# An Itô Formula for an Accretive Operator

## Abstract

**:**

## 1. Introduction

## 2. Statement of the Theorems

**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 $\parallel {u}_{t}-{v}_{t}\parallel \le \u03f5$.

**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 $exp[-tL]$.

**Definition 2.**The Itô transform ${L}^{f}$ of L is the operator densely defined on ${C}_{b}({T}^{d}\times R)$

**Theorem 2.**If L is m-accretive on ${C}_{b}\left({T}^{d}\right)$, its Itô-transform is m-accretive on ${C}_{b}({T}^{d}\times R)$.

- -
- $exp[-tL]$ acting on ${C}_{b}\left({T}^{d}\right)$.
- -
- $exp[-t{L}^{f}]$ acting on ${C}_{b}({T}^{d}\times R)$.

**Theorem 3.**(Itô formula) We have the relation

## 3. Proof of the Theorems

- -
- $L\otimes {I}_{1}$ is densely defined. Let g be a bounded continuous function on ${T}^{d}\times R$. By using a suitable partition of unity on R, we can write$$g(x,y)=\sum {g}^{n}(x,y)$$
- -
- Clearly Equation (2) is satisfied.
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- It remains to show Equation (3). If g belong to ${C}_{b}({T}^{d}\times R)$ we can find $x\to h(x,y)$ such that$$\begin{array}{c}\hfill h(x,y)+\lambda {L}_{x}h(x,y)=g(x,y)\end{array}$$$$\begin{array}{c}\hfill \parallel g(,y)-g(.,{y}^{\prime}){\parallel}_{\infty}\ge \parallel h(.,y)-h(.,{y}^{\prime}){\parallel}_{\infty}\end{array}$$

## Acknowledgements

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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