Fuzzy Stochastic Differential Equations Driven by a Fuzzy Brownian Motion ()
1. Introduction
In our previous research, we have dealt with Fuzzy Itô Integral Driven by a Fuzzy Brownian Motion and in the present work, we develop Fuzzy stochastic differential equations driven by a fuzzy Brownian motion [1] .
Historically, the earliest approach for deterministic fuzzy differential equations was based on generalization of the Hukuhara derivative of a set-valued function. This was made by Puri and Ralescu in [2] and used by Kaleva in [3] [4] .
Deterministic fuzzy stochastic differential equations have been developed due to investigations of dynamic systems that have several applications in the modeling of classical problems in control theory, physics, biology, engineering economics and finance. In this kind of study, random disturbances are the only source of uncertainty. To resolve these situations, stochastic analysis methods must be used. However, in most real problems, we face a second source of uncertainty: that of nature vague (imprecise, fuzzy, etc). In this paper, we will deal with fuzzy stochastic differential equations by taking into account the fuzzy Brownian motion. By concerning fuzzy stochastic differential equations containing a classical Brownian motion (not fuzzy), the following works can be viewed, Malinowski [5] [6] and [7] . Under certain conditions, we study the existence and uniqueness of solutions of fuzzy stochastic differential equations containing a fuzzy Brownian motion.
The novelty of the paper is the definition of the fuzzy Itô process driven by a fuzzy Brownian motion, the expansion and the generalization of the fuzzy itô formula. These are done in Section 3 and Section 4.
2. Preliminaries and Notations
This section presents some definitions and elementary concepts of fuzzy stochastic processes that will be used in the sequel. We assume that the reader is familiar with the fuzzy arithmetic, fuzzy functions and fuzzy random variables notions, for more details, see [8] - [13] and references therein.
Let us begin by the notion of the Hukuhara Generalized Derivative.
Definition 2.1. (Bede and Stefanini [14] ) The Hukuhara generalized derivative of the fuzzy function
at
is defined by
(2.1)
If
verifying (2.1) exists, we say that
is Hukuhara generalized differentiable (gH-differentiable) at
.
Definition 2.2. (Bede and Stefanini [14] ) Let
and
, with
and
all differentiable in
. We say that
1)
is
-differentiable in
if
(2.2)
2)
is
-differentiable in
if
(2.3)
Definition 2.3 (Hinge point, Stefanini and Bede [14] ) We say that the point
is a hindge point considering the differentiability of
, if in all neighbourhood V of
, it exists
such that
Type (1) at the point
(2.2) is verified while (2.3) isn’t and at the point
(2.3) is satisfied while (2.2) isn’t;
Type (2) at the point
(2.3) is verified while (2.2) is not and at the point
(2.2) is satisfied while (2.3) isn’t.
Definition 2.4. (Parametric representation of the Hukuhara partial derivative of order 1, Allahviranloo and al. [9] ) Let
,
and
partially derivable functions with respect to x. We say that
1)
est
-differentiable with respect to x at
if
(2.4)
2)
is
-differentiable with respect to x at
if
(2.5)
Definition 2.5. (Filtration and filtered probability space, H. Duminil-Copin [15] )
Let
, we denote by
and let
be a probability space. A family
of σ-algebra on
is called a filtration if for all
,
. A filtration is continuous if for all
,
.
is called filtered probabilised space.
Definition 2.6. (Break Time, H. Duminil-Copin [15] )
Let
be a probability space and
a filtration. A random variable T with values in
is called a break time if
is in
for all
.
Definition 2.7. (Adapted fuzzy process, Malinowski [5] )
A fuzzy stochastic process
is
-adapted if for all
, the function
is
-mesurable for all
. The fuzzy stochastic
process
is mesurable
is
-mesurable for all
, the function
is mesurable with respect to the σ-algebra
defined by
(2.6)
Definition 2.8. Let
be a fuzzy stochastic process.
is
-continuous if almost all (with respect to the probability measure P) his trajectories, that means
are
-continuous fuzzy functions.
(2.7)
Definition 2.9. (Kumwimba and al. [1] ) The simple fuzzy Itô integral
driven by a Brownian motion is defined on
by
(2.8)
where
Remark 2.10. (Kumwimba and al. [1] ) In the expression (2.8),
and
are fuzzy numbers for all
. Due to the stability of
,
and
in
, The simple fuzzy Itô integral with respect to a fuzzy Brownian motion is a fuzzy number.
3. Fuzzy Itô Process
Let us define first the Fuzzy Itô process in the following manner.
Definition 3.1.
Let
be a one-dimensional fuzzy Brownian motion defined upon a filtered
.
We call fuzzy Itô process, a fuzzy process
with values in
such that
(3.1)
with
1)
-mesurable.
2) For all
,
.
3) For all
,
where
.
In the differential form, (3.1) is denoted by
(3.2)
Progressively, we can now present a one-dimensional fuzzy Itô process.
Now, let us give the fuzzy version of the integration by parts using generalised Hukuhara (Bede and Stefanini [14] ).
Theorem 3.2.
Let
be a continuous fuzzy function with bounded variation in
. Then
(3.3)
Moreover, (3.3) is a fuzzy mumber.
Proof. We know that
is a fuzzy number. Let
, we have
and
and
are differentiable-continuous. We thus obtain two cases (from the definition 2.2):
for
-differentiable and
for
-differentiable.
Suppose that
est
-differentiable.
Then, from the classical integration by parts, we obtain
for all
,
.
Due to the definition of Hukuhara difference and from this inequality
.
This
By expressing the above integral in a sum of little integrals, for all
we obtain
Thus the equality (3.3) is obtained from the equality of fuzzy mumbers.
4. Fuzzy Itô Formula
The most important result in the fuzzy stochastic integration is the fuzzy Itô formula. The Itô formula is also named variables change formula.
4.1. Fuzzy Itô Formula Driven by a Fuzzy Brownian Motion
Theorem 4.1
Let
be a
class Hukuara differentiable fuzzy function in
and let
a fuzzy Brownian motion
. So
is the fuzzy stochastic process defined by the formula
(4.1)
where
.
Proof. We only prove the cases where the fuzzy functions
and
are either
-differentiable or either
-differentiable and they do not contain the hinge points. Notice that the other cases can be treated similarly.
For the first case, the fuzzy functions
,
and
have parametric representations for
;
(4.2)
(4.3)
and
(4.4)
The lower and the upper bounds of these α-cuts are given by
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
and
(4.10)
In the same way, we could take
. From the classical Itô formula of the function
(Aggoun, Elliott [16] ) we have
(4.11)
Thus, the equations from (4.5) to (4.10) by taking the minimum and the maximum in (4.11) give for all
and
Hence, for all
,
Hence, for every
,
which establishes by definition, the equality sought.
4.2. Fuzzy Itô Formula of a Fuzzy Itô Process
Let us start this part by the following theorem.
Theorem 4.2.
Let
Fuzzy d’Itô process such that
(4.12)
Let
. Then
(4.13)
is also a fuzzy Itô process and
(4.14)
where
is computed according
(4.15)
Proof. Let
and
in
, such that
Then we have for all
, two cases according that
with
and
such that
. These
are abtained in this manner
Moreover,
where
and
Without loss of generalities, we can deal with the case of the second order generalized Hukuhara differential of type (1) from the definition 2.4. We also consider that the domain
does not have a hinge point. Thus, we have the following expressions
(4.16)
(4.47)
(4.18)
(4.19)
(4.20)
(4.21)
Hence, for all
with
, from the classical ItÃ’ formula applied to the function
, we thus obtain
(4.22)
By taking the maximum and the minimum to every terms from the Equation (4.22) and from the Equations (4.16)-(4.21) we get
and
Then, we have
Because all the derivatives are taken in the sense of generalized Hukuara and due to the definition 2.4, we have for every
Thus we obtain for every
Remark 4.3. The general case should take into account that the domain would just contain some hinge points. This case could be reduced to the treated case by subdividing the domain in several areas in which we do not have the hinge point inside but only at the edges of these subdomains. Some properties are useful in the following.
Theorem 4.4. Let
. Then, we have
(4.23)
Proof. From the definition of D, we have for all
We have expanded out and have assumed that
.
5. Fuzzy Stochastic Differential Equations
Notice that the notion of fuzzy differential equations defined in an uncertain framework has already been the subject of several investigations (see Buckley and Feuring [17] and Puri and Ralescu [2] ). In the same way, we will define in this section, the notion of fuzzy stochastic differential equations driven by a fuzzy Brownian motion. Using the Hukuhara differential, we have the following definition.
Definition 5.1. A fuzzy stochastic differential Equation (FSDE), on a filtered probability space
and driven by a fuzzy Brownian motion is given by
(5.1)
where
and
are borel-measurable fuzzy stochastic functions defined from
with values in
.
In the integral form, the fuzzy stochastic differential Equation (FSDE) driven by a fuzzy Brownian motion (5.1) is denoted
(5.2)
The solution of (5.1) is defined like that:
Definition 5.2. A solution of a fuzzy stochastic differential Equation (FSDE) driven by a fuzzy Brownian motion (5.1) is a fuzzy stochastic process
defined on
such that:
1) There exists a one-dimensional fuzzy Brownian motion,
with
;
2)
is
-adapted and continuous in t
, that is
for
with probability 1;
3)
et
satisfy (5.2).
Definition 5.3. There is uniqueness of a trajectory if for any probability space
, every filtration
and every fuzzy Brownian motion
, two fuzzy solutions
and
such that
almost everywhere, verifies almost everywhere
We say that
is a strong fuzzy solution wether
is adapted to the canonical fuzzy filtration
.
There is a weak unicity since two solutions of the fuzzy stochastic differential equation have necessarily the same law.
Theorem 5.4. Let
and
in
such that
(5.3)
or
(5.4)
for all
and
a.s.. Then for all
,
(5.5)
Proof. Due to the Doob inequality and the classical Itô integral isometry we obtain from the hypothesis (5.3)
And from the hypothesis (5.4), we get
Before stating and proving the existence and the uniqueness theorem of solutions of a FSDE, let us present a version of the Gronwall Lemma that will help us to prove this theorem.
Lemma 5.5. (Gronwall Lemma, Bouleau [18] , Lamberton and Lapeyre [19] ) if f is a continuous function, such that for all
, we have
, then
.
We can now state and prove the following important theorem:
Theorem 5.6. Assume that the fuzzy functions
and
satisfy the following conditions:
1) There exists a constant
such that
(5.6)
(5.7)
for every
and
;
2) For all fuzzy stochastic processes
, the fuzzy stochastic process
has the parametric representation
defined by
;
3) For all
et
in
, we assume that
(5.8)
or
(5.9)
for all
a.e.
Then the fuzzy stochastic differential equation driven by a fuzzy Brownian motion (5.1) admits a unique solution.
Proof. Let
, for all
, we define
. Then from the condition (1) of the theorem, we get
and thus,
. From the definition of the fuzzy stochastic integral with respect to a fuzzy brownian motion,
is well-defined. So we can define a continuous fuzzy stochastic process
with
, and that for all
. Now suppose that fuzzy continuous processes
are defined and verify
. So by the condition (1) of the theorem, we get
and then,
.
Which shows the fuzzy Itô integral
is well-defined. Hence we can define a continuous fuzzy stochastic process
(5.10)
From mathematical induction, we obtain a sequence of fuzzy stochastic processes
in
. Due to the Theorems 4.4 and 5.4, and from the equalities (2) and (4) of the Theorem and under our hypothesis, we get
where
. From the same above reason, we get
So we have
We get from the Chebyshev inequality
where
is appropriately a constant that only depend on
and T. From the fact that
is a closed subspace of the metric space
and from the Borel-Cantelli Lemma, we see that
uniformly converges with probability one on
. T is arbitrary,
determine a continuous fuzzy process that is clearly a solution of (5.1).
To show the unicity of the solution, let
and
two solutions of (5.1). So by above similar computations, we have
for all
. We get from the Lemma 5.5 (Gronwall Lemma) that
for all
. Hence, for
, we have
almost everywhere for all
. As
and
are D-continuous in t almost everywhere, we can conclude that
for all
almost everywhere.
6. Conclusion
The fuzzy stochastic differential equations treated in this paper constitute our great contribution in this field. We essentially based ourselves on the use of the Hukuhara differential by treating for certain cases of a single type of parametric representation between a fuzzy function and its different fuzzy Hukuhara derivatives. As an application of our results, we will soon treat the fuzzy Ornstein-Uhlenbeck process and the fuzzy Geometric Brownian Motion. By applying the methods presented in this article, it would be possible to deal with the fuzzy version of Partial Derivative Equations of Posson and Klein-Gordon with Neumann conditions as a generalized Problem of Two-Dimensional.