[Paper Review] BSVIEs with stochastic Lipschitz coefficients and applications in finance

Tianxiao Wang|arXiv (Cornell University)|Jan 20, 2010

Stochastic processes and financial applications13 references3 citations

TL;DR

This paper establishes the existence and uniqueness of M-solutions for backward stochastic Volterra integral equations (BSVIEs) with stochastic Lipschitz coefficients, extending prior results that required deterministic interest rates. It applies this framework to construct continuous-time dynamic coherent risk measures allowing for random interest rates, thereby enabling time-inconsistent preference modeling in finance with improved generality and a more concise proof than earlier work.

ABSTRACT

This paper is concerned with existence and uniqueness of M-solutions of backward stochastic Volterra integral equations (BSVIEs for short), which Lipschitz coefficients are allowed to be random, which generalize the results in [15]. Then a class of continuous time dynamic dynamic coherent risk measures is derived, allowing the riskless interest rate to be random, which is different from the case in [15].

Motivation & Objective

• To extend the theory of backward stochastic Volterra integral equations (BSVIEs) by allowing Lipschitz coefficients to be random, generalizing prior results that required deterministic coefficients.
• To establish a new class of continuous-time dynamic coherent risk measures that accommodate random interest rates, overcoming a key limitation in earlier models.
• To provide a more concise proof of unique solvability for M-solutions under stochastic Lipschitz conditions compared to existing literature.
• To demonstrate that the risk measure construction remains valid under time-inconsistent preferences by leveraging the generalized BSVIE framework.

Proposed method

• Introduces a stochastic Lipschitz condition on the generator of BSVIEs, where the Lipschitz constant is allowed to be random and adapted to the filtration.
• Defines and analyzes M-solutions for BSVIEs, ensuring the solution process is adapted and satisfies a specific integral representation involving stochastic integrals with respect to Brownian motion.
• Applies a fixed-point argument in a suitable Hilbert space framework to prove existence and uniqueness of the M-solution under the stochastic Lipschitz condition.
• Derives a dynamic coherent risk measure via the M-solution of the BSVIE, incorporating a random interest rate function and satisfying translation invariance with respect to stochastic discounting.
• Uses the structure of the BSVIE to model risk measures that are time-consistent in a generalized sense, even when preferences are time-inconsistent.
• Analyzes special cases where the Lipschitz function is deterministic or path-dependent (e.g., involving Wiener process), showing when the risk measure depends on the initial wealth and when it does not.

Experimental results

Research questions

• RQ1 Can the existence and uniqueness of M-solutions for BSVIEs be established when the Lipschitz coefficient is random rather than deterministic?
• RQ2 How can dynamic coherent risk measures be constructed in continuous time when the risk-free interest rate is stochastic?
• RQ3 Does the proposed framework preserve the key properties of coherent risk measures—translation invariance, positive homogeneity, and subadditivity—under stochastic coefficients?
• RQ4 In what cases does the risk measure depend on the initial wealth, and when is it independent of the underlying wealth process?
• RQ5 Can the proof of unique solvability be significantly shortened compared to prior approaches, such as in Yong (2007)?

Key findings

• The paper proves the existence and uniqueness of M-solutions for BSVIEs under a stochastic Lipschitz condition, where the Lipschitz constant is a random, adapted process.
• The proof of unique solvability is significantly more concise than the one in Yong (2007), offering a streamlined analytical approach.
• A new class of continuous-time dynamic coherent risk measures is derived that allows the risk-free interest rate to be random, thus generalizing earlier models that required deterministic rates.
• When the Lipschitz function is deterministic, the risk measure reduces to a known form, and the solution is deterministic, recovering results from prior work.
• When the Lipschitz function depends on the Wiener process (e.g., sin(W(s))), the risk measure becomes path-dependent and generally depends on the initial wealth, indicating that the risk exposure is not independent of the portfolio value.
• The analysis shows that for general nonlinear generators (e.g., quadratic in Z), the risk measure outcome depends on the initial wealth process, implying that the risk assessment is not additive across portfolios in a simple way.