This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.
GuidoumA.C., & BoukhetalaK. (2020). Performing parallel monte carlo and moment equations methods for itô and stratonovich stochastic differential systems: R package Sim.DiffProc. Journal of Statistical Software, 96(2), 1-82. doi: 10.18637/jss.v096.i02.
5.
HendersonD., & PlaschkoP. (2006). Stochastic differential equations in science and engineering. World Scientific. doi: 10.1142/5806.
6.
KlebanerF.C. (2005). Introduction to stochastic calculus with applications (2nd ed.). Imperial College Press. doi: 10.1142/p386
MertonR.C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141-183. doi: 10.2307/3003143.
9.
ØksendalB. (2003). Stochastic differential equations: An introduction with applications (6th ed.). Springer Berlin, Heidelberg. doi: 10.1007/978-3-642-14394-6.
10.
OlverF.W.LozierD.W.BoisvertR.F., & ClarkC.W. (2010). Nist handbook of mathematical functions (1st ed.). New York, United States: Cambridge University Press.
11.
R Core Team. (2023). R: A language and environment for statistical computing [Computer software manual]. Vienna, Austria. Retrieved from https://www.R-project.org/.
12.
RiskenH. (1996). The fokker-planck equation: Methods of solution and applications (2nd ed., Vol. 18). Springer-Verlag. doi: 10.1007/978-3-642-61544-3.
13.
VasicekO. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177-188. doi: 10.1016/0304-405X(77)90016-2.
14.
WillinkR. (2005). Normal moments and hermite polynomials. Statistics & Probability Letters, 73(3), 271-275. doi: 10.1016/j.spl.2005.03.015.
