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Communication

# An Itô Formula for an Accretive Operator

by
Rémi Léandre
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

## 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