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Conditional Expectation expression in mean-field SDEs and its applications

Samaneh Sojudi, Mahdieh Tahmasebi

TL;DR

The paper develops a novel conditional-expectation representation for mean-field SDEs with jumps by leveraging Malliavin calculus on Poisson space, enabling the expression of conditional expectations as weighted ordinary expectations. Central contributions include (i) a rigorous Malliavin-weight framework with weights that remain well-defined in the jump-mean-field setting, (ii) localization-based variance reduction to improve Monte Carlo efficiency, and (iii) a practical Malliavin Monte Carlo scheme for pricing American options under high-dimensional jump-diffusion mean-field dynamics, illustrated through numerical experiments with single- and multi-asset examples and Kou-type jumps. The approach yields substantial accuracy improvements over traditional techniques and provides a scalable pathway for pricing and hedging in complex mean-field environments. Collectively, the work advances stochastic analysis and computational finance by enabling efficient conditional-estimation in distribution-dependent jump models and by delivering actionable tools for American option valuation in high dimensions.

Abstract

This study developed a novel formulation of conditional expectations within the framework of a jump-diffusion mean-field stochastic differential equation. We introduce an integrated approach that combines unconditioned expectations with rigorously defined weighting factors, employing Malliavin calculus on Poisson space and directional derivatives to enhance estimation accuracy. \noindent The proposed method is applied to the numerical pricing of American put options in a jump-diffusion mean-field setting, addressing the challenges proposed by early-exercise features. Comprehensive numerical experiments demonstrate substantial improvements in pricing accuracy compared with conventional techniques.

Conditional Expectation expression in mean-field SDEs and its applications

TL;DR

The paper develops a novel conditional-expectation representation for mean-field SDEs with jumps by leveraging Malliavin calculus on Poisson space, enabling the expression of conditional expectations as weighted ordinary expectations. Central contributions include (i) a rigorous Malliavin-weight framework with weights that remain well-defined in the jump-mean-field setting, (ii) localization-based variance reduction to improve Monte Carlo efficiency, and (iii) a practical Malliavin Monte Carlo scheme for pricing American options under high-dimensional jump-diffusion mean-field dynamics, illustrated through numerical experiments with single- and multi-asset examples and Kou-type jumps. The approach yields substantial accuracy improvements over traditional techniques and provides a scalable pathway for pricing and hedging in complex mean-field environments. Collectively, the work advances stochastic analysis and computational finance by enabling efficient conditional-estimation in distribution-dependent jump models and by delivering actionable tools for American option valuation in high dimensions.

Abstract

This study developed a novel formulation of conditional expectations within the framework of a jump-diffusion mean-field stochastic differential equation. We introduce an integrated approach that combines unconditioned expectations with rigorously defined weighting factors, employing Malliavin calculus on Poisson space and directional derivatives to enhance estimation accuracy. \noindent The proposed method is applied to the numerical pricing of American put options in a jump-diffusion mean-field setting, addressing the challenges proposed by early-exercise features. Comprehensive numerical experiments demonstrate substantial improvements in pricing accuracy compared with conventional techniques.
Paper Structure (15 sections, 11 theorems, 104 equations, 12 figures, 3 tables)

This paper contains 15 sections, 11 theorems, 104 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mean-field SDEs with jumps
  3. Conditional expectation expression in mean-field SDEs
  4. Conditional expectation
  5. Mean-field SDEs with jump model
  6. Variance reduction
  7. Localisation
  8. Practical Variance Reduction for Mean-Field Models with Jump Components
  9. Implementations and simulations
  10. Numerical Expriments
  11. (i) The gradient term
  12. (ii) The weighted term
  13. Malliavin Derivative Operator
  14. Proof of Theorem \ref{['th2']}
  15. Algorithm of Pricing American Options

Key Result

Theorem 3

For $p > 1$ and $\vartheta = (h, v) \in \mathcal{H}^\infty \times \mathcal{V}^\infty$, the process $D_{\vartheta} X_t \in W^{1, \vartheta, p}$, defined in the Appendix A.

Figures (12)

  • Figure 1: On top parameters set to $\lambda = 10, T = 1, N = 2^{12}$ Example \ref{['E1']}.
  • Figure 2: On top parameters set to $\lambda = 10, T = 1, N = 2^{12}$ Example \ref{['E1']}.
  • Figure 3: On top parameters set to $\lambda = 10, T = 1, N = 2^{12}$ Example \ref{['E1']}.
  • Figure 4: Overview of Errors in Example \ref{['E1']}.
  • Figure 5: On top parameters set to $\lambda = 10, T = 1, N = 2^{12}$ Example \ref{['E2']}.
  • ...and 7 more figures

Theorems & Definitions (21)

  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • ...and 11 more