2024 Vol.7 Apr.N02

• Title: Parameter estimation for stochastic heat equation
• Name: hongsheng qi
• Company: Department of Mathematics, College of Science, Bengbu University 1866 Caoshan Rd., Bengbu 233030, China
• Abstract:

  Stochastic thermal equation is a kind of partial differential equation describing the heat conduction process of physical system, and the accurate estimation of its parameters is the key to many engineering and scientific applications. However, due to the randomness and complexity of the equation, parameter estimation faces many challenges. Therefore, this paper analyzes the parameter estimation of stochastic heat equation. Gaussian white noise is transformed into discrete observation form by mathematical model, and quasi-likelihood estimation is constructed based on observation data. According to the consistency and asymptotic normality of the estimation, the estimation gradually approaches the real parameters, and its distribution will approach the normal distribution. The parameter estimation of stochastic thermal equation provides an effective method to deal with spatial Gaussian white noise, which is helpful to solve practical problems in related fields.

  
  

• Keyword: Gaussian white noise; Stochastic heat equation; Discrete space observations; Quasi-likelihood estimators; Asymptotic normality
• DOI: 10.12250/jpciams2024090309
• Citation form: hongsheng qi .Parameter estimation for stochastic heat equation [J]. Computer Informatization and Mechanical System,2024,Vol.7,pp.37-41

[1] R.M. Dudley, Sample functions of the Gaussian process, Ann. Probab., 1(1973), 66-103.

[2] R.M. Dudley, Uniform Central Limit Theorems, Cambridge University Press, 1999.

[3] D. Nualart, The Malliavin Calculus and Related Topics. Probability and Its Applications (New York), second ed., Springer-Verlag, Berlin, 2006.

[4] D. Xia and L. Yan, Some properties of the solution to fractional heat equation with a fractional  Brownian noise. Advances in Difference Equations, 4 (2017) .

[5] X. Sun, L. Yan, X. Yu, Quadratic covariations for the solution to a stochastic heat equation with time-space white noise. Advances in Difference Equations, 4 (2020).

[6] J. Liu, L. Yan, Solving a nonlinear fractional stochastic partial differential equation with fractional noise. J. Theor.Probab. 4 29(2016), 307-347.

[7] Bourguin S, Tudor C A., On the law of the solution to a stochastic heat equation with fractional noise in time[J]. Random Operators and Stochastic Equations, 23(2015), 179-186.

[8] Ouahhabi H, Tudor C A., Additive Functionals of the Solution to Fractional Stochastic Heat Equation[J].Journal of Fourier Analysis Applications, 19(2013), 777-791.

[9] Balan, R.M., Tudor, C.A., The stochastic heat equation with fractional-colored noise: existence of the solution. ALEA Lat. Am. J. Probab. Math. Stat., 4 (2008), 57-87.

[10] J. Posp´ıˇsil, R. Tribe, Parameter estimates and exact variations for stochastic heat equations driven by spaceCtime white noise, Stoch. Anal. Appl. 25 (3) (2007) 593C611.

[11] Cialenco I, Kim H J, Parameter estimation for discretely sampled stochastic heat equation driven by space-only noise[J].Stochastic Processes and their Applications, 2022, 143.

[12] R. Altmeyer, I. Cialenco, G. Pasemann, Parameter estimation for semilinear SPDEs from local measurements, 2020, Preprint, arXiv:2004.14728.

[13] S.J. Taylor, Exact asymptotic estimates of Brownian path variation, Duke Math. J., 39(1972), 219-241.

[14] I. Nourdin, G. Peccati, Normal Approximations with Malliavin Calculus, from Steins Method To Universality, in: Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, 2012.