# A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

## Abstract

**:**

## 1. Introduction

## 2. Essentials on Gauss-Markov Processes and FET

- (i)
- ${S}_{1}(t)<{S}_{2}(t)$, $\forall t\in T$
- (ii)
- ${S}_{1}({t}_{0})<X({t}_{0})\equiv {x}_{0}<{S}_{2}({t}_{0})$, ${t}_{0}\in T$.

#### Closed-Forms Results

## 3. Symmetry Properties

#### Transition Distribution Function in a Two-Sided Region

- (i)
- ${S}_{1}(t,\tau )<{S}_{2}(t,\tau )\phantom{\rule{1.7cm}{0ex}}\forall t\ge \tau $;
- (ii)
- ${\mathrm{lim}}_{t\downarrow \tau}{S}_{1}(t,\tau )<y(\tau )<{\mathrm{lim}}_{t\downarrow \tau}{S}_{2}(t,\tau ).$

**Theorem**

**1.**

## 4. Characterization of the Transition Density in a Two-Sided Region

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 4.1. Proof of Theorem 1

**Proof.**

- (i)
- $\tilde{\beta}(x,t|y({t}_{0}),{t}_{0})=0$ for $x={S}_{i}(t,{t}_{0})$ ($i=1,2$), $\forall t>{t}_{0}$;
- (ii)
- $\tilde{\beta}(x,t|y({t}_{0}),{t}_{0})\ge 0$, $\forall x\in ({S}_{1}(t,{t}_{0}),{S}_{2}(t,{t}_{0}))$, $\forall t>{t}_{0}$;
- (iii)
- $\tilde{\beta}(x,t|y({t}_{0}),{t}_{0})$ satisfies the delta condition $\forall x\in ({S}_{1}(t,{t}_{0}),{S}_{2}(t,{t}_{0}))$, i.e.,$$\underset{t\downarrow {t}_{0}}{\mathrm{lim}}\tilde{\beta}(x,t|y({t}_{0}),{t}_{0})=\delta (x-y({t}_{0}))$$

## 5. Pdf of FET

**Proposition**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

## 6. Two Specific Examples

#### 6.1. The Wiener Process

#### 6.2. The Ornstein-Uhlenbeck Process

## Funding

## Conflicts of Interest

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Pirozzi, E.
A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries. *Symmetry* **2020**, *12*, 279.
https://doi.org/10.3390/sym12020279

**AMA Style**

Pirozzi E.
A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries. *Symmetry*. 2020; 12(2):279.
https://doi.org/10.3390/sym12020279

**Chicago/Turabian Style**

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2020. "A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries" *Symmetry* 12, no. 2: 279.
https://doi.org/10.3390/sym12020279