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Communication

An Itô Formula for an Accretive Operator

Laboratoire de Mathématiques, Université de Franche-Comté, route de Gray, Besançon 25030, France
Axioms 2012, 1(1), 4-8; https://doi.org/10.3390/axioms1010004
Submission received: 21 November 2011 / Revised: 12 March 2012 / Accepted: 13 March 2012 / Published: 21 March 2012
(This article belongs to the Special Issue Axioms: Feature Papers)

Abstract

:
We give an Itô formula associated to a non-linear semi-group associated to a m-accretive operator.

1. Introduction

Let us recall the Itô formula in the Stratonovich Calculus [1]. Let B t be a one dimensional Brownian motion and f be a smooth function on R. Then
f ( B t ) = f ( B 0 ) + 0 t f ( B s ) d B s
where we consider the Stratonovich differential.
In [2,3], we have remarked that the couple ( B t , f ( B t ) ) is a diffusion on R × R whose generator can be easily computed. This leads to an interpretation inside the semi-group theory of the Itô formula. Various Itô formulas were stated by ourself for various partial differential equations where there is no stochastic process [4,5,6,7,8,9]. See [9] for a review. For an Itô formula associated to a bilaplacian viewed inside the Fock space, we refer to [10].
There is roughly speaking following Hunt theory a stochastic process associated to a linear semi-group when the infinitesimal generator of the semi-group satisfied the maximum principle.
For nonlinear semi-group, the role of maximum principle is played by the notion of accretive operator. The goal of this paper is to state an Itô formula for a nonlinear semi-group associated to a m-accretive operator on C b ( T d ) , the space of continuous functions on the d-dimensional torus T d endowed with the uniform metric . .

2. Statement of the Theorems

Let ( E , . ) be a Banach space. Let L be a non-linear operator densely defined on E. We suppose L 0 = 0 . We recall that L is said to be accretive if for λ 0
e 1 e 2 + λ ( L ( e 1 ) L ( e 2 ) ) e 1 e 2
It is said to be m-accretive if for λ > 0
I m ( I + λ L ) = E
Let us recall what is a mild solution of the non-linear parabolic equation
t u t + L u t = 0 ; u 0 = e
We consider a subdivision 0 t 1 < < t N = 1 . We say that u t i is an ϵ-discretization of Equation (4) if:
t i + 1 t i < ϵ
u t i u t i 1 t i + 1 t i + L u i = 0
Definition 1. v is said to be a mild solution of Equation (4) if for all ϵ there exist an ϵ-discretization u of Equation (6) such that u t v t ϵ .
Let us recall the main theorem of [11,12]:
Theorem 1. If L is m-accretive, there exists for all e in E a unique mild-solution of Equation (4). This generates therefore a non-linear semi-group exp [ t L ] .
We consider the d-dimensional torus. We consider E = C b ( T d ) and let L be an m-accretive operator whose domain contains C b ( T d ) , the space of smooth functions on T d with bounded derivatives at each order which is continuous from C b ( T d ) into C b ( T d ) .
Let f C b ( T d ) . We consider g C b ( T d × R ) .
We consider the diffeomorphism ψ f of T d × R :
ψ f ( x , y ) = ( x , y + f ( x ) )
It defines a continuous linear isometry Ψ f of C b ( T d × R )
Ψ f [ g ] ( x , y ) = g ψ f ( x , y )
Definition 2. The Itô transform L f of L is the operator densely defined on C b ( T d × R )
L f = ( Ψ f ) 1 ( L I 1 ) Ψ f
Let us give the domain of L I 1 . C b ( T d × R ) is constituted of function g ( x , y ) .
L I 1 [ g ] ( x , y ) = L x g ( x , y )
where we apply the operator L on the continuous function x g ( x , y ) supposed in the domain of L for all y. We suppose moreover that ( x , y ) L x g ( x , y ) is bounded continuous. The domain contains clearly C b ( T d × R ) .
Theorem 2. If L is m-accretive on C b ( T d ) , its Itô-transform is m-accretive on C b ( T d × R ) .
We deduce therefore two non-linear semi-groups if L is m-accretive:
-
exp [ t L ] acting on C b ( T d ) .
-
exp [ t L f ] acting on C b ( T d × R ) .
Let g be an element of C b ( T d × R ) . We consider g f ( x ) = g ( x , f ( x ) ) . We get:
Theorem 3. (Itô formula) We have the relation
exp [ t L ] [ g f ] ( x ) = exp [ t L f ] [ g ] ( x , f ( x ) )
This formula is an extension in the non-linear case of the classical Itô formula for the Brownian motion. If we take L = 1 / 2 2 x 2 acting densely on C b ( R ) , we have
exp [ t L ] [ g ] ( x ) = E [ g ( B t + x ) ]
where t B t is a Brownian motion on R starting from 0. ( B t + x , f ( B t + x ) + y ) is a diffusion on R × R whose generator is L f .

3. Proof of the Theorems

Proof of Theorem 2. L I 1 is clearly m-accretive on C b ( T d × R ) . Let us show this result.
-
L I 1 is densely defined. Let g be a bounded continuous function on T d × R . By using a suitable partition of unity on R, we can write
g ( x , y ) = g n ( x , y )
where g n ( x , y ) = 0 if y does not belong to [ n 1 , n + 1 ] . By an approximation by convolution we can find a smooth function g n , ϵ ( x , y ) close from g ( x , y ) for the supremum norm and with bounded derivative of each order. x L x g n , ϵ is continuous in x and the joint function ( x , y ) L x g n , ϵ ( x , y ) is bounded continuous in ( x , y ) by the hypothesis on L.
-
Clearly Equation (2) is satisfied.
-
It remains to show Equation (3). If g belong to C b ( T d × R ) we can find x h ( x , y ) such that
h ( x , y ) + λ L x h ( x , y ) = g ( x , y )
g ( , y ) g ( . , y ) h ( . , y ) h ( . , y )
Therefore ( x , y ) h ( x , y ) is jointly bounded continuous.
Since Ψ f is a linear isometry of C b ( T d × R ) which transform a smooth function into a smooth function,
L f = ( Ψ f ) 1 ( L I 1 ) Ψ f
is clearly still m-accretive.       ☐
Proof of Theorem 3. Let us consider t i = i / N to simplify the exposition. Let us consider an ϵ-discretization u . of the parabolic equation associated to L f . This means that
u t i ( Ψ f ) 1 ( I d + 1 + 1 / N ( L I 1 ) ) i Ψ f g
I d + 1 is the identity on C b ( T d × R ) . But
( I d + 1 + 1 / N ( L I 1 ) ) = ( I d + 1 / N L ) I 1
such that
( ( I d + 1 / N L ) i I 1 ) Ψ f u t i = Ψ f g
By doing y = 0 in the previous equality, we deduce that
( 1 + L / N ) i u t i f = g f
Therefore u t i f is an ϵ-discretization to the parabolic equation associated to L.       ☐

Acknowledgements

We thank M. Mokhtar-Karroubi and B. Andreianov for helpful discussion.

References

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MDPI and ACS Style

Léandre, R. An Itô Formula for an Accretive Operator. Axioms 2012, 1, 4-8. https://doi.org/10.3390/axioms1010004

AMA Style

Léandre R. An Itô Formula for an Accretive Operator. Axioms. 2012; 1(1):4-8. https://doi.org/10.3390/axioms1010004

Chicago/Turabian Style

Léandre, Rémi. 2012. "An Itô Formula for an Accretive Operator" Axioms 1, no. 1: 4-8. https://doi.org/10.3390/axioms1010004

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