On fuzzy type-1 and type-2 stochastic ordinary and partial differential equations and numerical solution

Methodologies and Application


This paper develops the mathematical framework and the solution of a system of type-1 and type-2 fuzzy stochastic differential equations (T1FSDE and T2FSDE) and fuzzy stochastic partial differential equations (T1FSPDE and T2FSPDE). The theory of fuzzy stochastic differential equations is developed with fuzzy initial values, fuzzy boundary values and fuzzy parameters. Some natural phenomena which perturbs randomly in the influence of a white noise can be modelled as stochastic dynamic systems whose initial conditions and/or parameters may be imprecise in nature as well. The imprecision of initial values and/or parameters is generally modelled by fuzzy sets. In this paper, the concept fuzzy stochastic differential equations is developed with the introduction of fuzzy stochastic process, fuzzy stochastic random variable and fuzzy Brownian motion. The generalized \(\hbox {L}^{\mathrm{p}}\)-integrability, based on the extension of the class of differentiable and integrable fuzzy functions, is applied and fuzzy stochastic differential equations are represented as fuzzy integral equations. A novel numerical scheme for simulations of fuzzy stochastic differential equations have also been developed. Some illustrative examples have been provided for different T1FSDE, T1FSPDE, T2FSDE and T2FSPDE models related to mathematical finance and problems in mathematical biology.


Fuzzy stochastic differential equation Fuzzy Ito integral equation Type-1 and Type-2 \(\hbox {L}^{\mathrm{p}}\)-integrability Fuzzy Euler–Maruyama scheme Fuzzy Option pricing Fuzzy Stochastic Lotka–Voltera model with diffusion 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.


  1. Balasubramaniam P, Muralisankar S (2004) Existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions. Comput Math Appl 47:1115–1122MathSciNetCrossRefMATHGoogle Scholar
  2. Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations. Fuzzy Set Systems 151:581–599MathSciNetCrossRefMATHGoogle Scholar
  3. Bongiorno EG (2012) A note on fuzzy set-valued Brownian motion. Stat Probab Lett 82(4):827–832MathSciNetCrossRefMATHGoogle Scholar
  4. Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions. Springer, BerlinCrossRefMATHGoogle Scholar
  5. Castillo O, Melin P (2008) Type-2 fuzzy logic: theory and applications. Springer, pp 29–43Google Scholar
  6. Chen B, Chen W, Zhang W (2012) Robust filter for nonlinear stochastic partial differential systems in sensor signal processing: fuzzy approach. IEEE Trans Fuzzy Syst 20(5):957–970CrossRefGoogle Scholar
  7. Chen B, Lee B, Guo L (2003) Optimal tracking design for stochastic fuzzy systems. IEEE Trans Fuzzy Syst 11(6):796–813CrossRefGoogle Scholar
  8. Chen L, Wu R, Pan D (2011) Mean square exponential stability of impulsive stochastic fuzzy cellular neural networks with distributed delays. Exp Syst Appl 38:6294–6299CrossRefGoogle Scholar
  9. Chen M, Fu Y, Xue X, Wu C (2008) Two-point boundary value problems of undamped uncertain dynamical systems. Fuzzy Sets Syst 159:2077–2089MathSciNetCrossRefMATHGoogle Scholar
  10. Deb K, Agrawal RB (1995) Simulated binary crossover for continuous search space. Compl Syst 9:115–148MathSciNetMATHGoogle Scholar
  11. Deb K, Kumar A (1995) Real-coded genetic algorithm with simulated binary crossover: studies on multimodal and multiobjective problems. Complex Syst 9:431–454Google Scholar
  12. Gao Q, Feng G, Wang Y, Qin J (2013) Universal fuzzy models and universal fuzzy controllers for stochastic nonaffine nonlinear systems. IEEE Trans Fuzzy Syst 21(2):328–341CrossRefGoogle Scholar
  13. Gong C, Su B (2009) Delay dependent robust stabilization for uncertain stochastic fuzzy system with time-varying delays. Int J Innov Comput Inf Control 5:1429–1440Google Scholar
  14. Guo M, Xue X, Li R (2003) The oscillation of delay differential inclusions and fuzzy biodynamic models. Math Comput Model 37:651–658MathSciNetCrossRefMATHGoogle Scholar
  15. Hamrawi H (2011) Type-2 fuzzy alpha-cuts. De Montfort University, Ph.D. thesisGoogle Scholar
  16. Hamrawi H, Coupland S, John R (2010) A novel alpha-cut representation for type-2 fuzzy sets. In IEEE international conference fuzzy system, pp 1–8Google Scholar
  17. Hiai F, Umegaki H (1977) Integrals, conditional expectation, and martingales of multivalued functions. J Multivar Anal 7:149–182MathSciNetCrossRefMATHGoogle Scholar
  18. Hofmann N, Platen E, Schweizer M (1992) Option Pricing Under Incompleteness and Stochastic Volatility. Math Finance 2(3):153–187CrossRefMATHGoogle Scholar
  19. Hüllermeier E (1997) An approach to modeling and simulation of uncertain dynamical systems. Int. J. Uncertain Fuzziness Knowl Based Syst 5:117–137CrossRefMATHGoogle Scholar
  20. Ito K (1946) On a stochastic integral equation. Proc Jpn Acad 22(2):32–35MathSciNetCrossRefMATHGoogle Scholar
  21. Ito K (1950) Stochastic differential equations in a differentiable manifold. Nagoya Math J 1:35–47MathSciNetCrossRefMATHGoogle Scholar
  22. Jing L, Chen B, Zhang B, Peng H (2013) A hybrid fuzzy stochastic analytical hierarchy process (FSAHP) approach for evaluating ballast water treatment technologies. Environ Syst Res 2:1–10CrossRefGoogle Scholar
  23. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317MathSciNetCrossRefMATHGoogle Scholar
  24. Kaleva O (1990) The Cauchy problem for fuzzy differential equations. Fuzzy Sets Syst 35:389–396MathSciNetCrossRefMATHGoogle Scholar
  25. Kim YK (2002) Measurability for fuzzy valued functions. Fuzzy Sets Syst 129:105–109MathSciNetCrossRefMATHGoogle Scholar
  26. Lakshmikantham V, Gnana Bhaskar T, Vasundhara DJ (2006) Theory of set differential equations in a metric space. Cambridge Scientific Publishers, CambrigdeMATHGoogle Scholar
  27. Last M, Eyal S (2005) A fuzzy-based lifetime extension of genetic algorithms. Fuzzy Sets Syst 149:131–147MathSciNetCrossRefMATHGoogle Scholar
  28. Li H, Liu Z (2008) Aprobabilistic neural-fuzzy learning system for stochastic modeling. IEEE Trans Fuzzy Syst 16(4):898–908CrossRefGoogle Scholar
  29. Li S, Guan L (2007) Fuzzy set-valued Gaussian processes and Brownian motions. Inf Sci 177:3251–3259MathSciNetCrossRefMATHGoogle Scholar
  30. Long S, Xu D (2011) Stability analysis of stochastic fuzzy cellular neural networks with time-varying delays. Neurocomputing 72:2385–2391CrossRefGoogle Scholar
  31. Maiti MK, Maiti M (2007) Two-storage inventory model with lot-size dependent fuzzy lead-time under possibility constraints via genetic algorithm. Eur J Oper Res 179:352–371CrossRefMATHGoogle Scholar
  32. Malinowski MT (2013) Some properties of strong solutions to stochastic fuzzy differential equations. Inform Sci 252:62–80MathSciNetCrossRefMATHGoogle Scholar
  33. Malinowski MT (2015) Fuzzy and set-valued stochastic differential equations with local Lipschitz condition. IEEE Trans Fuzzy Syst 23(5):1891–1898CrossRefGoogle Scholar
  34. Malinowski MT (2012) Strong solutions to stochastic fuzzy differential equations of Itôo type. Math Comput Model 55:918–928CrossRefMATHGoogle Scholar
  35. Mazandarani M, Najariyan M (2014) Differentiability of type-2 fuzzy number-valued functions. Commun Nonlinear Sci Numer Simul 19:710–725MathSciNetCrossRefGoogle Scholar
  36. Mendel JM (2007) Type-2 fuzzy sets and systems: an overview. IEEE Comput Intell Mag 2(2):20–29Google Scholar
  37. Mizukoshi MT, Barros LC, Chalco-Cano Y, Román-Flores H, Bassanezi RC (2007) Fuzzy differential equations and the extension principle. Inf Sci 177:3627–3635MathSciNetCrossRefMATHGoogle Scholar
  38. Möller B, Beer M (2004) Fuzzy Randomness: uncertainty in civil engineering and computational mechanics. Springer, BerlinCrossRefMATHGoogle Scholar
  39. Nieto JJ (1999) The Cauchy problem for continuous fuzzy differential equations. Fuzzy Sets Syst 102:259–262MathSciNetCrossRefMATHGoogle Scholar
  40. Oberguggenberger M, Pittschmann S (1999) Differential equations with fuzzy parameters. Math Comput Model Dyn Syst 5:181–202CrossRefMATHGoogle Scholar
  41. Park JY, Jeong JU (2013) On random fuzzy functional differential equations. Fuzzy Sets Syst 223:89–99MathSciNetCrossRefMATHGoogle Scholar
  42. Rao R, Pu Z (2013) Stability analysis for impulsive stochastic fuzzy \(p\)-Laplace dynamic equations under Neumann or Dirichlet boundary condition. Bound Value Probl 2013:133MathSciNetCrossRefMATHGoogle Scholar
  43. Rao R, Wang X, Zhong S, Pu Z (2013) LMI approach to exponential stability and almost sure exponential stability for stochastic fuzzy Markovian-jumping Cohen-Grossberg neural networks with nonlinear \(p\)-Laplace diffusion. J Appl Math 2013(396903):21MathSciNetMATHGoogle Scholar
  44. Romaán-Flores H, Rojas-Medar M (2002) Embedding of level-continuous fuzzy sets on Banach spaces. Inform Sci 144:227–247MathSciNetCrossRefMATHGoogle Scholar
  45. Song S, Wu C, Lee ES (2005) Asymptotic equilibrium and stability of fuzzy differential equations. Comput Math Appl 49:1267–1277MathSciNetCrossRefMATHGoogle Scholar
  46. Song S, Wu C, Xue X (2006) Existence and uniqueness of Cauchy problem for fuzzy differential equations under dissipative conditions. Comput Math Appl 51:1483–1492MathSciNetCrossRefMATHGoogle Scholar
  47. Sourirajan K, Ozsen L, Uzsoy R (2009) A genetic algorithm for a single product network design model with lead time and safety stock considerations. Eur J Oper Res 197:599–608CrossRefMATHGoogle Scholar
  48. Tseng C (2007) Robust fuzzy filter design for a class of nonlinear stochastic systems. IEEE Trans Fuzzy Syst 15(2):261–274CrossRefGoogle Scholar
  49. Vasant P (2012) Meta-heuristics optimization algorithms in engineering, business, economics, and finance. IGI GlobalGoogle Scholar
  50. Vasant P (2014) Handbook of research on artificial intelligence techniques and algorithms. IGI GlobalGoogle Scholar
  51. Vasant P (2016) Handbook of research on holistic optimization techniques in the hospitality, tourism, and travel industry. IGI GlobalGoogle Scholar
  52. Wang S, Watada J (2012) Fuzzy stochastic optimization. Springer, New YorkCrossRefMATHGoogle Scholar
  53. Wu L, Ho D (2009) Fuzzy filter design for Itô stochastic systems with application to sensor fault detection. IEEE Trans Fuzzy Syst 17(1):233–242CrossRefGoogle Scholar
  54. Xia Z, Li J (2012) Switching fuzzy filtering for nonlinear stochastic delay systems using piecewise Lyapunov-Krasovskii function. Int J Fuzzy Syst 14:530–539MathSciNetGoogle Scholar
  55. Xiong P, Huang L (2013) On \(p\text{ th }\) moment exponential stability of stochastic fuzzy cellular neural networks with time-varying delays and impulses. Adv. Differ. Equ. 2013:172CrossRefGoogle Scholar
  56. Yao J, Chen M, Lu H (2006) A fuzzy stochastic single-period model for cash management. Eur J Oper Res 170:72–90CrossRefMATHGoogle Scholar
  57. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-1. Inf Sci 8:199–249MathSciNetCrossRefMATHGoogle Scholar
  58. Zmeškal Z (2001) Application of the fuzzy-stochastic methodology to appraising the firm value as a European call option. Eur J Oper Res 135:303–310MathSciNetCrossRefMATHGoogle Scholar
  59. Zmeškal Z (2010) Generalised soft binomial American real option pricing model (fuzzy-stochastic approach). Eur J Oper Res 207:1096–1103MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyDurgapurIndia

Personalised recommendations