Maxim A. Alshanskiy     (Ural Federal University, Ekaterinburg, Russian Federation)


Under consideration is construction of a model of age-structured population reflecting random oscillations of the death and birth rate functions. We arrive at an Itô-type difference equation in a Hilbert space of functions which can not be transformed into a proper Itô equation via passing to the limit procedure due to the properties of the operator coefficients. We suggest overcoming the obstacle by building the model in a space of Hilbert space valued generalized random variables where it has the form of an operator-differential equation with multiplicative noise. The result on existence and uniqueness of the solution to the obtained equation is stated.


Brownian sheet, Cylindrical Wiener process, Gaussian white noise, Stochastic differential equation, Age-structured population model

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  1. Nasyrov F.S. On the derivative of local time for the Brownian sheet with respect to a space variable. Theory Probab. Appl., 1987. Vol. 32, no. 4. P. 649–658. DOI: 10.1137/1132097
  2. Alshanskiy M.A., Melnikova I.V. Regularized and generalized solutions of infinite-dimensional stochastic problems. Sbornik Mathematics, 2012. Vol. 202, no. 11. P. 1565–1592. DOI: 10.1070/SM2011v202n11ABEH004199
  3. Alshanskiy M.A. The Itô integral and the Hitsuda-Skorohod integral in the infinite dimensional case. Sib. Elektron. Mat. Izv., 2014. Vol. 11, no. 1. P. 185–199.
  4. Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov [Theory of Stochastic Processes]. Moscow: Fizmatlit Publ., 2005. 400 p. (in Russian)
  5. Da Prato G., Zabczyk J. Stochastic equations in infinite dimensions (2nd edition). Cambridge: Cambridge Univ. Press, 2014. 493 p. DOI: 10.1017/CBO9781107295513
  6. Gawarecki L., Mandrekar V. Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations. Berlin, Heidelberg: Springer-Verlag, 2011. 291 p. DOI: 10.1007/978-3-642-16194-0
  7. Ma W., Ding B., Zhang Q. The existence and asymptotic behaviour of energy solutions to stochastic age-dependent population equations driven by Levy processes. Appl. Math. Comput., 2015. No. 256, P. 656–665. DOI: 10.1016/j.amc.2015.01.077
  8. Melnikova I.V., Alshanskiy M.A. The generalized well-posedness of the Cauchy problem for an abstract stochastic equation with multiplicative noise. Proc. Steklov Inst. Math., 2013. Vol. 280 (Suppl. 1), P. 134–150. DOI: 10.1134/S0081543813020119
  9. Melnikova I.V., Alshanskiy M.A. Stochastic equations with an unbounded operator coefficient and multiplicative noise. Sib. Math. Journ., 2017. Vol. 58, no .6. P. 1052–1066. DOI: 10.1134/S0037446617060143
  10. Qi-Min Z., Wen-An L., Zan-Kan N. Existence, uniqueness and exponential stability for stochastic age-dependent population. Appl. Math. Comput., 2004. Vol. 154, no. 1. P. 183–201. DOI: 10.1016/S0096-3003(03)00702-1

DOI: http://dx.doi.org/10.15826/umj.2018.1.001

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