Abstract
Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton–Jacobi–Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.
Similar content being viewed by others
References
Asmussen, S., Glynn, P.W.: Stochastic simulation: algorithms and analysis. Springer, New York (2007)
Asmussen, S., Kroese, D.P.: Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Prob. 38, 545–558 (2006)
Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures. Studies in mathematics and its applications, North-Holland (1978)
Berglund, N., Gentz, B.: Metastability in simple climate models: pathwise analysis of slowly driven Langevin equations. Stoch. Dyn. 02, 327–356 (2002)
Blanchet, J., Glynn, P.: Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Probab. 18, 1351–1378 (2008)
Boué, M., Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26, 1641–1659 (1998)
Brooks, S.P.: Markov chain Monte Carlo method and its application. J. R. Stat. Soc. Ser D 47, 69–100 (1998)
Cerrai, S.: Second order PDE’s in finite and infinite dimension: a probabilistic approach, no. 1762 in Lecture Notes in Mathematics, Springer, (2001)
Cerrai, S.: Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative noise. Siam J. Math. Anal. 43, 2482–2518 (2011)
Da Prato, G., Zabczyk, J.: Ergodicity for infinite dimensional systems, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University Press, (1996)
Da Prato, G., Zabczyk, J.: Second order partial differential equations in Hilbert spaces, London Mathematical Society Lecture Note Series. Cambridge University Press, London (2002)
Doucet, A., De Freitas, N., Gordon, N. (eds.): Sequential Monte Carlo methods in practice. Springer, New York (2001)
Duane, S., Kennedy, A.D., Pendleton, B.J., Roweth, D.: Hybrid Monte Carlo. Phys. Lett. B 195, 216–222 (1987)
Dupuis, P., Spiliopoulos, K., Wang, H.: Rare event simulation for rough energy landscapes. In: Simulation Conference (WSC), Proceedings of the 2011 Winter, 2011, pp. 504–515
Dupuis, P., Spiliopoulos, K., Wang, H.: Importance sampling for multiscale diffusions. Multiscale Model. Simul. 10, 1–27 (2012)
Dupuis, P., Wang, H.: Importance sampling, large deviations, and differential games. Stoch. Stoch. Rep. 76, 481–508 (2004)
Dupuis, P., Wang, H.: Subsolutions of an Isaacs equation and efficient schemes for importance sampling. Math. Oper. Res. 32, 723–757 (2007)
Liu, W .E.D., Vanden-Eijnden, E.: Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math. 58, 1544–1585 (2005)
Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solutions. Springer, New York (2006)
Freidlin, M., Wentzell, A.: Random perturbations of dynamical systems, vol. 260 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2012)
Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs (1964)
Givon, D.: Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems. Multiscale Model. Simul. 6, 577–594 (2007)
Glasserman, P., Heidelberger, P., Shahabuddin, P.: Asymptotically optimal importance sampling and stratification for pricing path-dependent options. Math. Finance 9, 117–152 (1999)
Hartmann, C., Latorre, J., Pavliotis, G., Zhang, W.: Optimal control of multiscale systems using reduced-order models. J. Comput. Dyn. 1, 279–306 (2014)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)
Hedges, L.O., Jack, R.L., Garrahan, J.P., Chandler, D.: Dynamic order-disorder in atomistic models of structural glass formers. Science 323, 1309–1313 (2009)
Huang, C., Liu, D.: Strong convergence and speed up of nested stochastic simulation algorithm. Commun. Comput. Phys. 15, 1207–1236 (2014)
Jack, R.L., Sollich, P.: Effective interactions and large deviations in stochastic processes. Eur. Phys. J. Special Topics 224, 2351–2367 (2015)
Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus, 2nd edn. Springer, New York (1991)
Khasminskii, R.: Principle of averaging for parabolic and elliptic differential equations and for Markov processes with small diffusion. Theory Probab. Appl. 8, 1–21 (1963)
Krylov, N.: Controlled diffusion processes. Stochastic modelling and applied probability. Springer, Berlin (1980)
Latorre, J.C., Metzner, P., Hartmann, C., Schütte, C.: A structure-preserving numerical discretization of reversible diffusions. Commun. Math. Sci. 9, 1051–1072 (2010)
Liu, D.: Strong convergence of principle of averaging for multiscale stochastic dynamical systems. Commun. Math. Sci. 8, 999–1020 (2010)
Liu, J .S.: Monte Carlo strategies in scientific computing, 2nd edn. Springer, New York (2008)
Liu, J.S., Chen, R.: Sequential monte carlo methods for dynamic systems. J. Am. Statist. Assoc. 93, 1032–1044 (1998)
Majda, A., Franzke, C., Khouider, B.: An applied mathematics perspective on stochastic modelling for climate. Philos. Trans. A Math. Phys. Eng. Sci. 366, 2429–2455 (2008)
Øksendal, B.: Stochastic differential equations: an introduction with applications, 6th edn. Springer, Berlin (2010)
Pavliotis, G., Stuart, A.: Multiscale methods: averaging and homogenization. Springer, Berlin (2008)
Prinz, J.-H., Wu, H., Sarich, M., Keller, B., Senne, M., Held, M., Chodera, J. D., Schütte, C., Noé, F.: Markov models of molecular kinetics: Generation and validation. J. Chem. Phys.,134 (2011)
Schütte, C., Fischer, A., Huisinga, W., Deuflhard, P.: A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys. 151, 146–168 (1999)
Spiliopoulos, K.: Large deviations and importance sampling for systems of slow-fast motion. Appl. Math. Optim. 67, 123–161 (2013)
Spiliopoulos, K.: Nonasymptotic performance analysis of importance sampling schemes for small noise diffusions. J. Appl. Probab. 52, 797–810 (2015)
Spiliopoulos, K.: Rare event simulation for multiscale diffusions in random environments. Multiscale Model. Simul. 13, 1290–1311 (2015)
Spiliopoulos, K., Dupuis, P., Zhou, X.: Escaping from an attractor: Importance sampling and rest points, part I. Ann. Appl. Probab. 25, 2909–2958 (2015)
Vanden-Eijnden, E., Weare, J.: Rare event simulation of small noise diffusions. Comm. Pure Appl. Math. 65, 1770–1803 (2012)
Zhang, W., Wang, H., Hartmann, C., Weber, M., Schütte, C.: Applications of the cross-entropy method to importance sampling and optimal control of diffusions. SIAM J. Sci. Comput. 36, A2654–A2672 (2014)
Acknowledgements
The authors acknowledge financial support by the DFG Research Center Matheon, the Einstein Center for Mathematics ECMath and the DFG-CRC 1114 “Scaling Cascades in Complex Systems”. Special thanks also go to anonymous referees whose valuable comments and criticism have helped to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Appendices
Two useful inequalities
Claim 7.1
Consider functions \(x_1(t), x_2(t)\) on \(t \in [0,T]\) satisfying
with \(x_1(0) = 0, x_2(0) = 1\), \(a_{ij} > 0\), \(1\le i, j \le 2\). Further assume that \(x_1(t) \ge 0\) for all \(t\in [0,T]\). Then there is a constant \(C > 0\) depending on \(a_{ij}\) and T, such that
Proof
Applying Gronwall’s inequality to the equation of \(x_2\), we have
Applying Gronwall’s inequality to \(x_1\) and using (7.2), we find
Since the right hand side in the last inequality is monotonically increasing (as a function of t), it follows that
The first part of the assertion then follows by applying Gronwall’s inequality in integral form to \(\max \limits _{0\le s \le t} x_1(s)\), while the second part is obtained using (7.2). \(\square \)
For \(0< \epsilon < 1\), we set \(t_1 = -\frac{2\epsilon \ln \epsilon }{\lambda } > 0\) and introduce the function \(\gamma :[0,T] \rightarrow [0, 1]\) by
Claim 7.2
Consider functions \(x_1(t), x_2(t)\) on \(t \in [0, T]\) satisfying
where \(\gamma \) is given in (7.5), \(a_i \ge 0, 1\le i \le 3\), and \(x_1(0) = 0, x_2(0) = 1\). Further assume that \(x_1(t) \ge 0\) on \(t \in [0,T]\). Then there is a constant \(C > 0\) independent of \(\epsilon \), such that
Proof
As in Claim 7.1, we can obtain
Then, for \(t < t_1\), the second inequality above implies
Using (7.7) and Gronwall’s inequality again, we conclude that
Repeating the above argument for \(t\in [t_1, T]\), noticing that \(x_1(t_1) \le C\epsilon ^2\), \(x_2(t_1) \le C\epsilon ^2\), \(\gamma (t) \equiv 0, t \in [t_1, T]\), it follows that
The proof is completed by combining (7.10) and (7.11). \(\square \)
Properties of the stationary process
For fixed \(x \in {\mathbb {R}}^k\) and \(\tau \in {\mathbb {R}}\), we introduce the process
where \(w_s\) is a standard Wiener process in \({\mathbb {R}}^{m_2}\). In the following, we summarize some properties related to the above process that we called the fast subsystem in Sect. 3. See also [10, 33] for additional results.
Lemma 8.1
Under Assumptions 1–2, there exists a constant \(C > 0\), independent of \(\epsilon , x, y\), such that:
-
1.
\({{\mathbf {E}}}|\xi ^x_{\tau , s}|^4 \le e^{-\frac{\lambda (s-\tau )}{\epsilon }} |y|^4 + C\big (|x|^4 + 1\big )\).
-
2.
For \(\tau _1 \le \tau _2\), it holds
$$\begin{aligned}{{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4 \le C\big (|x|^4 + |y|^4+1\big )\,e^{-\frac{4\lambda (s-\tau _2)}{\epsilon }}\,, \quad s \ge \tau _2\,. \end{aligned}$$ -
3.
For \(x, x' \in {\mathbb {R}}^k\) and \(\tau _1 \le \tau _2\),
$$\begin{aligned} {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4\le e^{-\frac{2\lambda (s-\tau _2)}{\epsilon }} \big (|x|^4 + |y|^4 + 1\big ) + C |x' - x|^4\,, \quad s \ge \tau _2\,. \end{aligned}$$
Proof
-
1.
By Ito’s formula, we have
$$\begin{aligned} \frac{d{{\mathbf {E}}}|\xi ^x_{\tau , s}|^4}{ds} =&\, \frac{1}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s}|^2 \big (4\langle g(x, \xi ^x_{\tau , s}), \xi ^x_{\tau , s}\rangle + 2\Vert \alpha _2(x,\xi ^x_{\tau , s})\Vert ^2\big ) \\&+\, 4|\alpha ^T_2(x,\xi ^x_{\tau , s})\xi ^x_{\tau , s}|^2\Big ] \nonumber \\ \le&\, \frac{1}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s}|^2 \big (4\langle g(x, \xi ^x_{\tau , s}), \xi ^x_{\tau , s}\rangle + 6\Vert \alpha _2(x,\xi ^x_{\tau , s})\Vert ^2\big )\Big ]\,. \end{aligned}$$Applying inequality (3.13) in Remark 2 and inequality (5.9), we obtain
$$\begin{aligned} \frac{d{{\mathbf {E}}}|\xi ^x_{\tau , s}|^4}{ds} \le&-\frac{2\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau , s}|^4 + \frac{C}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s}|^2 (|x|^2 + 1)\Big ] \nonumber \\ \le&-\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau , s}|^4 + \frac{C}{\epsilon } \Big (|x|^4 + 1\Big )\,, \end{aligned}$$and the first statement follows from Gronwall’s inequality.
-
2.
For the second statement, using Ito’s formula and Assumption 2, it follows
$$\begin{aligned}&\frac{d{{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4}{ds} \nonumber \\&\quad = \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x, \xi ^x_{\tau _2, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \nonumber \\&\qquad + 2\Vert \alpha _2(x,\xi ^x_{\tau _2, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) + 4\big |\big (\alpha _2(x,\xi ^x_{\tau _2, s})\\&\qquad - \alpha _2(x,\xi ^x_{\tau _1, s})\big )^T\big (\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}\big )\big |^2\Big ] \nonumber \\&\quad \le \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x, \xi ^x_{\tau _2, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \\&\qquad + 6\Vert \alpha _2(x,\xi ^x_{\tau _2, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) \Big ] \nonumber \\&\quad \le -\frac{4\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4\,. \end{aligned}$$Therefore, integrating and using the first statement above, we obtain
$$\begin{aligned} {{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4 \le e^{-\frac{4\lambda (s-\tau _2)}{\epsilon }} {{\mathbf {E}}}|\xi ^x_{\tau _1, \tau _2} - y|^4 \le C\big (1+|x|^4 + |y|^4\big )e^{-\frac{4\lambda (s-\tau _2)}{\epsilon }}\,. \end{aligned}$$ -
3.
For the third statement, in a similar way, applying Ito’s formula, using Assumption 2, as well as Lipschitz property of functions g and \(\alpha _2\), we have
$$\begin{aligned}&\frac{d{{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4}{ds} \nonumber \\&\quad = \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x', \xi ^{x'}_{\tau _2, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \nonumber \\&\qquad + 2\Vert \alpha _2(x',\xi ^{x'}_{\tau _2, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) + 4\big |\big (\alpha _2(x',\xi ^{x'}_{\tau _2, s})\\&\qquad - \alpha _2(x,\xi ^x_{\tau _1, s})\big )^T\big (\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}\big )\big |^2\Big ] \nonumber \\&\quad \le \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x', \xi ^{x'}_{\tau _2, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \\&\qquad + 6\Vert \alpha _2(x',\xi ^{x'}_{\tau _2, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) \Big ] \nonumber \\&\quad \le \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x', \xi ^{x'}_{\tau _2, s}) - g(x', \xi ^x_{\tau _1, s}), \xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \\&\qquad + 12\Vert \alpha _2(x',\xi ^{x'}_{\tau _2, s})-\alpha _2(x',\xi ^x_{\tau _1, s})\Vert ^2\big ) \Big ] \nonumber \\&\qquad + \frac{1}{\epsilon } {{\mathbf {E}}}\Big [ |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \big (4\langle g(x', \xi ^{x}_{\tau _1, s}) - g(x, \xi ^x_{\tau _1, s}), \xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s} \rangle \\&\qquad + 12\Vert \alpha _2(x',\xi ^{x}_{\tau _1, s})-\alpha _2(x,\xi ^x_{\tau _1, s})\Vert ^2\big ) \Big ] \nonumber \\&\quad \le -\frac{4\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4\\&\qquad + \frac{C}{\epsilon } {{\mathbf {E}}}\big ( |\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^3|x'-x|\big ) + \frac{C}{\epsilon } {{\mathbf {E}}}\big (|\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2|x'-x|^2\big ) \nonumber \\&\quad \le -\frac{2\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4 + \frac{C}{\epsilon } |x' - x|^4\,, \end{aligned}$$where inequality (5.9) is used to obtain the last inequality. Gronwall’s inequality together with the first statement above then yield the assertion.
\(\square \)
Now consider the derivative process
with \(s\ge \tau \,, \xi ^x_{\tau , \tau ,x_i} = 0\), \(1\le i \le k\). In the above, we used \(D_{x_i}\) to denote derivatives with respect to scalar \(x_i \in {\mathbb {R}}\) and \(\nabla _y\) to denote derivatives with respect to a vector \(y \in {\mathbb {R}}^l\). We summarize its properties in the following result.
Lemma 8.2
Under Assumptions 1–2, there exists a constant \(C > 0\), independent of \(\epsilon , x, y\), such that \(\forall 1 \le i \le k\),
-
1.
For \(x \in {\mathbb {R}}^k\), \(s \ge \tau \), \({{\mathbf {E}}}|\xi ^x_{\tau , s, x_i}|^4 \le C\).
-
2.
For \(\tau _1 \le \tau _2\), \(x \in {\mathbb {R}}^k\),
$$\begin{aligned} {{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2 \le C\big (1 + |x|^2+|y|^2\big ) e^{-\frac{\lambda (s-\tau _2)}{\epsilon }}\,. \end{aligned}$$ -
3.
For \(\tau _1 \le \tau _2\), \(x, x' \in {\mathbb {R}}^k\),
$$\begin{aligned} {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 \le Ce^{-\frac{\lambda (s-\tau _2)}{\epsilon }} \left[ 1 + \frac{s - \tau _2}{\epsilon } \big (1 + |x|^2+|y|^2\big )\right] + C |x - x'|^2\,. \end{aligned}$$
Proof
-
1.
Using Ito’s formula, Assumption 1 (Lipschitz continuity of functions g and \(\alpha _2\)), inequality (3.11) in Remark 2, as well as inequality (5.9), we see that
$$\begin{aligned} \frac{d{{\mathbf {E}}}|\xi ^x_{\tau , s, x_i}|^4}{ds} \le&\ \frac{1}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s, x_i}|^2\Big (4\langle D_{x_i} g(x,\xi ^x_{\tau , s}) + \nabla _{y} g(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}, \xi ^x_{\tau , s, x_i}\rangle \\&+ 6\Vert D_{x_i} \alpha _2(x,\xi ^x_{\tau , s}) + \nabla _{y} \alpha _2(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}\Vert ^2\Big )\Big ] \nonumber \\ \le&\frac{1}{\epsilon } {{\mathbf {E}}}\Big [|\xi ^x_{\tau , s, x_i}|^2\Big (C|\xi ^x_{\tau , s, x_i}| + 4\langle \nabla _{y} g(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}, \xi ^x_{\tau , s, x_i}\rangle + C \\&+ 12\Vert \nabla _{y} \alpha _2(x,\xi ^x_{\tau , s}) \xi ^x_{\tau , s, x_i}\Vert ^2\Big )\Big ] \nonumber \\ \le&-\frac{2\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau , s, x_i}|^4 + \frac{C}{\epsilon } \end{aligned}$$and therefore \({{\mathbf {E}}}|\xi ^x_{\tau , s, x_i}|^4 \le C\) by Gronwall’s inequality.
-
2.
Now consider \(\xi ^x_{\tau _1, s,x_i}, \xi ^x_{\tau _2, s,x_i}\) with \(\tau _1 \le \tau _2\). Using Lipschitz condition of functions \(g, \alpha _2\), inequality (3.11) in Remark 2, as well as inequality (5.9), it follows
$$\begin{aligned}&\frac{d{{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2}{ds} \nonumber \\&\quad = \frac{2}{\epsilon } {{\mathbf {E}}}\langle D_{x_i} g(x, \xi ^x_{\tau _2, s}) - D_{x_i} g(x, \xi ^x_{\tau _1, s}) + \nabla _{y} g(x, \xi ^x_{\tau _2, s}) \xi ^x_{\tau _2, s, x_i} \\&\qquad - \nabla _{y} g(x, \xi ^x_{\tau _1, s}) \xi ^x_{\tau _1, s, x_i}, \xi ^x_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{1}{\epsilon } {{\mathbf {E}}}\Vert D_{x_i} \alpha _2(x, \xi ^x_{\tau _2, s}) - D_{x_i} \alpha _2(x, \xi ^x_{\tau _1, s}) + \nabla _{y} \alpha _2(x, \xi ^x_{\tau _2, s}) \xi ^x_{\tau _2, s, x_i} \\&\qquad - \nabla _{y} \alpha _2(x, \xi ^x_{\tau _1, s}) \xi ^x_{\tau _1, s, x_i}\Vert ^2 \nonumber \\&\quad \le \frac{C}{\epsilon }{{\mathbf {E}}}\Big (|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}||\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|\Big ) + \frac{2}{\epsilon } {{\mathbf {E}}}\langle \big (\nabla _{y} g(x, \xi ^x_{\tau _2, s})\\&\qquad -\nabla _{y} g(x, \xi ^x_{\tau _1, s})\big )\xi ^x_{\tau _1, s, x_i}, \xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{2}{\epsilon } {{\mathbf {E}}}\langle \nabla _{y} g(x, \xi ^x_{\tau _2, s})(\xi ^x_{\tau _2, s, x_i} {-} \xi ^x_{\tau _1, s, x_i}), \xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}\rangle {+} \frac{C}{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2\nonumber \\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert \big (\nabla _{y} \alpha _2(x, \xi ^x_{\tau _2, s}){-}\nabla _{y} \alpha _2(x, \xi ^x_{\tau _1, s})\big ) \xi ^x_{\tau _1, s, x_i}\Vert ^2\\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert \nabla _{y} \alpha _2(x, \xi ^x_{\tau _2, s})(\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i})\Vert ^2 \nonumber \\&\quad \le -\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon } \big ({{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^4\big )^{\frac{1}{2}} \big ({{\mathbf {E}}}|\xi ^x_{\tau _1, s, x_i}|^4)^{\frac{1}{2}} \\&\qquad + \frac{C}{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s} - \xi ^x_{\tau _1, s}|^2 \nonumber \\&\quad \le -\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon } (1 + |x|^2+|y|^2) e^{-\frac{2\lambda (s-\tau _2)}{\epsilon }}\,, \end{aligned}$$where the first assertion above and Lemma 8.1 have been used in the last inequality. Then Gronwall’s inequality entails
$$\begin{aligned} {{\mathbf {E}}}|\xi ^x_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i}|^2 \le C\big (1 + |x|^2+|y|^2\big ) e^{-\frac{\lambda (s-\tau _2)}{\epsilon }}\,. \end{aligned}$$ -
3.
Consider \(\xi ^{x}_{\tau _1, s,x_i}, \xi ^{x'}_{\tau _2, s,x_i}\) with \(\tau _1 \le \tau _2\). In a similar way, we have
$$\begin{aligned}&\frac{d{{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2}{ds} \nonumber \\&\quad = \frac{2}{\epsilon } {{\mathbf {E}}}\langle D_{x_i} g(x', \xi ^{x'}_{\tau _2, s}) - D_{x_i} g(x, \xi ^x_{\tau _1, s}) + \nabla _{y} g(x', \xi ^{x'}_{\tau _2, s}) \xi ^{x'}_{\tau _2, s, x_i} \\&\qquad - \nabla _{y} g(x, \xi ^x_{\tau _1, s}) \xi ^x_{\tau _1, s, x_i}, \xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{1}{\epsilon } {{\mathbf {E}}}\Vert D_{x_i} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) - D_{x_i} \alpha _2(x, \xi ^x_{\tau _1, s}) + \nabla _{y} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) \xi ^{x'}_{\tau _2, s, x_i} \\&\qquad - \nabla _{y} \alpha _2(x, \xi ^x_{\tau _1, s}) \xi ^x_{\tau _1, s, x_i}\Vert ^2 \nonumber \\&\quad \le \frac{2}{\epsilon } {{\mathbf {E}}}\langle D_{x_i} g(x', \xi ^{x'}_{\tau _2, s}) - D_{x_i} g(x', \xi ^x_{\tau _1, s}) + \nabla _{y} g(x', \xi ^{x'}_{\tau _2, s})\\&\qquad (\xi ^{x'}_{\tau _2, s, x_i} -\xi ^x_{\tau _1, s, x_i}), \xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{2}{\epsilon } {{\mathbf {E}}}\langle D_{x_i} g(x', \xi ^{x}_{\tau _1, s}) - D_{x_i} g(x, \xi ^x_{\tau _1, s}) + \big (\nabla _{y} g(x', \xi ^{x'}_{\tau _2, s}) \\&\qquad - \nabla _{y} g(x, \xi ^{x}_{\tau _1, s})\big ) \xi ^x_{\tau _1, s, x_i}, \xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}\rangle \nonumber \\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert D_{x_i} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) - D_{x_i} \alpha _2(x, \xi ^x_{\tau _1, s})\Vert ^2\\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert \nabla _{y} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) (\xi ^{x'}_{\tau _2, s, x_i} - \xi ^x_{\tau _1, s, x_i})\Vert ^2 \nonumber \\&\qquad + \frac{3}{\epsilon } {{\mathbf {E}}}\Vert \big (\nabla _{y} \alpha _2(x', \xi ^{x'}_{\tau _2, s}) - \nabla _{y} \alpha _2(x, \xi ^x_{\tau _1, s})\big ) \xi ^x_{\tau _1, s, x_i}\Vert ^2 \nonumber \\&\quad \le -\frac{2\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon } {{\mathbf {E}}}\big (|\xi ^{x'}_{\tau _2, s} - \xi ^x_{\tau _1, s}||\xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}|\big )\\&\qquad + \frac{C}{\epsilon } {{\mathbf {E}}}\big (|x' - x||\xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}|\big ) \nonumber \\&\qquad +\frac{C}{\epsilon } {{\mathbf {E}}}\big [\big (|x'-x|+ |\xi ^{x'}_{\tau _2, s}-\xi ^x_{\tau _1, s}|\big )|\xi ^x_{\tau _1, s, x_i}||\xi ^{x'}_{\tau _2, s, x_i}-\xi ^x_{\tau _1, s, x_i}|\big ]\\&\qquad + \frac{C}{\epsilon } |x-x'|^2+\frac{C}{\epsilon }{{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s}-\xi ^x_{\tau _1, s}|^2 \nonumber \\&\qquad + \frac{C}{\epsilon } {{\mathbf {E}}}\big [(|x'-x| + |\xi ^{x'}_{\tau _2, s}-\xi ^x_{\tau _1, s}|\big )^2|\xi ^x_{\tau _1, s, x_i}|^2\big ] \nonumber \\&\quad \le -\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon }\Big (|x' - x|^2 + {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^2 \\&\qquad + ({{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s} - \xi ^{x}_{\tau _1, s}|^4)^{\frac{1}{2}}\Big ) \nonumber \\&\quad \le -\frac{\lambda }{\epsilon } {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 + \frac{C}{\epsilon } \Big [(1 + |x|^2+|y|^2) e^{-\frac{\lambda (s-\tau _2)}{\epsilon }} + |x' - x|^2\Big ]\,, \end{aligned}$$and thus
$$\begin{aligned} {{\mathbf {E}}}|\xi ^{x'}_{\tau _2, s, x_i} - \xi ^{x}_{\tau _1, s, x_i}|^2 \le Ce^{-\frac{\lambda (s-\tau _2)}{\epsilon }} \Big [ 1 + \frac{s - \tau _2}{\epsilon } (1 + |x|^2+|y|^2)\Big ] + C |x' - x|^2\,. \end{aligned}$$
\(\square \)
The above results allow us to define the stationary process \(\xi _s^x=\xi _{-\infty ,s}^{x}\) with \(\xi _{s}^{x}\sim \rho _{x}(y)\,dy\) where \(\rho _{x}\) is the stationary probability density with respect to Lebesgue measure, and also the derivative process \(\xi ^x_{s, x_i}\) for \(1 \le i \le k\), satisfying that \(\forall f \in C^1_b({\mathbb {R}}^k\times {\mathbb {R}}^l)\) and \({\widetilde{f}}(x) = {{\mathbf {E}}}(f(x, \xi _s^x)) = \int _{{\mathbb {R}}^l} f(x,y) \rho _x(y) dy\), it holds
The processes \(\xi ^x_s\) and \(\xi ^x_{s, x_i}\) have the following properties:
Lemma 8.3
Under Assumptions 1 and 2, there is a constant \(C>0\), independent of \(\epsilon \), x and y, such that \(\forall f \in C_b^1({\mathbb {R}}^l)\):
-
1.
$$\begin{aligned} \Big | {{\mathbf {E}}}f(\xi ^x_{0,s}) - \int _{{\mathbb {R}}^l} f(y) \rho _x(y) dy\Big | \le \sup |f'| \Big (|x| + |y| + 1\Big ) e^{-\frac{\lambda s}{\epsilon }}\,. \end{aligned}$$(8.3)
-
2.
$$\begin{aligned}&\Big | {{\mathbf {E}}}\Big (f(\xi ^x_{0,s})\xi ^x_{0,s,x_i}\Big ) -{{\mathbf {E}}}\Big (f(\xi ^x_{s})\xi ^x_{s,x_i}\Big ) \Big |\nonumber \\&\quad \le C\Big (\sup |f| + \sup |f'|\Big ) \Big (1 + |x| + |y|\Big ) e^{-\frac{\lambda s}{2\epsilon }}\,. \end{aligned}$$(8.4)
Proof
We only prove the second inequality, as the first one follows in a similar fashion. Using Lemmas 8.1 and 8.2, we readily conclude that
\(\square \)
An analogous property for the stationary process \(\xi ^x_{s}\) is the following:
Lemma 8.4
Under Assumption 1 and 2, there exists constant \(C > 0\), independent of \(x, x'\), such that
-
1.
For \(x \in {\mathbb {R}}^k\), \({{\mathbf {E}}}|\xi ^x_{s, x_i}|^4 \le C\).
-
2.
For \(x , x'\in {\mathbb {R}}^k\), \({{\mathbf {E}}}|\xi ^{x'}_{s} - \xi ^x_{s}|^4 \le C |x - x'|^4\).
-
3.
For \(x, x' \in {\mathbb {R}}^k\), \({{\mathbf {E}}}|\xi ^{x'}_{s, x_i} - \xi ^{x}_{s, x_i}|^2 \le C |x - x'|^2\).
Proof
The conclusions follow directly by letting \(\tau _1, \tau _2\rightarrow -\infty \) in Lemma 8.1 and 8.2. \(\square \)
Rights and permissions
About this article
Cite this article
Hartmann, C., Schütte, C., Weber, M. et al. Importance sampling in path space for diffusion processes with slow-fast variables. Probab. Theory Relat. Fields 170, 177–228 (2018). https://doi.org/10.1007/s00440-017-0755-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-017-0755-3
Keywords
- Importance sampling
- Hamilton–Jacobi–Bellmann equation
- Monte Carlo method
- Change of measure
- Rare events
- Diffusion process