[Paper Review] On the first time that an Ito process hits a barrier

Gerardo Hernández‐del‐Valle|arXiv (Cornell University)|Sep 10, 2012

Stochastic processes and financial applications10 references3 citations

TL;DR

This paper derives explicit density formulas for the first hitting time of Ito processes with local drifts modeled via solutions to Burgers' equation, particularly focusing on processes of class $\mathcal{B}^1$—such as 3D Bessel processes and Brownian bridges—when they hit moving boundaries. The key contribution is a classification of such processes based on their drift structure and the use of $h$-transforms and heat polynomials to derive first-passage time densities in both unbounded and bounded state space settings.

ABSTRACT

This work deals with first hitting time densities of Ito processes whose local drift can be modeled in terms of a solution to Burgers equation. In particular, we derive the densities of the first time that these processes reach a moving boundary. We distinguish two cases: (a) the case in which the process has unbounded state space before absorption, and (b) the case in which the process has bounded state space before absorption. The reason as to why this distinction has to be made will be clarified. Next, we classify processes whose local drift can be expressed as a linear combination to solutions of Burgers equation. For example the local drift of a Bessel process of order 5 can be modeled as the sum of two solutions to Burgers equation and thus will be classified as of class $\mathcal{B}^2$. Alternatively, the Bessel process of order 3 has a local drift that can be modeled as a solution to Burgers equation and thus will be classified as of class $\mathcal{B}^1$. Examples of diffusions within class $\mathcal{B}^1$, and hence those to which the results described within apply, are: Brownian motion with linear drfit, the 3D Bessel process, the 3D Bessel bridge, and the Brownian bridge.

Motivation & Objective

• To derive exact densities for the first hitting time of Ito processes with local drifts modeled via solutions to Burgers' equation.
• To classify stochastic processes based on whether their drift can be expressed as a sum of solutions to Burgers' equation, introducing the $\mathcal{B}^k$ classification.
• To distinguish between first hitting time behavior in unbounded versus bounded state space settings for such processes.
• To establish a connection between heat polynomials and the drift structure of key diffusions like the 3D Bessel process and bridge.
• To demonstrate that the 3D Bessel bridge is the only process in $\mathcal{B}^2$ that is also in $\mathcal{B}^1$ under the heat polynomial construction.

Proposed method

• Utilizes Doob's $h$-transform to reweight measures and derive expectations under the original measure from a standard Brownian motion.
• Applies Ito's lemma to martingale representations involving solutions to the backward heat equation.
• Employs heat polynomials $v_n(x,t)$ and derived heat polynomials $w_n(x,t)$ to construct $h$-functions with specific drift properties.
• Uses the backward heat equation $-h_t = \frac{1}{2}h_{xx}$ as a foundation for constructing $h$-transforms that generate the desired drifts.
• Classifies processes by the number $k$ of solutions to Burgers' equation in their drift, defining classes $\mathcal{B}^k$, with $\mathcal{B}^1$ including Brownian motion with drift and Bessel processes.
• Analyzes the structure of $w_n(x,t)$ and $v_n(x,t)$ to derive the ratio $w_n'(x,t)/w_n(x,t)$, which determines the drift of the transformed process.

Experimental results

Research questions

• RQ1 What is the first hitting time density for an Ito process with local drift derived from a solution to Burgers' equation when it encounters a moving boundary?
• RQ2 How can stochastic processes be systematically classified based on the number of solutions to Burgers' equation that compose their drift?
• RQ3 What distinguishes the first hitting time behavior of processes with unbounded state space from those with bounded state space under the same drift structure?
• RQ4 Which well-known diffusions, such as the 3D Bessel bridge, belong to the $\mathcal{B}^1$ or $\mathcal{B}^2$ classes, and what is their unique structural property?
• RQ5 Is there a unique process that belongs to both $\mathcal{B}^1$ and $\mathcal{B}^2$ under the heat polynomial construction, and if so, what is its significance?

Key findings

• The first hitting time density for a $\mathcal{B}^1$ process with a moving boundary is derived explicitly using $h$-transforms and heat polynomial constructions.
• The 3D Bessel bridge is the only process that belongs to both $\mathcal{B}^1$ and $\mathcal{B}^2$ under the heat polynomial framework, as shown by the structure of $w_n(x,t)$ and $v_n(x,t)$.
• The drift of the 3D Bessel bridge is shown to be expressible as $\frac{w_0'(x,t)}{w_0(x,t)} = -\frac{x}{t}$, which satisfies the Burgers equation.
• The 3D Bessel process and Brownian bridge are both classified as $\mathcal{B}^1$, with their drifts arising from single solutions to Burgers' equation.
• The ratio $\frac{w_n'(x,t)}{w_n(x,t)}$ is derived as $\frac{w_0'(x,t)}{w_0(x,t)} + \frac{v_n'(x,-t)}{v_n(x,-t)}$, linking the drift to heat polynomial derivatives.
• For $n=1$, the ratio $\frac{w_1'(x,t)}{w_1(x,t)} = -\frac{x}{t} + \frac{1}{x}$, confirming the drift of the 3D Bessel bridge and validating its $\mathcal{B}^1$ classification.