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

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## Abstract

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.

## Keywords

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## Notes

### 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.

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