Paper: arXiv 2601.01250

Authors: Miryana Grigorova, James Wheeldon

Abstract

We study non-linear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and p default martingales. The driver of the BSDE with multiple default jumps can take a generalized form involving an optional finite variation process. We first show existence and uniqueness. We then establish comparison and strict comparison results for these BSDEs, under a suitable assumption on the driver. In the case of a linear driver, we derive an explicit formula for the first component of the BSDE using an adjoint exponential semimartingale. The representation depends on whether the finite variation process is predictable or only optional. We apply our results to the problem of pricing and hedging a European option in a linear complete market with two defaultable assets and in a non-linear complete market with p defaultable assets. Two examples of the latter market model are provided: an example where the seller of the option is a large investor influencing the probability of default of a single asset and an example where the large seller’s strategy affects the default probabilities of all p assets.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of non-linear BSDEs driven by Brownian motion and multiple default jumps with generalized optional driver processes. While highly technical with complete mathematical derivations, it is strictly theoretical and entirely lacks empirical testing, backtesting, or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Pricing & Hedging European Options<br/>in Markets with Multiple Defaults] --> B[Inputs: Brownian Motion + p Default Martingales<br/>+ Optional Finite Variation Process]
    B --> C[Formulate Generalized Non-linear BSDEs<br/>with Jump Components]
    C --> D[Mathematical Analysis:<br/>1. Prove Existence & Uniqueness<br/>2. Establish Comparison Theorems<br/>3. Derive Explicit Formulas for Linear Drivers]
    D --> E[Financial Application:<br/>Model Complete Markets with 2 to p Defaultable Assets<br/>including Large Investor Effects]
    E --> F[Outcomes: Explicit Pricing/Hedging Formulas<br/>& Impact Analysis of Large Investor Strategies<br/>on Default Probabilities]

Cited References (21)

  1. Enlargement of Filtrations: An Exposition of Core Ideas with Financial Examples — → 2303.03573
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  3. European Options in a Nonlinear Incomplete Market Model with Default (2020) — link
  4. Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs (2017) — link
  5. BSDEs with default jump (2016) — link
  6. Existence and uniqueness results for BSDEs with jumps: the whole nine yards (2016) — link
  7. On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples (2016) — link
  8. BILATERAL COUNTERPARTY RISK UNDER FUNDING CONSTRAINTS—PART I: PRICING (2015) — link
and 13 more
  1. Generalized Dynkin games and doubly reflected BSDEs with jumps (2013) — link
  2. BSDEs with jumps, optimization and applications to dynamic risk measures (2013) — link
  3. Mathematical Methods for Financial Markets (2009) — link
  4. Backward stochastic differential equations with jumps and related non-linear expectations (2006) — link
  5. PDE approach to valuation and hedging of credit derivatives (2005) — link
  6. Backward Stochastic Differential Equations in Finance (1997) — link
  7. Adapted solution of a backward stochastic differential equation (1990) — link
  8. Conjugate convex functions in optimal stochastic control (1973) — link
  9. Backward Stochastic Differential Equations (2011) — link
  10. Hedging of Defaultable Claims (2004) — link
  11. A Remark on default risk models (1999) — link
  12. Non-linear pricing theory and backward stochastic differential equations (1997) — link
  13. Limit Theorems for Stochastic Processes (1987) — link

1 of 21 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.