Moment equations and Hermite expansion for nonlinear stochastic differential equations with application to stock price models

Abstract

Exact moment equations for nonlinear Itô processes are derived. Taylor expansion of the drift and diffusion coefficients around the first conditional moment gives a hierarchy of coupled moment equations which can be closed by truncation or a Gaussian assumption. The state transition density is expanded into a Hermite orthogonal series with leading Gaussian term and the Fourier coefficients are expressed in terms of the moments. The resulting approximate likelihood is maximized by using a quasi Newton algorithm with BFGS secant updates. A simulation study for the CEV stock price model compares the several approximate likelihood estimators with the Euler approximation and the exact ML estimator (Feller, in Ann Math 54: 173–182, 1951).

Appendix: derivation of the moment equations

The conditional density p(yt|xs) fulfils the Fokker-Planck equation

$$\begin{aligned} \frac{\partial p(y, t | x, s)}{\partial t}=&-\sum_{i} \frac{\partial}{\partial y_{i}}\left[f_{i}(y, t) p(y, t | x, s)\right] \\ &+\frac{1}{2} \sum_{i j} \frac{\partial^{2}}{\partial y_{i} \partial y_{j}}\left[\Omega_{i j}(y, t) p(y, t | x, s)\right] \\ :=& F(y, l) p(y, t | x, s) \end{aligned}$$

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where F is the Fokker-Planck operator. Thus the first conditional moment μ(tt_i) = E[Y(t)∣Yⁱ] fulfils

$$\begin{aligned} \dot{\mu}\left(t | t_{i}\right) &=(\partial / \partial t) \int y p\left(y, t | Y^{i}, t_{i}\right) \mathrm{d} y=\int y F p\left(y, t | Y^{i}, t_{i}\right) \mathrm{d} y \\ &=\int(L y) p\left(y, t | Y^{i}, t_{i}\right) \mathrm{d} y=E\left[(L y)(Y(t)) | Y^{i}\right] \end{aligned}$$

where $L=\sum\nolimits_{j} f_{j}(y, t) \frac{\partial}{\partial y_{j}}+\frac{1}{2} \sum\nolimits_{j k} \Omega_{j k}(y, t) \frac{\partial^{2}}{\partial y_{j} \partial y_{k}}$the backward operator. Thus we obtain

$$\dot{\mu}\left(t | t_{i}\right)=\int f(y, t) p\left(y, t | Y^{i}, t_{i}\right) \mathrm{d} y=E\left[f(Y, t) | Y^{i}\right].$$

Higher order moments m_k:= E[M_k]:= E[(Y − μ)^k] fulfil the equations (scalar notation, condition suppressed)

$$\begin{aligned} \dot{m}_{k} &=(\partial / \partial t) \int(y-\mu)^{k} p(y, t) \mathrm{d} y \\ &=-\int k(y-\mu)^{k-1} \dot{\mu} p(y, t) \mathrm{d} y+\int\left(L(y-\mu)^{k}\right) p(y, t) \mathrm{d} y\\ &=-k E\left[(Y-\mu)^{k-1}\right] E[f(Y)]+k E\left[f(Y) *(Y-\mu)^{k-1}\right] \\ &+\frac{1}{2} k(k-1) E\left[\Omega(Y)(Y-\mu)^{k-2}\right] \\ & = k E\left[f(Y) *\left(M_{k-1}-m_{k-1}\right)\right]+\frac{1}{2} k(k-1) E\left[\Omega(Y) * M_{k-2}\right] \end{aligned}$$