Quantifying uncertainty with a derivative tracking SDE model and application to wind power forecast data

Abstract

We develop a data-driven methodology based on parametric Itô’s Stochastic Differential Equations (SDEs) to capture the real asymmetric dynamics of forecast errors, including the uncertainty of the forecast at time zero. Our SDE framework features time-derivative tracking of the forecast, time-varying mean-reversion parameter, and an improved state-dependent diffusion term. Proofs of the existence, strong uniqueness, and boundedness of the SDE solutions are shown by imposing conditions on the time-varying mean-reversion parameter. We develop the structure of the drift term based on sound mathematical theory. A truncation procedure regularizes the prediction function to ensure that the trajectories do not reach the boundaries almost surely in a finite time. Inference based on approximate likelihood, constructed through the moment-matching technique both in the original forecast error space and in the Lamperti space, is performed through numerical optimization procedures. We propose a fixed-point likelihood optimization approach in the Lamperti space. Another novel contribution is the characterization of the uncertainty of the forecast at time zero, which turns out to be crucial in practice. We extend the model specification by considering the length of the unknown time interval preceding the first time a forecast is provided through an additional parameter in the density of the initial transition. All the procedures are agnostic of the forecasting technology, and they enable comparisons between different forecast providers. We apply our SDE framework to model historical Uruguayan normalized wind power production and forecast data between April and December 2019. Sharp empirical confidence bands of wind power production forecast error are obtained for the best-selected model.

A Appendix

For a time horizon $T>0$, a parameter $\alpha > 0$, and $(\theta _t)_{t\in [0,T]}$ a positive deterministic function, let us consider the model given by

$$\begin{aligned} \left\{ \begin{array}{@{}rl@{}} \hbox {d}X_t \!\!\!&{}= \big (\dot{p}_t - \theta _t (X_t-p_t) \big ) \hbox {d} t +\sqrt{2 \alpha \theta _0 X_t (1-X_t)} \hbox {d}W_t, \,\, t \in [0,T]\\ X_0 \!\!\!&{}= x_0 \in [0,1], \end{array}\right. \nonumber \\ \end{aligned}$$

(29)

where $(p_t)_{t\in [0,T]}$ denotes the prediction function that satisfies $0\le p_t\le 1$ for all $t\in [0,T]$. This prediction function is assumed to be a smooth function of the time so that

$$\begin{aligned} \sup _{t\in [0,T]}\bigl ( |p_t| + |\dot{p}_t|\big ) <+\infty . \end{aligned}$$

The following proofs are based on standard arguments for stochastic processes that can be found e.g. in Alfonsi (2015) and Karatzas and Shreve (1998) that we adapted to the setting of our model (29).

Theorem 1

Assume that

$$\begin{aligned} \forall t\in [0,T],\;\; 0\le \dot{p}_t +\theta _tp_t\le \theta _t \le C_1 < \infty \,. \end{aligned}$$

(A)

Then, there is a unique strong solution to (29) s.t. for all $t\in [0,T]$, $X_t\in [0,1]\,.$

Proof

Let us first consider the following SDE for $t\in [0,T]$

$$\begin{aligned} X_t&=x_0+ \int _0^t\big (\dot{p}_s - \theta _s(X_s-p_s) \big ) \hbox {d} s \nonumber \\&\quad + \int _0^t\sqrt{2\alpha \theta _0 |X_s(1-X_s)|} \hbox {d}W_s, \quad 0\le x_0\le 1. \end{aligned}$$

(30)

According to Proposition 2.13, p. 291 of Karatzas and Shreve (1998), under assumption (A) there is a unique strong solution $X_t$ to (30). Moreover, as the diffusion coefficient is of linear growth, we have for all $p>0$

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{t\in [0,T]}|X_t|^p\right] <\infty . \end{aligned}$$

(31)

Then, it remains to show that for all $t\in [0,T]$, $X_t\in [0,1]$ a.s. For this aim, we need to use the so-called Yamada function $\psi _n$ that is a ${\mathcal {C}}^2$ function that satisfies a bunch of useful properties:

$$\begin{aligned}&|\psi _n(x)|\underset{n\rightarrow +\infty }{\rightarrow }|x|, \;\; x{\psi '}_n(x)\underset{n\rightarrow +\infty }{\rightarrow }|x|, \\&\quad |\psi _n(x)|\wedge |x{\psi '}_n(x)| \le |x|, \;\; {\psi '}_n(x) \le 1,\\&\quad \text {and}\;\; {\psi ''}_n(x)=g_n(|x|)\ge 0\;\; \text{ with } \;\; g_n(x)x\le \frac{2}{n}\;\; \text {for all}\;\; x\in {\mathbb {R}} . \end{aligned}$$

See the proof of Proposition 2.13, p. 291 of Karatzas and Shreve (1998) for the construction of such function. Applying Itô’s formula we get

$$\begin{aligned} \psi _n(X_t)&=\psi _n(x_0) +\int _0^t {\psi '}_n(X_s) (\dot{p}_s + \theta _s p_s - \theta _sX_s \big ) \hbox {d}s \\&\quad + \int _0^t{\psi '}_n(X_s)\sqrt{2\alpha \theta _0 |X_s(1-X_s)|} \hbox {d}W_s \\&\quad + \alpha \theta _0 \int _0^t g_n(|X_s|) |X_s(1-X_s)| \hbox {d}s. \end{aligned}$$

Now, thanks to (A), (31), and to the above properties of $\psi _n$ and $g_n$, we get

$$\begin{aligned} {\mathbb {E}}\left[ \psi _n(X_t)\right]&\le \psi _n(x_0) +\int _0^t \left( \dot{p}_s + \theta _sp_s - \theta _s {{\mathbb E}}[{\psi '}_n(X_s)X_s] \right) \hbox {d}s\\&\quad + \frac{2\alpha \theta _0}{n}\int _0^t {\mathbb {E}} \left[ |1-X_s|\right] \hbox {d}s. \end{aligned}$$

Therefore, letting n tends to infinity, we use Lebesgue’s theorem to get

$$\begin{aligned} {\mathbb {E}}\left[ |X_t|\right] \le x_0 +\int _0^t \left( \dot{p}_s + \theta _sp_s - \theta _s {\mathbb {E}}\left[ |X_s|\right] \right) \hbox {d}s. \end{aligned}$$

Besides, taking the expectation of (30), we get

$$\begin{aligned} {\mathbb {E}} \left[ X_t\right] =x_0+ \int _0^t\big (\dot{p}_s +\theta _sp_s - \theta _s {\mathbb {E}}\left[ X_s\right] \big ) \hbox {d}s, \end{aligned}$$

and thus we have

$$\begin{aligned} {\mathbb {E}}\left[ |X_t| -X_t \right] \le \int _0^t \theta _s \mathbb E\left[ X_s - |X_s| \right] \hbox {d}s. \end{aligned}$$

Then, Gronwall’s lemma gives us ${\mathbb {E}}\left[ |X_t|\right] =\mathbb E \left[ X_t\right]$ and thus for any $t\in [0,T]$ $X_t\ge 0$ a.s. The same arguments work to prove that for any $t\in [0,T]$ $Y_t:=1-X_t\ge 0$ a.s. since the process $(Y_t)_{t\in [0,T]}$ is solution to

$$\begin{aligned} \hbox {d}Y_t= \big ( \theta _t(1-p_t) -\dot{p}_t - \theta _tY_t \big ) \hbox {d}t -\sqrt{2\alpha \theta _0Y_t(1-Y_t)} \hbox {d}W_t. \end{aligned}$$

Then similarly, we need to assume that $\dot{p}_t +\theta _tp_t\ge 0$. This completes the proof. $\square$

Theorem 2

Let that assumptions of Theorem 1 hold with $x_0\in ]0,1[$. Let $\tau _0:=\inf \{t\in [0,T],\; X_t=0\}$ and $\tau _1:=\inf \{t\in [0,T],\; X_t=1\}$ with the convention that $\inf \emptyset =+\infty$. Assume in addition that for all $t\in [0,T]$, $p_t\in ]0,1[$ and that

$$\begin{aligned} 0< \alpha \theta _0 \le \dot{p}_t +\theta _t p_t \le \theta _t - \alpha \theta _0 \le C_1 < \infty \,. \end{aligned}$$

(A′)

Then, $\tau _0=\tau _1=+\infty$ a.s.

Proof

For $t\in [0,\tau _0[$, we have

$$\begin{aligned} \frac{\hbox {d}X_t}{X_t}= \left( \frac{\dot{p}_t +\theta _t p_t}{X_t} - \theta _t \right) \hbox {d}t +\sqrt{\frac{2\alpha \theta _0 (1-X_t)}{X_t}} \hbox {d}W_t \end{aligned}$$

so that

$$\begin{aligned} X_t= & {} x_0\exp \left( \int _0^t \left( \frac{\dot{p}_s +\theta _sp_s}{X_s} - \theta _0 \alpha \right) \hbox {d}s\right. \\&\left. \quad +\alpha \theta _0 t- \int _0^t\theta _s \hbox {d}s + M_t\right) , \end{aligned}$$

where $M_t=\int _0^t\sqrt{\frac{2\alpha \theta _0 (1-X_s)}{X_s}} \hbox {d}W_s$ is a continuous martingale. Then as for all $t\in [0,T]$, we have $\dot{p}_t +\theta _tp_t- \theta _0 \alpha \ge 0$, we deduce that

$$\begin{aligned} X_t\ge x_0\exp \left( \alpha \theta _0 t- \int _0^t\theta _s \hbox {d}s + M_t\right) . \end{aligned}$$

By way of contradiction let us assume that $\{\tau _0<\infty \}$, then letting $t\rightarrow \tau _0$ we deduce that

$$\begin{aligned} \lim _{t\rightarrow \infty } {{\mathbf {1}}}_{\{\tau _0<\infty \}}M_{t\wedge \tau _0}= - {\mathbf{1}}_{\{\tau _0<\infty \}}\infty \, \text { a.s.} \end{aligned}$$

This leads to a contradiction since we know that continuous martingales likewise the Brownian motion cannot converge almost surely to $+\infty$ or $-\infty$. It follows that $\tau _0=\infty$ almost surely. Next, recalling that the process $(Y_t)_{t\ge 0}$ given by $Y_t=1-X_t$ is solution to

$$\begin{aligned} \hbox {d}Y_t= \big ( \theta _t(1-p_t) -\dot{p}_t - \theta _t Y_t \big ) \hbox {d}t -\sqrt{2 \alpha \theta _0 Y_t(1-Y_t)} \hbox {d}W_t, \end{aligned}$$

we deduce using similar arguments as above $\tau _1=\infty$ a.s. provided that $\theta _t(1-p_t) -\dot{p}_t -\alpha \theta _0 \ge 0$. $\square$

Remark 6

Observe that the condition (A$^\prime$) can be expressed as the following assumption on $\theta _t$ given $( p_t, \dot{p}_t)$:

$$\begin{aligned} \theta _t\ge \max \left( \frac{\alpha \theta _0+\dot{p}_t}{1-p_t},\frac{\alpha \theta _0-\dot{p}_t}{p_t}\right) . \end{aligned}$$

(B)

Remark 7

As the diffusion coefficient of $X_t$ given by $x \mapsto \sqrt{2 \alpha \theta _0 x(1-x)}$ is strictly positive for all $x \in ]0,1[$, the condition (B) ensures that the transformation between $Z_t$ and $X_t$ is bijective, so that we deduce the properties of existence and uniqueness of $Z_t$ from those of $X_t$. The application of Itô’s formula in Sect. 4 is subjected to the condition (B) that avoids the process $X_t$ hits the boundaries of the interval ]0, 1[, otherwise the Lamperti transform is not applicable.