# A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier

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## Abstract

**:**

## 1. Introduction

## 2. The Formulation of the Problem

**Remark**

**1.**

**Lemma**

**1.**

**Remark**

**2.**

**Proof.**

**Remark**

**3.**

## 3. The HJB Equation

**Lemma**

**2.**

**Proof.**

**Remark**

**4.**

## 4. The Solution

**Theorem**

**1.**

**Proof.**

## 5. Application

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Technical Lemmas

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

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**Figure 1.**Plotting $\beta $ as a function of $\alpha $. The marked points are $\beta (0)=1$ and $\beta (1)\approx 0.839924$.

**Figure 2.**Plots of the value function ${V}^{*}(x,t)$, computed as in (22) associated with the process $\{{X}^{s}\}$ in (24) for d = 2 and $\alpha =3$. The black curves represent the optimal stopping boundary $b(t)={(1-t)}^{2}$. (

**a**) ${V}^{*}(x,t)$ as a funtion of $t\in [0,1]$ for fixed values of $x\in \{0,0.2,0.4,0.6,0.8\}$. (

**b**) ${V}^{*}(x,t)$ as a funtion of $(x,t)\in {[0,1]}^{2}$.

**Figure 3.**Four simulations of ${X}_{s}$, $s\in [0,1]$, as defined in (26) with $t=0$, $x=0$, for six different values of $\alpha $. The light grey areas represent one standard deviation of ${X}_{s}$ above and below its expected value (null in the simulations). The black solid curves represent the optimal stopping boundaries.

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D’Auria, B.; Ferriero, A.
A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier. *Mathematics* **2020**, *8*, 123.
https://doi.org/10.3390/math8010123

**AMA Style**

D’Auria B, Ferriero A.
A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier. *Mathematics*. 2020; 8(1):123.
https://doi.org/10.3390/math8010123

**Chicago/Turabian Style**

D’Auria, Bernardo, and Alessandro Ferriero.
2020. "A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier" *Mathematics* 8, no. 1: 123.
https://doi.org/10.3390/math8010123