Author’s Accepted Manuscript

Stochastic differential equations with fuzzy drift and diffusion

Bjo¨rn Sprungk, K. Gerald van den Boogaart

PII: S0165-0114(13)00098-5

DOI: http://dx.doi.org/10.1016/j.fss.2013.02.011

Reference: FSS6286

To appear in: Fuzzy Sets and Systems

Cite this article as: Bjo¨rn Sprungk and K. Gerald van den Boogaart, Stochastic differential equations with fuzzy drift and diffusion, Fuzzy Sets and Systems, http://dx.do i.org/10.1016/j.fss.2013.02.011

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Stochastic Differential Equations with fuzzy Drift and

Diffusion

Bjo¨rn Sprungka, K. Gerald van den Boogaartb aTU Bergakademie Freiberg, Fakulta¨t fu¨r Mathematik und Informatik, Institut fu¨r

Numerische Mathematik und Optimierung, 09596 Freiberg, Germany bTU Bergakademie Freiberg, Fakulta¨t fu¨r Mathematik und Informatik, Institut fu¨r

Stochastik, 09596 Freiberg, Germany

Abstract

A new framework for the fuzzification of stochastic differential equations is presented. It allows for a detailed description of the model uncertainty and the non-predictable stochastic law of natural systems, e.g. in ecosystems even the probability law of the random dynamic changes due to unobservable influences like anthropogenic disturbances or climate variation. The fuzziness of the stochastic system is modelled by a fuzzy set of stochastic differential equations which is identified with a fuzzy set of initial conditions, timedependent drift and diffusion functions. Using appropriate function spaces the extension principle leads to a consistent theory providing fuzzy solutions in terms of fuzzy sets of processes, fuzzy states, fuzzy moments and fuzzy probabilites.

Keywords: Fuzzy stochastic differential equations, fuzzy stochastic processes, model uncertainty

Preprint submitted to Fuzzy Sets and Systems February 28, 2013

Stochastic Differential Equations with fuzzy Drift and

Diffusion

Bjo¨rn Sprungka, K. Gerald van den Boogaartb aTU Bergakademie Freiberg, Fakulta¨t fu¨r Mathematik und Informatik, Institut fu¨r

Numerische Mathematik und Optimierung, 09596 Freiberg, Germany bTU Bergakademie Freiberg, Fakulta¨t fu¨r Mathematik und Informatik, Institut fu¨r

Stochastik, 09596 Freiberg, Germany 1. Introduction

Stochastic differential equations (SDEs) dXt = f(t,Xt) dt+ g(t,Xt) dWt, X0 = ξ(ω) (1) have become a powerful tool to model processes arising in nature, engineering or economic sciences which are not deterministic, but subject to random fluctuation (e.g. [1, 2]). Random disturbances on the modelled system are represented by a stochastic differential term g(t,X) dW , where W denotes a multivariate Brownian motion and g(t,Xt) dWt is understood in the sense of an Itoˆ integral (see [1, 2, 3]). But still there remains a ”structural uncertainty”, fluctuations of the predicted behaviour of the system due to uncertain values or even unknown nonrandom fluctuations of model parameters.

For example in stochastic models of ecosystems long term parameters like the capacity for given species might vary due to climate change. In this paper we will present a fuzzy based framework for model and parameter uncertainty in stochastic differential equations. Like for fuzzy (ordinary) differential equations there already exist several approaches.

Jung and Kim [4] defined a stochastic integral for set-valued stochastic processes which is used in [5] to define fuzzy set-valued stochastic differential equations. Unfortunately this approach leads to unbounded fuzzy setvalued random variables, since set-valued stochastic integrals are typically unbounded (see [6, 7]). Thereon, Ogura [8], Li and Li [9] and Zhang et al. [7, 10] presented approaches for set-valued and fuzzy set-valued SDEs but with single-valued diffusion terms g(t,Xt) in order to get bounded results.

Preprint submitted to Fuzzy Sets and Systems February 28, 2013

However, single-valued diffusions of fuzzy-set valued processes seem not to be an appropriate tool for modelling parameter uncertainty, e.g. in economics this would correspond to a single, precisely known volatility of a fuzzy option price, which is indeed the weak point of economic modelling.

Another approach to SDEs with set-valued coefficients is the theory of stochastic inclusions (see [11, 12, 13]). There, the set of solutions is given by all processes X whose increments are in the closure of the corresponding

Aumann integrals:

Xt −Xs ∈ cl {∫ t s

F (r,Xr) dr + ∫ t s

G(r,Xr) dWr) } . (2)

Here W is one-dimensional, F : [0, T ]×Rn → Kk(Rn) and G : [0, T ]×Rn →

Kk(Rn) are compact set-valued mappings and the integrals are defined as

Aumann integrals. This theory maps set-valued coefficients to sets of processes and avoids the unboundedness-problem of the approaches mentioned before.

We will present a much simpler but also more flexible approach, which is in substance the following: go back to the classical extension principle, carry out some sensitivity analysis for SDE models, use fuzzy sets of inputs (or data) and ask for the fuzzy set of outputs (or solutions). If this operation is continuous in a well-defined sense, bounded fuzzy data cannot lead to unbounded fuzzy solutions. In particular, our approach allows to describe more complicated types of uncertainty, e.g. dependencies between drift and the diffusion coefficients, changing model parameters or fuzzy information about range and speed of change of time depending parameters.

This article is organized as follows. In section 2 we state the necessary basics of stochastic processes and stochastic differential equations. In section 3 our approach to SDEs with fuzzy initial conditions and coefficients is presented. Therefore, a theory of fuzzy sets of processes is discussed and some properties of the solutions of fuzzy sets of SDEs, so called fuzzy SDEs, are stated. Moreover, it is shown how SDEs depending on fuzzy parameters fit into our concept of fuzzy SDEs. In section 4 the presented approach is applied to an illustrative problem from stochastic population dynamics and in section 5 some conclusions are drawn. 2 2. Stochastic processes and differential equations