(4.20)

Thus,

τ² m n=1

∇

uⁿ−Uⁿ

2

2

= ∇

_t^m

0 e_ndt

2 2≤τ

Ω

u^m−U^m _m

n=1

uⁿ−Uⁿdx +c30τ²

Ω

m n=1

f(u)ⁿ−fUⁿ _m

n=1

uⁿ−Uⁿ

dx +c31τ²

m n=1

fUⁿ,

m n=1

uⁿ−Uⁿ

≤I+II+III.

(4.21)

Clearly,

I≤e^m_H−1(Ω)

_m

n=1

In

u(t)_H¹_0(Ω)dt+τ m n=1

Uⁿ_H¹_0(Ω)

≤c32e^m_H−1(Ω)

≤c33τ^1/2.

(4.22)

We also get II≤





Ω

_m

n=1

In

f(u)−fUⁿdt 2

dx





1/2

×





Ω

_m

n=1

In

u(t)−Uⁿdt 2

dx





1/2

≤T² _m

n=1

In

f(u)−fUⁿ²₂dt 1/2

× _m

n=1

In

u(t)−Uⁿ²₂dt 1/2

≤T² _m

n=1

In

f(u)−f(Uⁿ²₂dt 1/2

×

2u²_L2(0,T,H0¹(Ω))+ 2τ m n=1

Uⁿ²₂ 1/2

≤c34τ^1/2.

(4.23) The last inequality follows by using simultaneously theL^∞-estimate ofu(t) (see [6]),Uⁿ, and the error bound given in (a). Arguing exactly as in the previous estimate, we get

III≤T² _m

n=1

In

fUⁿ²₂dt 1/2

×

2u²_L2(0,T,H0¹(Ω))+ 2τ m n=1

Uⁿ²₂ 1/2

. (4.24) Using again the hypothesis (H1) and the estimates above, we obtain

III≤c35τ^1/2. (4.25)

Finally collecting these results, it follows that ∇

_T

0 e_ndt

2

2≤c36τ^1/2. (4.26)

This completes the proof.

Corollary4.2. Under hypotheses(H1)-(H2), problem (1.3) generates a continuous semi- groupSτdefined bySτUⁿ⁻¹=Uⁿ.

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Abderrahmane El Hachimi: UFR Math´ematiques Appliqu´ees et Industrielles, D´epartement de Math´ematiques et Informatique, Facult´e des Sciences, Universit´e Chouaib Doukkali, B.P. 20, El Jadida, Morocco

E-mail address:[email protected]

Moulay Rchid Sidi Ammi: UFR Math´ematiques Appliqu´ees et Industrielles, D´epartement de Math´ematiques et Informatique, Facult´e des Sciences, Universit´e Chouaib Doukkali, B.P. 20, El Ja- dida, Morocco

E-mail address:[email protected]

Special Issue on

Intelligent Computational Methods for Financial Engineering

Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)

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