[Paper Review] A Stochastic Target Problem for Branching Diffusions

Idris Kharroubi, Antonio Ocello|arXiv (Cornell University)|Jun 27, 2022

Stochastic processes and financial applications25 references4 citations

TL;DR

This paper formulates and solves a stochastic target problem for branching diffusion processes, where the goal is to determine the minimal initial condition ensuring all particles in a controlled branching system reach a terminal target set with probability one. Using a dynamic programming principle and a novel branching property that expresses the value function as a pointwise maximum over particle labels, the authors establish that the value function is the unique viscosity solution to a Hamilton-Jacobi-Bellman variational inequality, providing a theoretical foundation for super-replication in crypto-asset derivatives markets.

ABSTRACT

We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition for which a control allows the underlying branching process to reach a target set at a finite terminal time for each of its branches. This problem is motivated by an example from fintech where we look for the super-replication price of options on blockchain based cryptocurrencies. We first state a dynamic programming principle for the value function of the stochastic target problem. We then show that the value function can be reduced to a new function with a finite dimensional argument by a so called branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational inequality.

Motivation & Objective

• To model and solve an optimal stochastic target problem for controlled branching diffusion processes.
• To characterize the minimal initial condition ensuring all particles in the branching system reach a target set at terminal time with probability one.
• To develop a dynamic programming framework applicable to branching processes with non-Markovian dependencies across particles.
• To establish a viscosity solution characterization for the value function under a novel branching property involving pointwise maxima rather than products.
• To provide a theoretical solution to the super-replication problem for options on blockchain-based cryptocurrencies, where branching dynamics emerge from protocol-level mechanisms.

Proposed method

• Derives a dynamic programming principle (DPP) for the value function using measurable selection theorems and conditioning arguments on the law of the controlled process.
• Introduces a new branching property that expresses the value function at a given initial configuration as the pointwise maximum over the optimal values at each particle's starting position.
• Reduces the infinite-dimensional value function to a finite-dimensional object indexed by particle labels, enabling analysis via viscosity solutions.
• Establishes that the value function is the unique viscosity solution to a Hamilton-Jacobi-Bellman (HJB) variational inequality with two components: a second-order nonlinear operator and a monotonicity constraint across labels.
• Uses a probabilistic approach based on controlled martingale problems and regular conditional probability distributions, avoiding reliance on smooth approximations of the value function.
• Applies results from stochastic control and PDE theory to characterize the value function as a solution to a PDE with non-standard structure, lacking polynomial terms typical in classical branching PDEs.

Experimental results

Research questions

• RQ1 What is the minimal initial condition required for a controlled branching diffusion to reach a target set at terminal time with probability one across all particles?
• RQ2 How can the dynamic programming principle be established for stochastic target problems in branching diffusion processes with dependent particle dynamics?
• RQ3 What is the correct generalization of the classical branching property when the value function depends on the maximum over particle labels rather than a product?
• RQ4 How can the value function be characterized as a viscosity solution to a PDE when the standard smoothness assumptions fail due to irregularity from the maximum operation?
• RQ5 Can this framework be applied to solve the super-replication problem for options on blockchain-based cryptocurrencies, where asset creation mimics branching processes?

Key findings

• The value function is shown to satisfy a dynamic programming principle based on probabilistic conditioning and measurable selection, without requiring smooth approximations.
• A novel branching property is established, expressing the value function as the pointwise maximum over the value functions at each particle's initial position, rather than a product as in classical branching processes.
• The value function is proven to be the unique viscosity solution to a Hamilton-Jacobi-Bellman variational inequality with two components: a second-order nonlinear operator and a label-wise monotonicity constraint.
• The viscosity solution framework is adapted to handle the label-dependent structure and the resulting irregularity from the maximum operation.
• The approach provides a rigorous mathematical foundation for super-replication pricing in fintech applications involving crypto-asset derivatives with branching dynamics.
• The method avoids reliance on regular solutions to approximating PDEs, relying instead on probabilistic arguments and conditional expectation techniques.