*3.3. Some Examples*

Let us finally consider some examples of processes *X* from (1) and present explicit expressions for the fundamental solutions <sup>Ψ</sup>*i*(*<sup>x</sup>*,*s*, *q*), *i* = 1, 2, of the homogeneous version of the second-order ordinary differential equation in (18).

**Example 1.** *Let μ*(*<sup>x</sup>*,*s*, *q*) = *β*(*<sup>s</sup>*, *q*) *and <sup>σ</sup>*(*<sup>x</sup>*,*s*, *q*) = *<sup>ν</sup>*(*<sup>s</sup>*, *q*)*, for all* (*<sup>x</sup>*,*s*, *q*) ∈ *E*<sup>3</sup> *and some continuously differentiable functions β*(*<sup>s</sup>*, *q*) *and <sup>ν</sup>*(*<sup>s</sup>*, *q*) > 0 *on* [−∞, <sup>∞</sup>]<sup>2</sup>*, so that the process X from (1) represents locally a Brownian motion with linear drift. In this case, we have* <sup>Ψ</sup>*i*(*<sup>x</sup>*,*s*, *q*) = *<sup>e</sup>γi*(*<sup>s</sup>*,*q*)*<sup>x</sup> with*

$$\gamma\_i(s,q) = -\frac{\beta(s,q)}{\nu^2(s,q)} - (-1)^i \sqrt{\frac{\beta^2(s,q)}{\nu^4(s,q)} + \frac{2(a+\lambda)}{\nu^2(s,q)}}\tag{55}$$

*for every i* = 1, 2*, so that <sup>γ</sup>*2(*<sup>s</sup>*, *q*) < 0 < *<sup>γ</sup>*1(*<sup>s</sup>*, *q*)*, for all q* ≤ *s.*

**Example 2.** *Let μ*(*<sup>x</sup>*,*s*, *q*) = *β*(*<sup>s</sup>*, *q*) − *<sup>δ</sup>*(*<sup>s</sup>*, *q*)*x and <sup>σ</sup>*(*<sup>x</sup>*,*s*, *q*) = *<sup>ν</sup>*(*<sup>s</sup>*, *q*)*, for all* (*<sup>x</sup>*,*s*, *q*) ∈ *E*<sup>3</sup> *and some continuously differentiable functions β*(*<sup>s</sup>*, *q*)*, <sup>δ</sup>*(*<sup>s</sup>*, *q*) = 0*, and <sup>ν</sup>*(*<sup>s</sup>*, *q*) > 0 *on* [−∞, <sup>∞</sup>]<sup>2</sup>*, so that the process X from (1) represents locally a mean-reverting Ornstein-Uhlenbeck process. In this case, we have*

$$\Psi\_1(\mathbf{x}, \mathbf{s}, q) = M \left( \frac{\mathfrak{a} + \lambda}{2\delta(\mathbf{s}, q)}, \frac{1}{2}, \frac{(\beta(\mathbf{s}, q) - \delta(\mathbf{s}, q)\mathbf{x})^2}{\delta(\mathbf{s}, q)\nu^2(\mathbf{s}, q)} \right) \tag{56}$$

*and*

$$\Psi\_2(\mathbf{x}, \mathbf{s}, q) = \mathcal{U}\left(\frac{\mathbf{a} + \lambda}{2\delta(\mathbf{s}, q)}, \frac{1}{2}, \frac{(\beta(\mathbf{s}, q) - \delta(\mathbf{s}, q)\mathbf{x})^2}{\delta(\mathbf{s}, q)\nu^2(\mathbf{s}, q)}\right) \tag{57}$$

*where we denote by*

$$M(\boldsymbol{\varrho}, \boldsymbol{\psi}; \boldsymbol{z}) = 1 + \sum\_{k=1}^{\infty} \frac{(\boldsymbol{\varrho})\_k}{(\boldsymbol{\psi})\_k} \frac{z^k}{k!} \tag{58}$$

*and*

$$M(\boldsymbol{\varphi}, \boldsymbol{\psi}; \boldsymbol{z}) = \frac{\Gamma(1 - \boldsymbol{\psi})}{\Gamma(\boldsymbol{\varphi} + 1 - \boldsymbol{\psi})} M(\boldsymbol{\varphi}, \boldsymbol{\psi}; \boldsymbol{z}) + \frac{\Gamma(\boldsymbol{\psi} - 1)}{\Gamma(\boldsymbol{\varphi})} \boldsymbol{z}^{1 - \boldsymbol{\psi}} M(\boldsymbol{\varphi} + 1 - \boldsymbol{\psi}, 2 - \boldsymbol{\psi}; \boldsymbol{z}) \tag{59}$$

*Kummer's confluent hypergeometric functions of the first and second kind, respectively, for ψ* = 0, −1, −2, ...*,* (*ϕ*)*k* = *ϕ*(*ϕ* + <sup>1</sup>)···(*ϕ* + *k* − 1) *and* (*ψ*)*k* = *ψ*(*ψ* + <sup>1</sup>)···(*ψ* + *k* − <sup>1</sup>)*, k* ∈ N*. Note that the series in (58) converges under all z* > 0 *(see, e.g., Abramovitz and Stegun 1972, chp. XIII; Bateman and Erdély 1953, chp. VI), and* Γ *denotes Euler's gamma function. Note that the functions in (58) and (59) admit the integral representations*

$$M(\varphi, \psi; z) = \frac{\Gamma(\psi)}{\Gamma(\varphi)\Gamma(\psi - \varphi)} \int\_0^1 \epsilon^{zv} v^{\varphi - 1} (1 - v)^{\psi - \varphi - 1} \, dv,\tag{60}$$

*for ψ* > *ϕ* > 0 *and all z* ∈ R*, and*

$$\mathcal{U}(\varphi,\psi;z) = \frac{1}{\Gamma(\psi)} \int\_0^\infty e^{-zv} v^{\psi-1} (1+v)^{\psi-\varphi-1} \, dv,\tag{61}$$

*for ψ* > 0 *and all z* > 0*, respectively (see, e.g., Abramovitz and Stegun 1972, chp. XIII and Bateman and Erdély 1953, chp. VI).*

**Example 3.** *Let μ*(*<sup>x</sup>*,*s*, *q*)=(*β*(*<sup>s</sup>*, *q*) − *<sup>ν</sup>*<sup>2</sup>(*<sup>s</sup>*, *q*)/2)*e*<sup>−</sup>*<sup>x</sup>* − *<sup>δ</sup>*(*<sup>s</sup>*, *q*) *and <sup>σ</sup>*(*<sup>x</sup>*,*s*, *q*) = *<sup>ν</sup>*(*<sup>s</sup>*, *<sup>q</sup>*)*e*<sup>−</sup>*<sup>x</sup>*/2*, for all* (*<sup>x</sup>*,*s*, *q*) ∈ *E*<sup>3</sup> *and some continuously differentiable functions β*(*<sup>s</sup>*, *q*)*, <sup>δ</sup>*(*<sup>s</sup>*, *q*) = 0*, and <sup>ν</sup>*(*<sup>s</sup>*, *q*) > 0 *such that β*(*<sup>s</sup>*, *q*) ≥ *<sup>ν</sup>*<sup>2</sup>(*<sup>s</sup>*, *q*)/2 *on* [−∞, <sup>∞</sup>]<sup>2</sup>*, so that the process X from (1) represents locally the logarithm of a mean-reverting Feller square root diffusion process. In this case, we have*

$$\Psi\_1(\mathbf{x}, \mathbf{s}, q) = M \left( \frac{\mathfrak{a} + \lambda}{\delta(\mathbf{s}, q)}, \frac{2\beta(\mathbf{s}, q)}{\upsilon^2(\mathbf{s}, q)}, \frac{2\delta(\mathbf{s}, q)e^{\mathbf{x}}}{\upsilon^2(\mathbf{s}, q)} \right) \tag{62}$$

*and*

$$\Psi\_2(x,s,q) = \mathcal{U}(\frac{\mathfrak{a}+\lambda}{\delta(s,q)}, \frac{2\beta(s,q)}{\nu^2(s,q)}, \frac{2\delta(s,q)e^x}{\nu^2(s,q)})\tag{63}$$

*where the functions <sup>M</sup>*(*ϕ*, *ψ*; *z*) *and <sup>U</sup>*(*ϕ*, *ψ*; *z*) *are Kummer's confluent hypergeometric functions of the first and second kind given by (58) and (59) above, respectively.*
