We extend the results of Sect. 2 to one-dimensional diffusions governed by the SDE in (1). Although the density is defined in infinite-dimensional space, in this section we justify both intuitively and formally that the diffusion can be approximated to arbitrary precision by considering a finite-dimensional projection of it.
The intuition behind using the Faber–Schauder basis is that, under mild assumptions on the drift function b, any diffusion process behaves locally as a Brownian motion. Expanding the diffusion process with the Faber–Schauder functions, this notion translates to the existence of a level N such that the random coefficients at higher levels which are associated with the Faber–Schauder basis are approximately independent standard normal and independent from \(\xi ^N\) under the measure \({\mathbb {P}}\).
Define the function \(Z_t:{\mathbb {R}}^+ \times C[0,T] \rightarrow {\mathbb {R}}^+\) given by
$$\begin{aligned} Z_t(X) = \exp \left( \int _0^t b(X_s) \mathrm {d}X_s - \frac{1}{2}\int _0^t b^2(X_s) \mathrm {d}s\right) \end{aligned}$$
(8)
where the first integral is understood in the Itô sense and \(X\equiv (X_s,\, s \in [0,T])\).
Assumption 3.1
\(Z_t\) is a \({\mathbb {Q}}\)-martingale.
For sufficient conditions for verifying that this assumption applies, we refer to Remark 3.6, Remark 3.9 and Liptser et al. (2013), Chapter 6.
Theorem 3.2
(Girsanov’s theorem) If Assumption 3.1 is satisfied,
$$\begin{aligned} \frac{ \mathrm {d}\mathbb {P}^u}{ \mathrm {d}\mathbb {Q}^u }(X) = Z_T(X). \end{aligned}$$
(9)
Moreover, a weak solution of the stochastic differential equation exists which is unique in law.
Proof
This is a standard result in stochastic calculus (see Liptser et al. 2013, Sect. 6). \(\square \)
As we consider diffusions on [0, T] with T fixed, we denote \(Z(X) := Z_T(X)\). Due to the appearance of the stochastic Itô integral in Z(X), we cannot substitute for X its truncated expansion in the Faber–Schauder basis. Clearly, whereas the approximation has finite quadratic variation, X has not. Assuming that b is differentiable and applying Itô’s lemma to the function \(B(x) = \int _0^x b(s) \mathrm {d}s\), the stochastic integral can be replaced and Eq. (8) is rewritten as
$$\begin{aligned} Z(X) = \exp \left( B(X_T) - B(X_0) - \frac{1}{2}\int _0^T \left( b^2(X_s) + b'(X_s)\right) \mathrm {d}s \right) , \end{aligned}$$
(10)
where \(b'\) is the derivative of b.
Definition 3.3
Let X be a diffusion governed by (1). Let \(X^N\) be the process derived from X by setting to zero all coefficients of level exceeding N in its Faber–Schauder expansion [see Eq. (2)]. Set
$$\begin{aligned} Z^N(X)= & {} \exp \left( B\left( X^N_T\right) - B\left( X^N_0\right) - \frac{1}{2}\int _0^T \left[ b^2\left( X^N_s\right) \right. \right. \\&+\left. \left. b'\left( X^N_s\right) \right] \mathrm {d}s \right) . \end{aligned}$$
We define the approximating measure \({\mathbb {P}}_N\) by the change of measure
$$\begin{aligned} \frac{ \mathrm {d}{\mathbb {P}}^u_N}{ \mathrm {d}{\mathbb {Q}}^u}(X) = \frac{Z^N(X)}{c_N}, \end{aligned}$$
(11)
where \(c_N = {\mathbb {E}}_{\mathbb {Q}}\left( Z^N(X)\right) \).
Note that the measure \({\mathbb {P}}^u_N\) associated with the approximated stochastic process is still on an infinite-dimensional space and such that the joint measure of random coefficients \(\xi ^N\) is different from the one under \({\mathbb {Q}}^u\), while the remaining coefficients stay independent standard normal and independent from \(\xi ^N\). This is equivalent to approximating the diffusion process at finite dyadic points with Brownian noise fill-in in between every two points. We now fix the final point \(v_T\) by setting \({\bar{\xi }} = v_T\). Define the approximated stochastic bridge with measure \({\mathbb {P}}^{u, v_T}_N\) in an analogous way of equation (11), so that
$$\begin{aligned} \frac{ \mathrm {d}{\mathbb {P}}^{u, v_T}_N}{ \mathrm {d}{\mathbb {Q}}^{u, v_T}}(X) = \frac{Z^N(X)}{c^{v_T}_N}. \end{aligned}$$
(12)
where \({c^{v_T}_N} = {\mathbb {E}}_{\mathbb {Q}^{u, v_T}}\left( Z^N(X)\right) \). The following is the main assumption made.
Assumption 3.4
The drift b is continuously differentiable, and \(b^2 + b'\) is bounded from below.
Theorem 3.5
If Assumptions 3.1 and 3.4 are satisfied, then \({\mathbb {P}}^{u, v_T}_N\) converges weakly to \({\mathbb {P}}^{u, v_T}\) as \(N \rightarrow \infty \).
Proof
In the following, we lighten the notation by omitting the initial point u from the notation, which will be assumed fixed to \(u = x_0\). We wish to show that \(\mathbb {P}^{v_T}_N\) converges weakly to \(\mathbb {P}^{v_T}\) as \(N \rightarrow \infty \). This is equivalent to showing that \(\int f \mathrm {d}\mathbb {P}^{v_T}_N \rightarrow \int f \mathrm {d}\mathbb {P}^{v_T}\) for all bounded and continuous functions f. Write \(c^{v_T}_\infty = p(0,x_0,T, v_T)/q(0,x_0,T, v_T)\). By equation (3) and (9),
$$\begin{aligned} \mathbb {E}_{\mathbb {Q}^{v_T}} Z(X) = \mathbb {E}_{\mathbb {Q}^{v_T}} \frac{d \mathbb {P}^{x_0}}{d \mathbb {Q}^{x_0}} = c_{\infty }^{v_T} \mathbb {E}_{\mathbb {Q}^{v_T}} \left[ \frac{ d \mathbb {P}^{v_T}}{d \mathbb {Q}^{v_T}}\right] = c_{\infty }^{v_T} \end{aligned}$$
and we have that
$$\begin{aligned}&\left| \int f \mathrm {d}\mathbb {P}^{v_T}_N - \int f \mathrm {d}\mathbb {P}^{v_T}\right| \nonumber \\&= \left| \int f \left( \frac{Z^N}{c^{v_T}_N} - \frac{Z}{c^{v_T}_\infty } \right) \mathrm {d}\mathbb {Q}^{v_T} \right| \nonumber \\&\le \Vert f\Vert _\infty \int \left| \frac{Z^N(X)}{c^{v_T}_N} - \frac{Z(X)}{c^{v_T}_\infty }\right| \mathrm {d}\mathbb {Q}^{v_T}(X)\nonumber \\&\le \Vert f\Vert _\infty \left( \frac{1}{c^{v_T}_N} \int \left| Z^N(X)-Z(X)\right| \mathrm {d}\mathbb {Q}^{v_T}(X)\right. \nonumber \\&\quad +\left. \int Z(X) \left| \frac{1}{c^{v_T}_N} - \frac{1}{c^{v_T}_{\infty }} \right| \mathrm {d}\mathbb {Q}^{v_T}(X) \right) \nonumber \\&\le \Vert f\Vert _\infty \left( \frac{1}{c^{v_T}_N} \int \left| Z^N(X)-Z(X)\right| \mathrm {d}\mathbb {Q}^{v_T}(X) + \left| \frac{c^{v_T}_\infty }{c^{v_T}_N}-1 \right| \right) \end{aligned}$$
(13)
where we used Assumption 3.1 for applying the change of measure between the conditional measures. Notice that \(Z^N(X) = Z(X^N)\). The mapping \(X \mapsto Z(X)\), as a function acting on C(0, T) with uniform norm, is continuous, since B, b and \(b'\) are continuous. Therefore, it follows from the Lévy–Ciesielski construction of Brownian motion (see Sect. 1.1.1) and the continuous mapping theorem that
$$\begin{aligned} Z^N(X) \rightarrow Z(X) \qquad {\mathbb {Q}}^{v_T}-a.s. \end{aligned}$$
Now, notice that, under conditional measures \(\mathbb {Q}^{v_T}\) and \(\mathbb {P}^{v_T}\), the term \(B(X_T) - B(X_0)\) is fixed. By the assumptions on b and \(b'\), Z is a bounded function and by dominated convergence, we get that
$$\begin{aligned} \lim _{N \rightarrow \infty } \mathbb {E}_\mathbb {Q}^{v_T} |Z^N(X)-Z(X)| = 0 \end{aligned}$$
giving convergence to zero of the first term in (13). This implies that also the constant \(c_N := \mathbb {E}_\mathbb {Q}^{v_T} |Z^N(X)| \) converges to \(\mathbb {E}_\mathbb {Q}^{v_T} |Z(X)| = c^{v_T}_\infty \) so that all the terms in (13) converge to 0. \(\square \)
We now list some technical conditions for the process to satisfy Assumptions 3.1 and 3.4.
Remark 3.6
If \(|b(x)| \le c(1 + |x|)\), for some positive constant c, then Assumption 3.1 is satisfied.
Proof
See Liptser et al. (2013), Sect. 6, Example 3 (b). \(\square \)
Remark 3.7
If b is globally Lipschitz and continuously differentiable, then Assumptions 3.1 and 3.4 are satisfied.
Proof
Assumption 3.4 is trivially satisfied. By Remark 3.6, also Assumption 3.1 is satisfied. \(\square \)
In Sect. 5.3, we will present an example where the drift b is not globally Lipschitz, yet Assumption 3.4 is satisfied.
Assumption 3.8
There exists a non-decreasing function \(h :[0,\infty ) \rightarrow [0,\infty )\) such that \({B(x) \le h(|x|)}\) and
$$\begin{aligned} \int _0^{\infty } \exp (h(x) - x^2/(2T)) \, d x < \infty . \end{aligned}$$
The above integrability condition is, for example, satisfied if \(h(|x|) = c(1 + |x|)\) for some \(c > 0\).
Remark 3.9
If Assumptions 3.4 and 3.8 hold, then Assumption 3.1 is satisfied.
Proof
By Sect. 3.5 in Karatzas and Shreve (1991), \((Z_t)\) is a local martingale. Say \(b'(x) + b^2(x) \ge -2 C\), where \(C \ge 0\). Using the assumptions, we have
$$\begin{aligned} Z_t= & {} \exp \left( B(X_t) - B(X_0) - \tfrac{1}{2} \int _0^t \{ b'(X_s) + b^2(X_s) \} \, ds \right) \\\le & {} A\exp (C t) \exp (h(|X_t|)), \end{aligned}$$
with constant \(A = \exp (-B(X_0))\). Then,
$$\begin{aligned}&\sup _{t \in [0,T]} Z_t \le A\sup _{t \in [0,T]} \exp (C t) \exp (h(|X_t|)) \le A\exp (C T) \\&\quad \exp \left( h \left( \max _{t \in [0,T]}| X_t|\right) \right) . \end{aligned}$$
By Lemma 3.10,
$$\begin{aligned} {\mathbb {E}} \sup _{t \in [0,T]} Z_t \le A \exp (C T)\, {\mathbb {E}} \exp (h (\max _{t \in [0,T]}| X_t|)) < \infty . \end{aligned}$$
Then, for a sequence of stopping times \((\tau _k)\) diverging to infinity such that \((Z_t^{\tau _k})_{0 \le t \le T}\) is a martingale for all k, we have
$$\begin{aligned} \mathbb {E}Z_0 = \mathbb {E}Z^{\tau _k}_0 = \mathbb {E}Z^{\tau _k}_t \rightarrow \mathbb {E}Z_t \end{aligned}$$
as \(k \rightarrow \infty \) by dominated convergence. \(\square \)
Lemma 3.10
Suppose \(h:[0,\infty ) \rightarrow [0,\infty )\) is non-decreasing. Let \(N_T = \max _{0 \le t \le T} |X_t|\) where \((X_t)\) is a Brownian motion. Then,
$$\begin{aligned} {\mathbb {E}} \exp h(N_T) \le 4 \int _0^{\infty } \frac{1}{\sqrt{2 \pi T}} \exp (h(x) - x^2/(2T)) \, d x. \end{aligned}$$
Proof
The maximum \(M_T = \max _{0 \le t \le T} X_t\) of a Brownian motion is distributed as the absolute value of a Brownian motion and thus has density function \(\frac{2}{\sqrt{2 \pi T}} \exp (-x^2/(2T))\), see Karatzas and Shreve (1991), Sect. 2.8. We have \({\mathbb {P}}(N_T \ge y) \le 2 {\mathbb {P}}(M_T \ge y)\) from which the result follows. \(\square \)
Finally, we mention that Theorem 3.5 can be generalised in the following way to diffusions without a fixed end point.
Proposition 3.11
If Assumption 3.4 is satisfied and B is bounded, then \({\mathbb {P}}_N\) converges weakly to \({\mathbb {P}}\).
The proof follows the same steps of the one of Theorem 3.5. In this case, we need to pay attention on B, as for unconditioned process, the final point is not fixed. If B is bounded, then Assumption 3.8 is satisfied. By Remark 3.9, also Assumption 3.1 is satisfied so that we can apply Theorem 3.2 for the change of measure. Finally, by the assumptions on b and B, the function Z is bounded and by dominated convergence, we get that
$$\begin{aligned} \lim _{N \rightarrow \infty } \mathbb {E}_\mathbb {Q}|Z^N(X)-Z(X)| = 0. \end{aligned}$$