Exact Solutions of Stochastic Differential Equations: Gompertz, Generalized Logistic and Revised Exponential

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1 Methodol Comput Appl Probab 1) 1:61 7 DOI 1.17/s Exact Solutions of Stochastic Differential Equations: Gompertz, Generalized Logistic and Revised Exponential Christos H. Skiadas Received: 1 June 9 / Accepted: 16 June 9 / Published online: 3 June 9 Springer Science + Business Media, LLC 9 Abstract Exact analytic solutions of some stochastic differential equations are given along with characteristic futures of these models as the Mean and Variance. The procedure is based on the Ito calculus and a brief description is given. Classical stochastic models and also new models are provided along with a related bibliography. Stochastic models included are the Gompertz, Linear models with multiplicative noise term, the Revised Exponential and the Generalized Logistic. Emphasis is given in the presentation of stochastic models with a sigmoid form for the mean value. These models are of particular interest when dealing with the innovation diffusion into a specific population, including the spread of epidemics, diffusion of information and new product adoption. Keywords Stochastic simulation Analytic solutions Stochastic modeling Stochastic Gompertz model Stochastic generalized Logistic model Revised exponential Stochastic simulation AMS Subject Classification 37H1 37L55 6H5 6H7 6H1 6H35 65C3 91B7 1 Introduction The Stochastic Differential Equations SDE) play an important role in numerous physical phenomena. The numerical methods for solving these equations show low accuracy especially for the cases with high non-linear drift terms. It is therefore very important to search and present exact solutions for SDE. The resulting solutions are also important to check for the accuracy of existing numerical methods. C. H. Skiadas B) Technical University of Crete, Chania, Crete, Greece [email protected]

2 6 Methodol Comput Appl Probab 1) 1:61 7 The first attempts for solving SDEs where based on proposing an integrating factor that could transform a SDE to a linear form that could be solved explicitly. A systematic method for reducing a non-linear SDE to a linear one was due to Kloeden and Platen 1999) and Kloeden et al. 199, 3). They proposed a suitable transformation function for the reduction of a particular SDE. This method is suitable for the cases presented here. The main theoretical issues are given in the following. 1.1 Ito Stochastic Differential Equations A stochastic process xt) confirms an Ito stochastic differential equation of the form if: dxt) = ax, t) dt + bx, t) dwt) 1) if for all t and t the following stands Gardiner 199), xt) = xt ) + t ax, s) ds + 1. Solution Methods of Stochastic Differential Equations t bx, s) dws) ) The method that will be presented and applied further down is based on the Ito norm Ito 1951, 1944) and is used for the reduction of an autonomous nonlinear stochastic differential equation in the form of Kloeden and Platen 1999): dyt) = ayt)) dt + byt)) dwt) 3) into a linear for xt) stochastic differential equation, dxt) = a 1 xt) + a ) dt + b 1 xt) + b ) dwt) 4) Then the solution of this last equation is given by where x t = Φ t x + a b 1 b ) Φ 1 t ds + b Φ t = exp a 1 t 1 b 1 t + b 1w t Φ 1 t dw s The mean and the variance are computed by the following forms for the Mean and dm t dt = a 1 m t + a dv t = a 1 + b 1 dt )V t + m t a + b 1 b ) + b for the Variance. By the use of a suitable transformation function xt) = Uyt)) the reduction method was initially presented by Gihman and Skorokhod 197) both for autonomous and for non-autonomous stochastic differential equations. In this case,

3 Methodol Comput Appl Probab 1) 1: only the reduction method for autonomous stochastic differential equations will be presented. In applying this formula to Ito in the transformation function Uy) the following results: duy) = ϑuy) ϑy dy + 1 ϑ Uy) ϑy dy) The above method will be used for the solution of some nonlinear stochastic diffusion equations which are presented in the following providing closed form solutions. The Gompertzian Stochastic Model The deterministic Gompertzian Model Gompertz 185)hastheform: dx t = bx t ln x t dt where b is a constant. This a growth model and the maximum growth rate is achieved when x inf = exp 1). This Gompertz function was proposed as a model to express the law of human mortality and can be used for population estimates. It is a sigmoid S-shaped) model as regards the form x t plotted against t. The Gompertz model is also applied in innovation diffusion modeling and in new product forecasting. In these cases the fluctuations are related to the magnitude x t of the measurable characteristic part of the system and it is assumed that the noise term could be expressed by a multiplicative 1-dimensional white noise process. Thus the S.D.E. model resulting from the above deterministic one must have the form see Skiadas et al. 1994): dx t = bx t ln x t dt + cx t dw t where x t is the unknown stochastic process, b and c are constants and w t is 1- dimensional Wiener process..1 First Method The solution of the last stochastic differential equation is obtained by applying the Ito formula to the transformation function y t = ln x t so that, dy t = d ln x t = x 1 t dx t 1 x dx t ) By substituting x t from the above Gompertz stochastic differential equation and rearranging yields: dy t = d ln x t = by t 1 c )dt + cdw t The last equation is a stochastic linear differential equation and it is solved using the previous formulas to give y t = ln x t = ln x exp bt) c 1 exp bt)) + c exp bt) b expbs)dw s

4 64 Methodol Comput Appl Probab 1) 1:61 7 and the solution for x t is x t = exp ln x exp bt) c 1 exp bt)) + c exp bt) b. Second Method expbs)dw s This second method of solution of the Gompertz stochastic differential equation is obtained by applying the Ito formula to the transformation function: So that, y t = gx t, t) = expbt) ln x t dy t = b expbt) ln x t dt + expbt) 1 dx t 1 x t expbt) 1 x dx t ) t By substituting x t from the above Gompertz stochastic differential equation and rearranging yields: Thus dexpbt) ln x t ) = c expbt)dw t 1 c expbt)dt dexpbt) ln x t ) = c expbt)dw t 1 c expbt)dt By integrating the last formula from to t and rearranging the solution of the Gompertz stochastic differential equation results: x t = exp exp bt) ln x + c exp bt) expbs)dw s c 1 exp bt)) b This is precisely the same as the form computed by the first method. 3 Stochastic Models with Multiplicative Error Term A stochastic model with multiplicative error term is expressed by the stochastic differential equation dx t = b t x t dt + cx t dw t By using the transformation yt, x t ) = ln x t and by applying ITO s rule results: dln x t ) = b t dt + cdw t 1 c dt Integrating the last equation from t to t and taking antilogarithms, the following analytic solution for x t is obtained: x t = x t exp b s ds 1 ) t c t t ) + cw t w t )

5 Methodol Comput Appl Probab 1) 1: The mean value of x t is expressed by Ex t =x t exp The Variance is expressed by 3.1 Special Cases The Case: b t = b/t Then Varx t =x t exp t t ) b s ds [exp b s ds c t t )) 1 ] dx t = b t x tdt + cx t dw t and x t = x exp b ln t/t ) 1 ) c t t ) + cw t w t ) 3.1. The Case: b t = b/t Then dx t = b t x tdt + cx t dw t and x t = x exp b1/t 1/t ) 1 ) c t t ) + cw t w t ) This is stochastic process providing paths with a sigmoid form mean value see Skiadas et al. 1993). The mean value of x t is expressed by Ex t =x t exp b1/t 1/t ) The Variance is expressed by Varx t =x t exp b1/t 1/t ) [ exp c t t )) 1 ] 4 The Generalized Logistic Stochastic Model A deterministic version of this model was developed by Richards 1959) based on a previous simpler model proposed by Von-Bertalanffy for the description of the increase of weight as a function of the metabolism process of animals. Other deterministic forms of Generalized models can be found in Skiadas 1985, 1986, 1987).

6 66 Methodol Comput Appl Probab 1) 1: The Deterministic Model The deterministic Generalized Logistic model model is expressed by the differential equation xt ) m ) dx t = bx t 1 dt F where b, m and F are parameters. By dividing both sides of the last equation by F and placing y t = x t /F results dy t = by t 1 yt ) m) dt To solve this differential equation the method of change of variables is needed by using z t = y m t. Then the last differential equation reduces to the linear differential equation which is easily solved to give dz t = bmz t 1)dt 5) lnz t 1) = lnz 1) = bmt 6) where z = zt = ) = y m = x /F) m Finally by transforming to y t andthentox t the solution of the deterministic Generalized Logistic model results y t =[1 + y m 1) exp bmt)] 1/m 7) x t = F [ 1 + x /F) m 1) exp bmt) ] 1/m This is a sigmoid form model with saturation level achieved at the upper limit F. The parameter b accounts for the speed of the product adoption process. The inflection point is achieved at x inf = 1/m + 1)) 1/m. 4. The Stochastic Model The stochastic Generalized Logistic model with a multiplicative noise term is given by the stochastic differential equation xt ) m ) dx t = bx t 1 dt + cx t dw t F As for the deterministic model above by dividing both sides of the last equation by F and placing y t = x t /F results dy t = by t 1 yt ) m) dt + cy t dw t For the solution of the last stochastic differential equation the reduction method will be used. The change of variables is achieved by using the same integration factor as for the deterministic case z t = y m t. Then the following Ito formula is applied to the transformation function z t : dz t = ϑy t ϑy dy t + 1 ϑ y t ϑy dy t) 8)

7 Methodol Comput Appl Probab 1) 1: By introducing in the last form the values for y t and dy t from the previous forms and rearranging the following form of the transformed stochastic differential equation results: c ) ) dz t = mm + 1) bm z t + bm dt cmz t dw t This is a linear autonomous stochastic differential equation. The solution arises after using the following general form for the solution of a linear stochastic differential equation of the type: dr t = a 1 r t + a )dt + b 1 r t )dw t where a 1, a and b 1 are parameters. The solution is given by: r t = Φ t r + a Φt 1 ds where Φ t = exp a 1 b 1 /)t + b 1w t Considering that in our case a 1 = c mm + 1)/ bm, a = bm and b 1 = cm; Φ t is given by: Then the resulting solution for z t is Φ t = exp bm+ c m/)t cmw t ) z t = Φ t z + bm [ c ) ] z t = exp m b t cw t z + bm Φs 1 ds exp c m/ + bm)s + cmw s ))ds Finally, the solution of the Generalized Logistic stochastic differential equation arises after the application of the reversal transformations y t = z 1/m t and x t = Fy t and is in the form of: 1/m y t = Φ 1/m t y m + bm Φs 1 ds x t = FΦ 1/m t x /F) m + bm The resulting mean value for zero noise c = is 1/m Φs 1 ds x t = F [ 1 + x /F) m 1) exp bmt) ] 1/m 9) That is precisely the solution of the deterministic case.

8 68 Methodol Comput Appl Probab 1) 1: The Stochastic Logistic Model The Logistic model, a model with very many applications in several fields, results as a special case of the Generalized Logistic model when the parameter m = 1. The stochastic version of this model is given by: xt )) dx t = bx t 1 dt F See analytic solution and related applications in Giovanis and Skiadas 1995) and for a more general model in Skiadas and Giovanis 1997) Then from the solution for the Generalized Logistic model the formula the the solution of the Logistic model results immediately by introducing m = 1. where x t = FΦt 1 x /F) 1 + b 1 Φs 1 ds Φ t = exp b + c /)t cw t x t = F exp b c /)t + cw t x /F) 1 + b The resulting value for zero noise c = is x t = F expbt) x /F) 1 + b exp b c /)s + cw s ds 1 1 expbs)ds x t = F [ 1 + x /F) 1 1) exp bt) ] 1 1) 4.4 The Mean Value To find the mean value of the Logistic stochastic model first we observe that the following relation holds for the expectation of a stochastic process gw) and Eexpgw)) =exp Egw)) Thus the following two relations result E exp We thus obtain Eexpcw t )=exp Ecw) cw s )ds = exp = exp E c t ) = exp E cw s ) / ds = exp EΦ t =exp bt) c t ) c s/)ds

9 Methodol Comput Appl Probab 1) 1: Fig. 1 The stochastic logistic model and E exp b ) Φt 1 ds = E exp b = exp b ) b c /)s + cw s )ds ) bs)ds = expbt) 1 Finally the mean value of the stochastic Logistic model is: Ex t =F/ x /F) 1 exp bt) + 1 exp bt) ) 4.5 The Variance F Ex t = 1 + x /F) 1 1 ) exp bt) To calculate the Variance of the stochastic Logistic model first we estimate the autocorrelation function that is Ex t x t =F expbt) x /F) 1 + expbt) 1 The Variance is calculated by the following form Thus Varx t =Ex t x t Ex t ) Varx t =F expbt) x /F) 1 + expbt) 1 expc t) 1) We can now give an illustrative example of the Stochastic Logistic Model including several stochastic paths presented in Fig. 1. The parameters selected are: c =.5, x =, F = 1, b =.. 5 Conclusion Exact solutions of several stochastic models are given along with the related analysis. The solution methods are based on the Ito theory for the solution of stochastic

10 7 Methodol Comput Appl Probab 1) 1:61 7 differential equations. Some sigmoid form models are given in the deterministic and the stochastic form and an illustrative example of the stochastic Logistic model is presented. The provided exact analytic forms could be very useful for testing the existing and new approximate methods for the solution of stochastic differential equations. References Gardiner CW 199) Handbook of stochastic methods for physics, chemistry and natural science, nd edn. Springer, Berlin Gompertz B 185) On the nature of the function expressive of the law of human mortality, and on the mode of determining the value of life contingencies. Philos Trans R Soc 115: Gihman II, Skorokhod AV 197) Stochastic differential equations. Springer, Berlin Giovanis AN, Skiadas CH 1995) Forecasting the electricity consumption by applying stochastic modeling techniques: the case of Greece. In: Janssen J, Skiadas CH, Zopounidis C eds) Advances in applying stochastic modeling and data analysis. Kluwer, Dordrecht Ito K 1944) Stochastic integral. Proc Imp Acad Tokyo : Ito K 1951) On stochastic differential equations. Mem Am Math Soc 4:1 51 Kloeden PE, Platen E 1999) Numerical solution of stochastic differential equations. Springer, Berlin Kloeden PE, Platen E, Schurz H 3) Numerical solution of SDE through computer experiments. Springer, Berlin Kloeden PE, Schurz H, Platten E, Sorensen M 199) On effects of discretization on estimators of drift parameters for diffusion processes. Research Report no. 49, department of theoretical statistics. Institute of Mathematics, University of Aarhus Richards FJ 1959) A flexible growth function for empirical use. J Exp Bot 1:9 3 Skiadas CH 1985) Two generalized rational models for forecasting innovation diffusion. Technol Forecast Soc Change 7:39 61 Skiadas CH 1986) Innovation diffusion models expressing asymmetry and/or positively or negatively influencing forces. Technol Forecast Soc Change 3: Skiadas CH 1987) Two simple models for the early and middle stage prediction of innovation diffusion. IEEE Trans Eng Manage 34:79 84 Skiadas CH, Giovanis AN 1997) A stochastic bass innovation diffusion model studying the growth of electricity consumption in Greece. Appl Stoch Models Data Anal 13:85 11 Skiadas CH, Giovanis AN, Dimoticalis J 1993) A sigmoid stochastic growth model derived from the revised exponential. In: Janssen J, Skiadas CH eds) Applied stochastic models and data analysis. World Scientific, Singapore, pp Skiadas CH, Giovanis AN, Dimoticalis J 1994) Investigation of stochastic differential models: the gompertzian case. In: Gutierez R, Valderama Bonnet MJ eds) Selected topics on stochastic modeling. World Scientific, Singapore