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<p>In the present paper, by introducing a family of coupled forward-backward stochastic differential equations (FBSDEs for short), a probabilistic interpretation for a system consisting of <i>m</i> second order quasilinear (and possibly degenerate) parabolic partial differential equations and (<i>m</i> × <i>d</i>) algebraic equations is given in the sense of the classical solution. For solving the problem, an <i>L</i><sup><i>p</i></sup>-estimate (<i>p</i> &gt; 2) for coupled FBSDEs on large time durations in the monotonicity framework is established, and a new method to analyze the regularity of solutions to FBSDEs is introduced. The new method avoids the use of Kolmogorov’s continuity theorem and only employs <i>L</i><sup>2</sup>-estimates and <i>L</i><sup>4</sup>-estimates to obtain the desired regularity.</p>

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Probabilistic Interpretation for a System of Quasilinear Parabolic Partial Differential-Algebraic Equations: The Classical Solution

  • 期刊论文

  • 发表日期：

    2025年11月19日

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  • Zhen Wu,
  • Bing Xie,
  • Zhiyong Yu

摘要

In the present paper, by introducing a family of coupled forward-backward stochastic differential equations (FBSDEs for short), a probabilistic interpretation for a system consisting of m second order quasilinear (and possibly degenerate) parabolic partial differential equations and (m × d) algebraic equations is given in the sense of the classical solution. For solving the problem, an Lp-estimate (p > 2) for coupled FBSDEs on large time durations in the monotonicity framework is established, and a new method to analyze the regularity of solutions to FBSDEs is introduced. The new method avoids the use of Kolmogorov’s continuity theorem and only employs L2-estimates and L4-estimates to obtain the desired regularity.