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Title:           Parameter estimation in stochastic differential equations via Wiener chaos expansion and stochastic gradient descent
Subject:         Applied Mathematical Modelling, 163 (2027) 117236. doi:10.1016/j.apm.2026.117236
Keywords:        Stochastic gradient descent,Parameter estimation,Wiener chaos expansion,Geometric Brownian motion,Microbial growth dynamics
Author:          Francisco Delgado-Vences
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     Item: 
      Title: Parameter estimation in stochastic differential equations via Wiener chaos expansion and stochastic gradient descent
      Children: 
       Item: 
        Title: 1 Introduction
        Children: 
         Item: 
          Title: 1.1 Deterministic gradient descent
         Item: 
          Title: 1.2 Stochastic gradient descent
         Item: 
          Title: 1.3 Comparison with variational data assimilation methods
         Item: 
          Title: 1.4 Overview of the proposed methodology
       Item: 
        Title: 2 On inverse problems
        Children: 
         Item: 
          Title: 2.1 Inverse problems and regularization
         Item: 
          Title: 2.2 Some statistical ILL-posed inverse problems
       Item: 
        Title: 3 Differential operators with a noise
        Children: 
         Item: 
          Title: 3.1 Framework
         Item: 
          Title: 3.2 Stochastic inverse problem
         Item: 
          Title: 3.3 Well-posedness of the minimization problem
       Item: 
        Title: 4 Classical examples of differential operators
        Children: 
         Item: 
          Title: 4.1 Stochastic differential equations
          Children: 
           Item: 
            Title: 4.1.1 Wiener Chaos expansion for the SDE
           Item: 
            Title: 4.1.2 Inverse problem for the SDE
       Item: 
        Title: 5 Learning parameters for SDEs
        Children: 
         Item: 
          Title: 5.1 Stochastic differential equations
         Item: 
          Title: 5.2 Wiener Chaos expansion for the SDE
         Item: 
          Title: 5.3 Learning parameters using the WCE
          Children: 
           Item: 
            Title: 5.3.1 Convergence of stochastic gradient
       Item: 
        Title: 6 Numerical experiments and applications
        Children: 
         Item: 
          Title: 6.1 The Ornstein-Uhlenbeck process
          Children: 
           Item: 
            Title: 6.1.1 SGD and numerical validation for the OU process
         Item: 
          Title: 6.2 Geometric Brownian motion
         Item: 
          Title: 6.3 Stochastic logistic equation
         Item: 
          Title: 6.4 Numerical validation: GBM and logistic SDE
       Item: 
        Title: 7 Application to empirical data
        Children: 
         Item: 
          Title: 7.1 Empirical validation: S&P 500 index
         Item: 
          Title: 7.2 Empirical validation: microbial growth dynamics
          Children: 
           Item: 
            Title: 7.2.1 Data acquisition and scientific relevance
           Item: 
            Title: 7.2.2 Parameter estimation via WCE-SGD
       Item: 
        Title: 8 Concluding remarks
       Item: 
        Title: CRediT authorship contribution statement
       Item: 
        Title: Declaration of competing interest
       Item: 
        Title: Acknowledgements
       Item: 
        Title: Software and codes
       Item: 
        Title: Data availability
       Item: 
        Title: References
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    Author: Francisco Delgado-Vences
    Subject: Applied Mathematical Modelling, 163 (2027) 117236. doi:10.1016/j.apm.2026.117236
    Keywords: Stochastic gradient descent,Parameter estimation,Wiener chaos expansion,Geometric Brownian motion,Microbial growth dynamics
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