International Journal of Analysis and Applications ISSN 2291-8639 Volume 2, Number 1 (2013), 38-53 http://www.etamaths.com APPLICATION OF Ep-STABILITY TO IMPULSIVE FINANCIAL MODEL OYELAMI, BENJAMIN OYEDIRAN AND SAM OLATUNJI ALE Abstract. In this paper, we consider an impulsive stochastic model for an investment with production and saving profiles. The conditions for financial growth for the investment are investigated under impulsive action and results are obtained using the quantitative and Ep stability methods. The impulsive stochastic differential equation considered is assumed to be driven by a process with jump and non-linear gestation properties. One of the results established shows that, in the long run, it is impossible for a financial investment to grow or dominates the prescribed average financial investment but has a threshold value for which the investment cannot grow beyond. It is also established that an Ep− stable investment vector can be found which allows financial growth but this vector must be constrained to be in a given invariant set:It is advisable for the saving and depreciation to satisfy certain growth rates for proper income and investment growths. 1. Introduction Impulsive differential equations (IDEs) are systems that are subject to rapid changes in the variables describing them. Impulses are noted to take place in different ways e.g. in the form of “shocks”, “jumps”, “mechanical impacts” etc. ([1]) and they take place for short moments during process of evolution ([1], [10], [14- 17]). Many real life processes are impulsive in nature, examples are the biological bustling rhythms, the change in the states of the economy of some countries, the population under rapid changes, the outbreaks of earthquakes, eruption of epidemic in some ecological set-ups and so on ([1] & [16-17]). In the recent times, financial markets are places where funds are sourced for investment. Many financial derivatives are traded under organized market system and trade over the counter. In the stock exchange market financial derivatives like plain vanillas, bonds and exotics options are traded. The volume of trade increases everyday and there is the need to analyze the performance of the market using some models. We need to determine the fair price of an option and payoff for the buyer. We need to understand the complex cash flow structures in the financial markets and the risks involve in managing the financial portfolios etc ([4], [6], [9],[13], [18] and [19]). Most business organizations and many countries of the world usually set aside some substantial amount for investment or put in place some machineries to gen- erate funds for investment. The instrument for raising money for investment often 2010 Mathematics Subject Classification. 97M30. Key words and phrases. Ep-Stability, Impulsive Financial Model. c©2013 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 38 APPLICATION OF Ep-STABILITY 39 take the form of sinking funds, treasury bills, capital stocks, national reserves etc. The goal of entrepreneurs are to make the investment have appreciable financial growths. It must be noted that sometimes, funds so invested experience favourable fi- nancial growth, if the investment atmosphere allows such growth. Under certain situations which are often not easily predicted, the business many suffer unexpect- ed economic recession, in which case, profit is lot in the investment period. Some countries rely on natural resources such as fossil oil as the major export earner, for which the prices of crude oil tends to fluctuate every year, hence, the state of the economy of those countries tend to show impulsive behaviour ([14], [16]). Furthermore, in the international market today, the price of fossil oil is impul- sive because of rapid rise and falling of price of the oil within short period of times. Moreover, profits made on investment on crude oil goes up and down in yo-yo way. Therefore, the use of impulsive models would be useful for modelling prices of energy derivatives using different exotic options. The model also have potential applications for studying several real life problems in several fields of human en- deavour that can be modeled using ISDEs with gestation function being taken to be zero. In view of the above, an investment may also be affected by impulsive phenom- enon and how can it experience growth under impulsive effect? the search for the answer to this question is the motivation for the study in this paper. We will con- sider an impulsive stochastic model in studying the financial investment portfolio with production and saving profiles. We will use the Ep-stability method, that is, stability relative to p-moment to the study model. Application of p-moment to SDEs are found in the literature but for the ISDEs we can say it is relatively new ([7]) if we consider the volume of publications made on SDEs. It is worthy of note to say, the structure of the solution of ISDEs changes with e- quilibrium points of the model, hence the investment and national income equilibria depend on the process driving forces and the volatilities for the model. Therefore, the stability of the financial model cannot be deduce in the ordinary stochastic sense. Hence, we will exploit a broader stability concept, that is, stability relative to invariant sets to study the model. The importance of this approach were em- phasized in many publications (see for examples [11-12] and [14], and references therein), however, we will unify this approach with Ep-stability, such kind of ap- proach had appeared for stochastic processes [17]) but as for impulsive systems the unified approach seems to be relatively new. 2. Usefulness of Impulsive Phenomenon in the Financial Modeling In mathematics a variety of models exist for financial investment using applica- tions of some theories in chaos, control, neural network, ecophysics and so on. Most models from Economics, the central problem is the concern about the interactions of many complex variables for which fundamental “scarce” variable is the capita ([2],[4],[6],[9],[13] and [16]). The model we will consider is one of the simplest impulsive mathematical model in economics, a “macro-model” with gestation lag and depreciation, it can be used as a simple model of company (or country) growth process and to demonstrate some fundamental relationships that exist among variables that quantify a financial investment using impulsive variables. 40 OYELAMI AND ALE We note that the impulsive system theory offers viable techniques to handle dwindling effect of the investment, for example, through impulsive system theory, we can identify factors responsible for rapid and irregular growth of the investment and also factors responsible for sudden drop in the investment at fixed and non-fixed investment periods ([10],[14],[16] and [20]). We will consider an impulsive stochastic model of a financial investment with income, capita stock and depreciation vector containing gestation lag. 3. Preliminary Treatment We shall denote by (Ω,ξ,p) a probability space Ω, being the set of points with events, which is a δ-algebra of subsets of Ω such that Ω ∈ ξ and p denotes the probability measure. Let x(t) be a random process at time t with expectation (mean) Ex(t) = x̄(t) and the variance is δ2x(t) = (x(t) − x̄(t))2. The autocovariant vector R(t,τ) = E(x(t)x(τ)). We shall say ([5]) the sequence of random process {xn(t)} converges to x(t) in probability almost surely as n → ∞, if for any ε > 0, δ > 0 there exists a number N such that p(|xn(t) −x(t)| > δ) < ε,n > N; and {xn(t)} is said to be convergent to x(t) with probability 1 as n →∞ if lim n→∞ p(|xn(t) −x(t)| = 0) = 1. Crucial to our investigation is the following set C([−h, 0],<) which is the space of continuous random processes on [−h, 0] and taking values on < = (−∞, +∞) and let <+ = [0, +∞). PC([−h, 0],<) = {x : [−h, 0] →< is a piecewise continuous random process for t ∈ [−h, 0] such that it is left continuous at tk, k = 1, 2, . . .}. K = {a(r) : a ∈ C([−h, 0],<) are monotonically increasing in r and lim r→+∞ a(r) = ∞}. Let V (t,x(t)) be a piecewise continuous random process on <+ ×C[−h, 0] and there exist wi ∈ K,i = 1, 2 such that the following conditions are satisfied: ω̄1(|ϕ(0)|) ≤ V (t,ϕ) ≤ ω̄2(|ϕ(0)|), ω̄i ∈ K, i = 1, 2 |V (t,ϕ1) −V (t,ϕ2)| ≤ C1(|ϕ1 −ϕ2|), C1 = constant, ϕi ∈ C([−h, 0],<+), i = 1, 2. Then the Dini derivative D+ (·)V (t,x(t)) of the function V (t,ϕ) along the solution path (·) for the ISDE in equation (·) is defined as D+ (·)V (t,x(t)) = limδ→0 sup 1 δ [V (t + δ,x(t) + δf) −V (t,x(t))]. Consider the following ISDE dx(t) = f(t,x(t))dt + g(x(t−h))dw(t), t 6= tk, k = 0, 1, 2, . . . (1) 4x(tk) = I(x(tk)) 0 < t0 < t1 < t2 < · · · < tk; lim k→∞ tk = +∞ where APPLICATION OF Ep-STABILITY 41 f : <+ × Ω → Ω; g : Ω → Ω, I : Ω → ∞. The expectation of V (t,x(t)) is EV (t,x(t)) and the variance δ2V (t,x(t)) and the p-moment EpV (t,x) about the origin. 3.1. Comparison system. Consider the following comparison impulsive stochas- tic differential equations (CISDE) corresponding to the eqn (1) dx(t) = e(t,u(t))dt + h(u(t))dw(t), t 6= γk, k = 0, 1, 2, . . . (2) 4u(γk) = I(u(γk)) 0 < γ0 < γ1 < γ2 < · · · < γk; lim k→∞ γk = +∞ where e : <+ × Ω → Ω; h,I : Ω → Ω. Assume that f,e,h and I are smooth enough as to guarantee the existence and uniqueness of solutions of eqn (1) and eqn (2) (see [10] and [19]). We will make use of the following definitions: Definition 1 ([11],[12] & [14]) Let x(t) be the solution of eqn (1) passing through (t0,x(t0 + 0) = x0) then we say that the solution x(t) = 0 of the eqn (1). 1. Ep-uniformly stable (u.s.) with respect to the invariant A = {x ∈ Ω : |x| ≤ r} if (a) |x0| = r implies |Exp(t)| = r,t ≥ t0; (b) ∀ ε > 0 and t0 ∈<+ there exists a real number δ = δ(ε) > 0 such that r − δ < |x0| < δ + r implies r −ε|Exp(t)| < r + ε, t ≥ t0; 2. Ep-uniformly asymptotically stable with respect to the invariant A if there exist real numbers δ0 > 0, and T = T(ε) > 0 such that r − δ0 < |x0| < r + δ + 0 implies r −ε < |Exp(t)| < r + ε,t ≥ t0 + T Remark 1 Ep-stability is the unification of invariant stability and stability with respect to p-moment ([10-11] & [14]). If Ex(t) = x̄(t) for p = 1 and r = 0, Ep-stability reduces to the usual stability in the Lyapunov’s sense for the impulsive stochastic equations. If the underlying variable is deterministic then the system is simply the impulsive ordinary differential equations. We will make use of the following auxiliary results. Let x be a random variable such that E(x) = µ,E(x−µ)2 = σ2,E|x−µ|r = βr,E(|x−µ|) = 0 and δ̃ = µ/σ. 4. Statement of the Problem Consider a simple impulsive stochastic differential model (isdm) dx(t) = δ−1y(t)dt− b + α1(t)g(x(t−h)) + σ1dw1(t), t 6= tk, k = 0, 1, 2, . . . (3) dy(t) = δ−1y(t)dt + α2(t)V (t) + σ2dw2(t), t 6= tk, k = 01, 2, . . . (4) dz(t) = βdx(t), t 6= tk, k = 0, 1, 2, . . . (5) 4x(tk) = βkx(tk) (6) Satisfying the initial conditions x(t0 + 0) = x0,y(t0 + 0) = y0 and z(t0 + 0) = z0 (7) 42 OYELAMI AND ALE 0 < t1 < t2 < t3 < · · · < tk, tk = +∞, as k →∞ where (1) x(t) is the investment variable (2) y(t) is the national (company’s) income (3) z(t) is the capital stock (4) v(t) is the fluctuation variable (5) wi(t) are assume to be Brownian processes. It is assumed that x(t) is the random variable representing in totality the amount of the investment (both liquid and solid asserts) which experience a growth rate of δβ−1 = saving Capita output δ = saving ratio, δ−1 is the drift, b is given depreciation rate and g(x(t−h)) is the depreciation function with gestation lag h with the expectation Eg (x(t−h)). g(x(t − h)) is generally assume to be nonlinear continuous random process. The expectation of g(x(t − h)) denote by g(x(t − h)) is assumed to exists. αi(t) are some jump parameters for i = 1, 2. The parameters βk = 1, 2, 3, . . . , account for the impulses that happen during the investment period. These parameters can be investment for some period of times. V (t) is assume to be statistically independent with respect to the investment variable, hence E(y(t),V (t)) = 0 and δ2(V (t),V (t)) = 1. We define f1(t,x(t)) = δ −1x(t) − bα1(t)g(x(t−h)) + σ1dw1(t) and f2(t,y(t)) = δ −1y(t) − bα2(t)v(t) + σ2dw2(t) dx(t) = a1(t)dt + b1(t)dw(t) and dy(t) = a2(t)dt + b2(t)dw2(t). Then we define the stochastic differential equation corresponding to (isde) as df1 = (f1t + a1(t)f1x + 1 2 b21(t)f1xxdt + b1(t)f1xdw1(t) df2 = (f2t + a2(t)f2x + 1 2 b22(t)f2xxdt + b2(t)f2xdw2(t) 4x(tk) = βkx(tk) 4y(tk) = βky(tk) and integration give f1(t,x(t) = f1(0,x0) + ∫ t 0 (f1s + a1(s)f1x + 1 2 b21(s)f1xx)ds + ∫ t 0 b1(s)f1xdw1(s) + ∑ t0 0, τ := t + s. Then E|x2(t)| ≤ A4e4α −1he−2α −1t|Ex2(−h)| + 2α10A 2B2AB2e−2α −1h|Ex(−h)|σ1 ∫ t 0 e−α −1τEx2(τ)dτ ≤ A4e−4α −1he−2α −1τC2 + 2AB 2e−2α −1h|Ex(−h)|exp ( σ1 1 −e−α −1τ α−1 ) ≤ C3. It follows that E|x2(t)| ≤ sup τ∈(−h,0] E|x2(τ)| ≤ max(C1,C2) < C4 Application of Chebyshev’s inequality yields, P(|x(t)| ≥ kE|x2(t)|) 1 k2 , k > 0 Then, lim t→∞ 1 t2 ∫ t 0 P(|x(s)| ≥ kE|x2(s)|) ≤ lim t→∞ 1 t2 ∫ t 0 1 k2 ds = lim t→∞ 1 t2 1 k2 t = 0. The result established above shows that, in the long run, “it is impossible for the investment to grow or dominates the prescribed average financial investment”. The behaviour of the financial investment under impulsive action is not neces- sarily be the same as the ordinary stochastic equations with prescribed probability distribution function, for example the Markov, Wienie, and Martingale processes etc. Let us assume that the impulsive processes have gamma distribution because of the relatively newness of the impulsive stochastic differential process and that their solutions behave as stated in the section 4.2. Although processes that allow jump behaviours as Poisson, Levy and Martingale can also be used to analyse the model. Theorem 1 APPLICATION OF Ep-STABILITY 47 Suppose the random variable x(t) in eqn (1) is a Stochastic process with gamma distribution with parameter (n + 1,µ) and define the following constants: H1 : A = [ ∏ t0 0, ∃δ = δ(x0,ε) such that |Ex(t)| < ε implies that |x0| < δ0, there is a finite number p > 0 such that r − ε < |Exp(t)| < r + ε for δ0 + r0 < |x0| < δ0 + r0. Let |x0|p < r −δ0,a2(ε−r) > β1(r − 2δ0), for a2 and β0 ∈ K Let m(t) be the solution of eqn (1) such that ṁ(t) ≤ g(t,m(t)), t 6= tk, k = 0, 1, 2, . . . m(t+k ) ≤ m(t) + ϕk(m(t)), t = tk, k = 0, 1, 2, . . . m(t+k ) ≤ u(t0) If r(t) = r(t,t0 + 0,u0) is a random process which is the maximal solution of the impulsive stochastic differential equation in eqn (1) then by standard results, V (t,x(t)) ≤ r(t) we can show that EV (t,x(t)) ≤ r(t). From eqn (2) EV (t,x(t)) ≤ EV (t0 + 0,x0) ≤ r(t) ≤ β1(E|x0|p) = β1(|x0|)p) ≤ β1(r − δ0) < β2(r −ε) Therefore, E|x(t)|p < ε + r for t ≥ t0, by similar estimation we have E|x(t)|p > ε−r for r − δ < |x0 < δ + r. Therefore the zero solution x(t) = 0 of the eqn (1) is E-stable with respect to the invariant set Ω. The general form of distribution function governing an impulsive stochastic mod- el is unknown, such distribution if exists may be continuous or discrete or even possess piecewise continuous property. Recently, the theory of Time Scale have been exploited to study systems which are either continuous or discrete or both simultaneously ([11-12]). In the quest for an ideal distribution for impulsive stochastic system and the correspond p-moments are open problems. Meanwhile, we define the characteristic function for x(t) taking into consideration impulsive tendency as Cx(ε) = E[x(t)e −iεt] From the characteristic function we can obtain the k-moment as E { xk(t) } = 1 (iε)k [ dkCx(ε) dεk ] , k = 1, 2, . . . The construction of the ideal distribution may be made by the formulation of im- pulsive analogue of the Chapman-Kolmogorov equation if the underlying stochastic process is a Markov process ([3]). The construction of an ideal distribution function for impulsive stochastic systems is one of the fundamental problems future research should focus on. Because of the peculiar nature of the problem of how to deter- mine Cx(ε) we resort to investigate the behaviour of the solution of the model using qualitative approach, hence the problem will be studied from stability point of view. Remark 3 We propose the monkey function which somehow shows the behaviour of a mon- key in the game and can be used to mimic the dynamics of the financial market. APPLICATION OF Ep-STABILITY 51 We define the monkey process as M(t1, t) =   fk(t) t ∈ [t1, tk), k = 1, 2, . . . fs(t) t ∈ [tk, tN ), fs(t) ∈ C∞(Ω) δ(tN − t) t = tN fj(t) t ∈ (tk+1, ts) 0 ≤ t1 < tk < tN ≤ ts   fk(t) is uniformly and identically distributed in the given interval fs(t) is a smooth random function in the given interval, and fj(t) is a continuous random process in the given interval δ(t) is the Dirac function of t The monkey function should be constructed to have the following properties: H1 : M(t1, t) ≥ 0 for t1 ≥ t H2 : ∫ t 0 M(t1,s)ds = 1 H3 : M(t1, 0) = δ(−t1) H4 : dM = M(t1, t + 4t) −M(t1, t) The construction of an ideal distribution for an impulsive stochastic system using the monkey function and the correspond p-moments are open problems. 5. Application of the E-stability Without loss of generality if the depreciation variable is chosen in such a way that g(x(t − h)) ≥ x(t) and the fluctuation variable is selected to be bounded (V (t) ≤ k = constant) then the comparison equations corresponding to the isdm is dx(t) ≤ (δ−1 − bα1)x(t)dt + σ1dw1(t), t 6= γk, k = 0, 1, 2, . . . dy(t) ≤ δ−1y(t)dt + K + σ2dw2(t), t 6= γk, k = 0, 1, 2, . . . 4x(γk) ≤ βkx(γk). If m1(t) and m2(t) are the maximal solution to the comparison’s equation above respectively. Then dm1(t) ≤ (δ−1 − bα1)m1(t)dt + σ1dw1(t), t 6= γk, k = 0, 1, 2, . . . dm2(t) ≤ δ−1m2(t)dt + K + σ1dw1(t), t 6= γk, k = 0, 1, 2, . . . 4x(tk) ≤ β1km1(γk) 4y(tk) ≤ β2km2(γk) m1(γk + 0) = x0 and m2(γk + 0) = y0 52 OYELAMI AND ALE whose solutions are found to be m1(t) = rm1 (t) = ∏ t0 0, as time t →∞, the investment is bounded above by lim t→∞ ∫ t t0 σ1 ∏ t0