@@ -159,29 +159,44 @@ For a description of argument and return types, see section
159159
160160### Probability density function
161161
162- If $\alpha \in \mathbb{R}^+$, $\tau \in \mathbb{R}^+$, $\beta \in [ 0, 1] $,
162+ If $\alpha \in \mathbb{R}^+$, $\tau \in \mathbb{R}^+$, $\beta \in ( 0, 1) $,
163163$\delta \in \mathbb{R}$, $s_ {\delta} \in \mathbb{R}^{\geq 0}$, $s_ {\beta} \in [ 0, 1)$, and
164164$s_ {\tau} \in \mathbb{R}^{\geq 0}$ then for $y > \tau$,
165165
166- \begin{eqnarray* }
167- \text{Wiener}(y|\alpha, \tau, \beta, \delta, s_ {\delta}, s_ {\beta}, s_ {\tau})
168-
169- &=& \frac{1}{s_ {\tau}}
170- \int_ {\tau}^{\tau + s_ {\tau}} \frac{1}{s_ {\beta}}\int_ {\beta -0.5s_ {\beta}}^{\beta + 0.5s_ {\beta}}
171- M \times p_3(y-\tau|a,\delta,\omega) \ d\omega \ d\tau,
172- \end{eqnarray* }
173- where $M$ and $p_3()$ are defined, by using $t:=y-\tau$, as
174- \begin{eqnarray* }
175- M := \frac{1}{\sqrt{1+s^2_ {\delta} t}}
176- \mathbb{e}^{a\delta\omega+\frac{\delta^2t}{2}+\frac{s^2_ {\delta} a^2
177- \omega^2-2a\delta\omega-\delta^2t}{2(1+s^2_ {\delta}t)}} \text{ and} \\ p_3(t|a,\delta,\beta) :=
178- \frac{1}{a^2} \mathbb{e}^{-a \delta \beta -\frac{\delta^2t}{2}} f(\frac{t}{a^2}|0,1,\beta), \end{eqnarray* }
179- where $f(t^* =\frac{t}{a^2}|0,1,\beta)$ has two forms
180- \begin{eqnarray* }
181- f_l(t^* |0,1,\beta) = \sum_ {k=1}^{\infty} k\pi \mathbb{e}^{-\frac{k^2\pi^2t^* }{2}}
182- \sin{(k \pi \beta)}\text{ and} \\
183- f_s(t^* |0,1,\beta) = \sum_ {k=-\infty}^{\infty} \frac{1}{\sqrt{2\pi (t^* )^3}}
184- (\beta+2k) \mathbb{e}^{-\frac{(\beta+2k)^2}{2t^* }}, \end{eqnarray* }
166+
167+ \begin{equation* }
168+ \begin{split}
169+ &\text{Wiener}(y\mid \alpha,\tau,\beta,\delta,s_ {\delta},s_ {\beta},s_ {\tau}) =
170+ \\
171+ &\frac{1}{s_ {\tau}}\int_ {\tau}^{\tau+s_ {\tau}}\frac{1}{s_ {\beta}}\int_ {\beta-\frac{1}{2}s_ {\beta}}^{\beta+\frac{1}{2}s_ {\beta}}\int_ {-\infty}^{\infty} p_3(y-{\tau_0}\mid \alpha,\nu,\omega)
172+ \\
173+ &\times \frac{1}{\sqrt{2\pi s_ {\delta}^2}}\exp\Bigl(-\frac{(\nu-\delta)^2}{2s_ {\delta}^2}\Bigr) \, d\nu \, d\omega \, d{\tau_0}=
174+ \\
175+ &\frac{1}{s_ {\tau}}\int_ {\tau}^{\tau+s_ {\tau}}\frac{1}{s_ {\beta}}\int_ {\beta-\frac{1}{2}s_ {\beta}}^{\beta+\frac{1}{2}s_ {\beta}} M\times p_3(y-{\tau_0}\mid \alpha,\nu,\omega) \, d\omega \, d{\tau_0},
176+ \end{split}
177+ \end{equation* }
178+
179+ where $p()$ denotes the density function, and $M$ and $p_3()$ are defined, by using $t:=y-{\tau_0}$, as
180+
181+ \begin{equation* }
182+ M \coloneqq \frac{1}{\sqrt{1+s_ {\delta}^2t}}\exp\Bigl(\alpha{\delta}\omega+\frac{\delta^2t}{2}+\frac{s_ {\delta}^2\alpha^2\omega^2-2\alpha{\delta}\omega-\delta^2t}{2(1+s_ {\delta}^2t)}\Bigr)\text{ and}
183+ \end{equation* }
184+
185+ \begin{equation* }
186+ p_3(t\mid \alpha,\delta,\beta) \coloneqq \frac{1}{\alpha^2}\exp\Bigl(-\alpha\delta\beta-\frac{\delta^2t}{2}\Bigr)f(\frac{t}{\alpha^2}\mid 0,1,\beta),
187+ \end{equation* }
188+
189+ where $f(t^* =\frac{t}{\alpha^2}\mid0,1,\beta)$ can be specified in two ways:
190+
191+ \begin{equation* }
192+ f_l(t^* \mid 0,1,\beta) = \sum_ {k=1}^\infty k\pi \exp\Bigl(-\frac{k^2\pi^2t^* }{2}\Bigr)\sin(k\pi \beta)\text{ and}
193+ \end{equation* }
194+
195+ \begin{equation* }
196+ f_s(t^* \mid0,1,\beta) = \sum_ {k=-\infty}^\infty \frac{1}{\sqrt{2\pi(t^* )^3}}(\beta+2k) \exp\Bigl(-\frac{(\beta+2k)^2}{2t^* }\Bigr).
197+ \end{equation* }
198+
199+ Which of these is used in the computations depends on which expression requires the smaller number of components $k$ to guarantee a pre-specified precision
185200
186201In the case where $s_ {\delta}$, $s_ {\beta}$, and $s_ {\tau}$ are all $0$, this simplifies to
187202\begin{equation* }
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