Branching-selection particle systems and inverse first passage problems

Technical Summary: Branching-Selection Particle Systems and Inverse First Passage Problems

Problem Statement
This paper addresses the generalized inverse first passage problem (IFPP). The classical IFPP, originating from a question by Shiryaev, asks whether, given a standard Brownian motion $(W_t)_{t \ge 0}$ and a target probability distribution $p$ on $[0, \infty]$, there exists a boundary $b(t)$ such that the first passage time $\tau := \inf\{t : W_t \le b(t)\}$ follows the distribution $p$.

The paper focuses on a generalization known as the soft-killed inverse first passage problem (SKIFPP). Here, the stopping time is defined as:
$$\tau := \inf \left\{ t : \Lambda \int_0^t \omega(W_s - b(s)) ds \ge U \right\}$$
where $U \sim \text{Exp}(1)$, $\Lambda \in (0, \infty)$, and $\omega: \mathbb{R} \to [0, 1]$ is a monotonic decreasing function. Unlike the classical case where default occurs immediately upon crossing a boundary, this formulation models default as a cumulative function of the path's history relative to the boundary. The objective is to construct a boundary $b(t)$ such that $\tau \sim p$, given a specific initial distribution for $W_0$ and a target survival probability $p([t, \infty))$.

Methodology
The author constructs a branching-selection particle system (specifically a variant of the $N$-branching Brownian motion, or $N$-BBM) to approximate the solution to the SKIFPP. The methodology proceeds as follows:

1. Particle System Construction:

   • Consider $N$ particles moving as independent Brownian motions.
   • Particles branch at a time-dependent rate $f(t)$.
   • Simultaneously with branching, a particle is deleted. The deletion is "soft-killed" based on a selection weighting function $\omega$. Specifically, at branching times, a particle at location $x$ is deleted with probability proportional to $\omega(x - b_N(t))$, where $b_N(t)$ is a moving boundary determined by the empirical distribution of the particles.
   • The boundary $b_N(t)$ is defined implicitly to ensure the total branching rate matches the prescribed rate $Nf(t)$:
     $$b_N(t) := \inf \left\{ x : \Lambda \langle \mu_N^t, \omega(\cdot - x) \rangle \ge f(t) \right\}$$
     where $\mu_N^t$ is the empirical measure of the $N$ particles.

2. Hydrodynamic Limit Analysis:

   • The paper analyzes the limit of the empirical measure $\mu_N^t$ as $N \to \infty$.
   • Using martingale methods (specifically Ito's formula and the Burkholder-Davis-Gundy inequality), the author derives the evolution equation for the test functions of the empirical measure.
   • Tightness of the sequence of laws is established in the space of measure-valued processes equipped with the Skorokhod topology.
   • The limit is shown to satisfy a free boundary problem (FBP) governed by a partial differential equation (PDE).

Key Results

1. Convergence to Free Boundary Problem (Theorem 1):
   The empirical measure $\mu_N^t$ of the constructed $(\omega, f, \Lambda, N)$-BBM converges weakly to a deterministic measure $\mu_t$ with density $u(x, t)$. This density is the unique weak solution to the following free boundary problem:
   $$\begin{cases} u_t = \frac{1}{2}u_{xx} + f(t)u - \Lambda \omega(x - b(t))u, & x \in \mathbb{R}, t > 0 \\ \int_{\mathbb{R}} u(y, t) dy = 1, & t > 0 \\ u(x, 0) = \rho(x), & x \in \mathbb{R} \end{cases}$$
   where the free boundary $b(t)$ is determined by the integral constraint $\int_{\mathbb{R}} \omega(x - b(t)) u(x, t) dx = f(t)/\Lambda$.

2. Connection to SKIFPP (Theorem 2):
   The paper demonstrates that if $(u, b)$ solves the above free boundary problem, then the boundary $b(t)$ is the unique solution to the SKIFPP. Specifically, if $\tau$ is defined by the stopping time involving $\omega$ and $b$, then the survival probability satisfies:
   $$P(\tau > t) = \exp\left( -\int_0^t f(s) ds \right)$$
   This result relies on existing uniqueness results for the SKIFPP (cited from [8] and [9]) and a scaling argument to map the PDE solution to the probabilistic stopping time.

3. Uniqueness and Regularity:
   The paper proves the uniqueness of the weak solution to the FBP for continuous $\omega$ (Proposition 9) and relies on prior literature for the case where $\omega(x) = \mathbb{1}_{x \le 0}$. It also establishes that the limiting measure has a density and no mass at infinity almost surely (Proposition 5).

Significance and Claims
The paper claims to provide a constructive probabilistic approximation for the generalized inverse first passage problem. By linking the SKIFPP to the hydrodynamic limit of a specific branching-selection particle system, the work offers a numerical and theoretical framework for solving these problems.

The author notes that this generalizes previous work by Klump [13] and De Masi et al. [4], which connected the classical IFPP (with exponential distributions) to the standard $N$-BBM. This paper extends that connection to the "soft-killed" setting with general monotonic weighting functions $\omega$ and time-dependent branching rates $f(t)$, thereby covering a broader class of credit default models where default is not instantaneous but accumulates over time. The significance lies in identifying the specific particle system dynamics (branching rate $f(t)$ and selection weight $\omega$) required to approximate the solution $b(t)$ for a given target distribution.