1 Introduction↩︎

Consider two stochastic differential equations (SDEs) on \(\mathbb{R}^d\), \[\begin{align} \label{eq:40X44Y41} dX_t=&F(t,X_t)dt+\sigma(t,X_t)dW_t\,,\\ dY_t=&G(t,Y_t)dt+\eta(t,Y_t)dW_t\,,\notag \end{align}\tag{1}\] driven by a \(d\)-dimensional Wiener process \(W_t\); we will view \(X_t\) as the baseline model and \(Y_t\) as a perturbation. Our goal is to compare observables (i.e., measurable functionals of the paths) of the two processes, where we allow both the diffusion and drift to be perturbed. Previous information-theoretic uncertainty quantification (UQ) methods [1]–[10] are effective at producing sensitivity bounds for stochastic systems that are well-behaved as \(t\to\infty\) (e.g., when applied to time averages), however they rely on information-theoretic divergences (e.g., the KL-divergence) which require absolute continuity to produce non-trivial results. Allowing \(X_t\) and \(Y_t\) to have differing diffusion terms, \(\eta\neq \sigma\), implies that the distributions of their solutions on path-space are not absolutely continuous, thus preventing us from bounding the difference in expectations under \(X_t\) and \(Y_t\) via a direct application of these previous information-theoretic methods. In this work we develop an approach which circumvents this issue via the introduction of auxiliary SDEs coupled to \(X_t\) and \(Y_t\), followed by a combination of stochastic calculus estimates, functional inequalities, and information-theoretic UQ bounds.

Under appropriate ergodicity assumptions on the baseline model \(X_t\) (in the form of \(\log\)-Sobolev inequalities) our results imply uniform-in-time sensitivity bounds on the time-average of a function \(h(t,x)\). Specifically, in Theorem 5 we obtain bounds of the form \[\begin{align} \label{eq:time95avg95bounds95preview} \sup_{T>0}\left| \mathbb{E}\left[\frac{1}{T}\int_0^T h(t,Y_t)dt\right]- \frac{1}{T}\int_0^T E_{\mu_X^*}[h_t] dt \right|=O\left(\|F-G\|_\infty+\|\sigma-\eta\|_\infty\right) \,, \end{align}\tag{2}\] where \(h_t\mathrel{\vcenter{:}}= h(t,\cdot)\), assuming that the drift and diffusion for \(X_t\) are time-homogeneous and the process is started in an invariant distribution, \(X_0\sim\mu_{X}^*\). In particular, 2 implies uniform convergence of time averages as the perturbations approach zero. We note that if one is interested in initial conditions other than the invariant distribution, our results can be combined with well-established methods for proving convergence to the invariant distribution.

If the drift and diffusion for both \(X_t\) and \(Y_t\) are time-homogeneous and they are started in invariant distributions, \(\mu_{X}^*\) and \(\mu_{Y}^*\) respectively, then in Theorem 7 we derive a bound on the \(1\)-Wasserstein distance between these invariant distributions, \[\begin{align} \label{eq:195Wasserstein95bounds95preview} W_1(\mu_{Y}^*,\mu_{X}^*)=O\left(\|F-G\|_\infty+\|\sigma-\eta\|_\infty\right) \,. \end{align}\tag{3}\] By considering Ornstein–Uhlenbeck (OU) processes, we show that the linear scaling of 3 in \(\|F-G\|_\infty\) and in \(\|\sigma-\eta\|_\infty\) are both optimal.

Our method can also be used to obtain sensitivity bounds on exponentially discounted observables of time-inhomogeneous SDEs, which are of interest in applications to economics, see, e.g., [11], [12], and control theory, see, e.g., [13]. Specifically, in Theorem 9 we derive bounds of the form \[\begin{align} \label{eq:discounted95cost95bounds95preview} \left| \mathbb{E}\left[\int_0^\infty h(t,Y_t) e^{-rt}dt\right]-\mathbb{E}\left[\int_0^\infty h(t,X_t) e^{-rt}dt\right]\right|=O\left(\|F-G\|_\infty+\|\sigma-\eta\|_\infty\right) \,. \end{align}\tag{4}\]

Finally, our method produces sensitivity bounds for solutions to linear parabolic PDEs of the form \[\begin{align} \label{eq:Feynman95Kac95pde95intro} &\partial_t u(t,x)=-A[u_t](t,x)-h(t,x)\,,\,\,\,\,u(T,x)=f(x)\,,\\ &A[u](t,x)\mathrel{\vcenter{:}}=\frac{1}{2}\sum_{i,j,k}\sigma^i_k(t,x)\sigma^j_k(t,x)\partial_{i}\partial_j u(x)+\sum_iF^i(t,x)\partial_iu(x)\,.\notag \end{align}\tag{5}\] More specifically, if \(u(t,x)\) solves 5 and \(v(t,x)\) solves 5 but with \(G\) in place of \(F\) and \(\eta\) in place \(\sigma\) then, in Theorem 11, we derive \(L^1\) error bounds of the form \[\begin{align} \label{eq:PDE95bounds95preview} \|v(t,\cdot)-u(t,\cdot)\|_{L^1(\mu_X^*)}= O\left(\|F-G\|_\infty+\|\sigma-\eta\|_\infty\right) \end{align}\tag{6}\] in the case where the SDE for \(X_t\) is time-homogeneous and has an invariant distribution, \(\mu_X^*\).

We provide an outline of the key ideas of our technique in Section 2. Applications of these results are explored in Section 3. Details proofs of the underlying estimates can be found in the Section 4. While we have previewed our results in the case of bounded perturbations, we emphasize that our method can still produce meaningful bounds when \(\|F-G\|_\infty=\infty\). In addition, we emphasize that our theorems are non-asymptotic in nature; below we provide explicit non-asymptotic formulas for the bounds. Our results do require \(\sigma-\eta\) to be bounded and \(\sigma\) to be invertible; thus the approach developed here requires the baseline model to be a uniformly elliptic diffusion. Generalizing beyond these assumptions is a task we leave for future work.

1.1 Related Work↩︎

The present work is related to a large body of research on perturbation bounds, also called sensitivity or condition number bounds, for Markov processes and their invariant measures. The majority of prior studies have focused on Markov chains on a discrete state-space [14]–[24], but some authors have also studied perturbation bounds for Markov chains on a countable state-space [25], [26] or a general Polish state space [27], and for continuous time, countable state-space Markov processes [28], [29]. Of particular relevance to this work are the approaches to sensitivity analysis for SDEs with perturbed drift [8], [30], [31] and especially the aforementioned information-theoretic uncertainty quantification (UQ) approach to sensitivity bounds for stochastic systems [1]–[10]. However, to the best of the author’s knowledge, sensitivity bounds of the type obtained here for SDEs with perturbed diffusion were out of reach of established techniques. Thus this work constitutes a novel method for obtaining sensitivity bounds on SDEs with simultaneously perturbed diffusion and drift.

2 Outline of the Proof↩︎

In this section we provide an outline of the key ideas underpinning our technique for deriving bounds of the form 2 , 3 , and 4 . The key new tool that enables consideration of perturbations to the diffusion is an appropriately defined auxiliary process \(\tilde{X}_t\) that has the same diffusion term as \(X_t\), and is constructed so that observables of \(\tilde{X}_t\) are close to those of both \(X_t\) and \(Y_t\). More specifically, we introduce two auxiliary processes \(\tilde{Y}_t\) and \(\tilde{X}_t\), coupled to \(X_t\) and \(Y_t\) respectively, as follows: \[\begin{align} dX_t=&F(t,X_t)dt+\sigma(t,X_t)dW_t\,,\tag{7}\\ d\tilde{Y}_t=&\left(G(t,\tilde{Y}_t)+\eta(t,\tilde{Y}_t)\sigma^{-1}(t,X_t)\left(F(t,X_t)-G(t,X_t)+\kappa(X_t-\tilde{Y}_t)\right)\right)dt +\eta(t,\tilde{Y}_t)dW_t \,,\tag{8}\\ {}\notag\\ d\tilde{X}_t=&\left(G(t,\tilde{X}_t)-\kappa(\tilde{X}_t-Y_t)\right)dt+\sigma(t,\tilde{X}_t)dW_t \,,\tag{9}\\ dY_t=&G(t,Y_t)dt+\eta(t,Y_t)dW_t\,, \tag{10} \end{align}\] where the coupling strength, \(\kappa\in[0,\infty)\), will be chosen later. As systems of SDEs, \((X_t,\tilde{Y}_t)\) and \((\tilde{X}_t,Y_t)\) have the same diffusion terms and the difference in drift is engineered so that Girsanov’s theorem implies that the distributions of \((X,\tilde{Y})|_{[0,T]}\) and \((\tilde{X},Y)|_{[0,T]}\) are absolutely continuous and leads to a formula for the relative entropy. The coupling between \(\tilde{X}_t\) and \(Y_t\) in Eq. 9 forces them to remain close pathwise, despite the difference in their diffusion terms. The average size of the coupling term can be thought of as the cost of replacing \(\eta\) with \(\sigma\) and will be shown to scale with the size of \(\eta-\sigma\). The derivation will then proceed as follows.

1. First we obtain bounds on the difference between observables of \(\tilde{X}_t\) and \(Y_t\) by using stochastic calculus estimates to compare these processes pathwise, assuming a sufficiently strong coupling strength \(\kappa\) in 9 . An effective pathwise comparison necessitates using the same driving-noise process, \(W_t\), in the equations for both \(\tilde{X}_t\) and \(Y_t\), as indicated in 9 10 .

2. Next we use the information-theoretic UQ method (see the references in Section 1.1 and especially [3], [8]) to bound the difference in expectations of observables of \(\{X_t\}_{t\in[0,T]}\) and \(\{\tilde{X}_t\}_{t\in[0,T]}\). The resulting bound is determined by two contributions.

   Moment Generating Function: The first involves the moment generating function (MGF) of the observable under the baseline process \(X_t\). For this we will employ standard tail-bounds (sub-Gaussian and sub-exponential bounds). The case of time-averaged observables requires additional care to ensure the bounds behave well as \(T\to\infty\). In that case we build on the strategy from [32]–[34] for obtaining concentration inequalities, and which was previously used in [8] to obtain sensitivity bounds for SDEs with perturbed drift. Here we generalize this technique to time-dependent functions, \(h(t,x)\), by combining it with the Feynman-Kac semigroup bound from [35].

   Relative Entropy: The second contribution to the information-theoretic UQ bound is the relative entropy between the distributions of \(\{(X_t,\tilde{Y}_t)\}_{t\in[0,T]}\) and \(\{(\tilde{X}_t,Y_t)\}_{t\in[0,T]}\) on path space. This will be bounded using Girsanov’s theorem, as was done in [3] for SDEs with perturbed drift; here it is key that the two systems have the same diffusion terms. We note that, as both components of each system must use the same driving noise process, the change of measure in Girsanov’s theorem necessarily changes the drift of both components when transforming from 7 8 to 9 10 . The form of Eq. 8 is designed to accommodate this change in drift, but does not otherwise directly impact any of the required bounds. Girsanov’s theorem leads to a formula for the relative entropy that has two main contributions, the first coming from the difference in drifts \(F-G\) and the second involving the size of the coupling term; the latter will be bounded via the estimates from stage [proof95step1] of the proof. In addition, the estimates from stage [proof95step1] are needed to prove Novikov’s condition, which justifies the use of Girsanov’s theorem; the change in drift inherently involves the coupling term, which is unbounded, thus making this justification non-trivial.

The above two-stage method is capable of producing sensitivity bounds that are well-behaved at large \(T\), as previewed in 2 . We emphasize that a more naive direct comparison of \(X_t\) and \(Y_t\) using the type of estimates described in stage [proof95step1] above leads to bounds that grow exponentially in \(T\), while the information-theoretic UQ methods are not directly applicable to 1 due to the difference in diffusion terms. Thus our approach crucially relies on the auxiliary processes \(\tilde{X}_t,\tilde{Y}_t\) and the combination of the two techniques described above.

3 Applications↩︎

In this section we show how our results can be applied to derive sensitivity bounds for time-averaged and discounted observables as well as for linear parabolic PDEs. We will work with SDEs satisfying the assumptions below. Additional required assumptions and proof details for the underlying estimates are given in Section 4.

3.1 Time-Averaged Observables↩︎

First we detail the results of our method when applied to time-averaged observables, assuming that the SDE for \(X_t\) is time homogeneous and starts in an invariant distribution, \(\mu_X^*\). For definitions of the sub-Gaussian and Bernstein MGF bounds used below see 26 and 27 respectively. For further background on such MGF bounds, including tools for obtaining them, see, e.g., Chapter 2 in [36], Chapter 2 in [38] and Chapter 5 in [39]. Here and in the following we define \(\|\sigma-\eta\|_{F,\infty}\mathrel{\vcenter{:}}=\sup_{t\geq 0,x\in\mathbb{R}^d}\|\sigma(t,x)-\eta(t,x)\|_{F}\).

We emphasize that our method produces a uniform-in-time bound ?? . A more elementary argument based on directly comparing the SDEs for \(X_t\) and \(Y_t\) via Gronwall’s inequality would produce bounds that grow exponentially in time and would become uninformative as \(T\to\infty\); similar comments apply to the other applications in this work.

3.1.1 Wasserstein Distance Between Invariant Distributions↩︎

When the SDE for \(Y_t\) is also time-homogeneous and is started in an invariant distribution, by optimizing over all \(1\)-Lipschitz functions in Theorem 5, we obtain the following bound on the \(1\)-Wasserstein distance between the invariant distributions in terms of the difference between the diffusions and drifts. In particular, this result implies 3 .

By considering the special case of OU processes, we can show that the linear scaling of ?? in \(\|F-G\|_\infty\) and in \(\|\sigma-\eta\|_{F,\infty}\) are both optimal. Consider the linear SDEs on \(\mathbb{R}\), \[\begin{align} dX_t=&-\frac{1}{2}(X_t-\mu_X) dt+\sigma dW_t\,,\,\,\,\,\,\,\, dY_t=-\frac{1}{2}(Y_t-\mu_Y) dt+\eta dW_t\,, \end{align}\] whose respective invariant distributions are the normal distributions \(\mu_{X}^*=N(\mu_X,\sigma^2)\), \(\mu_{Y}^*=N(\mu_Y,\eta^2)\). Now consider the following two cases.

1. \(\sigma=\eta\): In this case, using the Kantorovich-Rubinstein formula for \(W_1\), we lower bound the \(1\)-Wasserstein distance by the difference in expected values of the \(1\)-Lipschitz function \(h(x)=x\), which gives \[\begin{align} W_1(\mu_Y^*,\mu_X^*)\geq |E_{N(\mu_Y,\sigma^2)}[h]-E_{N(\mu_X,\sigma^2)}[h]|=|\mu_Y-\mu_X|=2\|F-G\|_\infty\,. \end{align}\] Hence the \(O(\|F-G\|_\infty)\) bound in ?? is the optimal rate.

2. \(\mu\mathrel{\vcenter{:}}=\mu_X=\mu_Y\): In light of Remark 6, we should not expect the bounds to hold uniformly in \(\sigma\), hence we consider a fixed value of \(\sigma\) and will derive a lower bound on the \(1\)-Wasserstein distance that scales with \(|\eta-\sigma|\) as \(\eta\to \sigma\). First recall the formula \(W_1(Q,P)=\int_{-\infty}^\infty|F_Q(t)-F_P(t)|dt\) for the \(1\)-dimensional \(1\)-Wasserstein metric [40], where \(F_Q\) and \(F_P\) denote the cumulative distribution functions of \(Q\) and \(P\) respectively. Using this along with Taylor’s theorem with integral remainder we can compute \[\begin{align} &W_1(\mu_{Y}^*,\mu_X^*)=\frac{1}{2}\int_{-\infty}^\infty\left|\mathrm{erfc}\left(\frac{t-\mu}{\sqrt{2}\sigma}\right)-\mathrm{erfc}\left(\frac{t-\mu}{\sqrt{2}\eta}\right)\right|dt\\ \geq & \left(\frac{2}{\pi}\right)^{1/2} \frac{\eta}{\sigma}|\sigma-\eta| \notag\\ & - \frac{1}{(2\pi)^{1/2}} (\sigma^{-1}-\eta^{-1})^2\left( \eta^{-1}+\sigma^{-1}\right)\int_{-\infty}^\infty \exp\left(- \left(\frac{1}{4\eta^2}+\frac{1}{2}\left(\frac{1}{\sigma}-\frac{1}{\eta}\right)^2\right)u^2\right) |u|^3 du\notag\\ =&\left(\frac{2}{\pi}\right)^{1/2} |\sigma-\eta| +O( |\sigma-\eta| ^2)\notag \end{align}\] as \(\eta\to \sigma\). Hence the \(O(\|\sigma-\eta\|_{F,\infty})\) bound in ?? is the optimal rate.

3.2 Discounted Observables↩︎

Next we detail the result of our method when applied to discounted observables. We emphasize that in this case we do not make any assumption regarding invariant initial distributions or even time-homogeneity nor do we require the use of functional inequalities; these are not necessary when utilizing a weight measure \(\rho(dt)\) whose tail decays sufficiently fast as \(t\to\infty\).

3.3 Sensitivity Bounds for Linear Parabolic PDEs↩︎

As our final application, we derive perturbation bounds for classical solutions to linear parabolic PDEs of the form 5 by applying our method to the Feynman-Kac formula; in particular, the following result implies 6 . We focus on perturbations to the drift, \(F\), and especially the diffusion, \(\sigma\), as perturbations to the other terms in the PDE are relatively trivial to handle since they do not change the underlying SDE.

We note that a large portion of the above argument can also be applied to \[\begin{align} {u}(t,x)\mathrel{\vcenter{:}}=&\mathbb{E}\left[f({X}^{t,x}_T) \exp\left(\int_t^T k(s,{X}_s^{t,x}) ds\right)+\int_t^T h(s,{X}_s^{t,x}) \exp\left(\int_t^s k(r,{X}^{t,x}_r)dr\right) ds\right]\,, \end{align}\] which, under appropriate assumptions, solves the PDE \[\begin{align} \partial_t u(t,x)=-A[u_t](t,x)-k(t,x)u(t,x)-h(t,x)\,,\,\,\,u(T,x)=f(x)\,. \end{align}\] However, effectively handling the corresponding MGF so that the bounds scale optimally in \(T-t\) requires techniques beyond those discussed in Section 4.2.1.

4 Proofs of Key Estimates↩︎

In this section we provide detailed proofs of the bounds that underpin the applications in Section 3 and which were outlined in Section 2.

4.1 Pathwise Comparison of \(\tilde{X}_t\) and \(Y_t\)↩︎

In the first stage of the proof, we show that the coupling term between \(\tilde{X}_t\) and \(Y_t\) is sufficient to keep them close to one another, with error that depends on the difference in diffusion. Working under Assumption 1, for \(K\in\mathbb{R}\), \(n\in\mathbb{Z}^+\), an application of Itô’s formula together with Assumption 1 . [assump:G95decomp] applied to \(Z_t\mathrel{\vcenter{:}}=\tilde{X}_t-Y_t\) yields \[\begin{align} \label{eq:Z952n95Ito} e^{Kt}\|Z_t\|^{2n}=&\|Z_0\|^{2n}+\int_0^t Ke^{Ks}\|Z_s\|^{2n}ds\\ &+\int_0^t2n e^{Ks} \|Z_s\|^{2(n-1)} (Z_s\cdot(G(s,\tilde{X}_s)-G(s,Y_s)) -\kappa \|Z_s\|^2)ds\notag\\ &+\int_0^t 2ne^{Ks} \|Z_s\|^{2(n-1)}Z_s\cdot (\sigma(s,\tilde{X}_s)-\eta(s,Y_s) )dW_s\notag\\ &+\int_0^t 2n(n-1)e^{Ks}\|Z_s\|^{2(n-2)} \|(\sigma(s,\tilde{X}_s)-\eta(s,Y_s))^T Z_s\|^2ds\notag\\ &+\int_0^t ne^{Ks}\|Z_s\|^{2(n-1)}\|\sigma(s,\tilde{X}_s)-\eta(s,Y_s)\|_F^2ds\notag\\ \leq&\|Z_0\|^{2n}- \left( 2n(\kappa+C_G-L_{\tilde{G}} )-K\right)\int_0^t e^{Ks} \|Z_s\|^{2n}ds\notag\\ &+\int_0^t 2ne^{Ks} \|Z_s\|^{2(n-1)}Z_s\cdot (\sigma(s,\tilde{X}_s)-\eta(s,Y_s) )dW_s\notag\\ &+\int_0^t 2n(n-1)e^{Ks}\|Z_s\|^{2(n-2)} \|(\sigma(s,\tilde{X}_s)-\eta(s,Y_s))^T Z_s\|^2ds\notag\\ &+\int_0^t ne^{Ks}\|Z_s\|^{2(n-1)}\|\sigma(s,\tilde{X}_s)-\eta(s,Y_s)\|_F^2ds\notag\,. \end{align}\tag{20}\] Based on Remark 4, we have \(\mathbb{E}\left[\int_0^t (2ne^{Ks} \|Z_s\|^{2(n-1)} \|(\sigma(s,\tilde{X}_s)-\eta(s,Y_s) )^TZ_s\|)^2 ds\right]<\infty\) for all \(t\) and hence \(\int_0^t 2ne^{Ks} \|Z_s\|^{2(n-1)}Z_s\cdot (\sigma(s,\tilde{X}_s)-\eta(s,Y_s) )dW_s\) is a martingale; see, e.g., Chapter 3 in [37]. In particular, it has expected value zero.

In the \(n=1\) case, taking the expectation and using the martingale property, we obtain \[\begin{align} e^{Kt}\mathbb{E}[\|Z_t\|^{2}]\leq & \mathbb{E}[\|Z_0\|^{2}]- \left( 2(\kappa+C_G-L_{\tilde{G}} )-K\right)\int_0^t e^{Ks} \mathbb{E}[\|Z_s\|^{2}]ds\\ &+\int_0^t e^{Ks} \mathbb{E}\left[\|\sigma(s,\tilde{X}_s)-\eta(s,Y_s)\|_F^2\right]ds\notag\\ \leq & \mathbb{E}[\|Z_0\|^{2}]- \left( 2(\kappa+C_G-L_{\tilde{G}} - (1+\epsilon)L_{\sigma}^2/2 )-K\right) \int_0^te^{Ks} \mathbb{E}[\|Z_s\|^2]ds\notag\\ &+(1+1/\epsilon) \int_0^te^{Ks} \mathbb{E}\left[\|\sigma(s,Y_s)-\eta(s,Y_s)\|_F^2\right]ds\notag \end{align}\] for all \(\epsilon>0\). With \(\epsilon,\kappa\) chosen according to Assumption 1 . [assump:epsilon95lambda] and letting \(K=K_{\kappa,\epsilon}\) we have \[\begin{align} \label{eq:E95Z95295bound} \mathbb{E}[\|Z_t\|^{2}]\leq&\mathbb{E}[\|Z_0\|^{2}]e^{-K_{\kappa,\epsilon}t} +(1+1/\epsilon)\mathbb{E}\left[\int_0^te^{-K_{\kappa,\epsilon}(t-s)}\|\sigma(s,Y_s)-\eta(s,Y_s)\|_F^2ds\right]\,. \end{align}\tag{21}\] In particular, we have the following bound in terms of the uniform norm of \(\sigma-\eta\): \[\begin{align} \label{eq:E95Z95295bound95uniform} \mathbb{E}[\|Z_t\|^{2}]\leq&\mathbb{E}[\|Z_0\|^{2}]e^{-K_{\kappa,\epsilon}t}+\frac{(1+1/\epsilon)\|\sigma-\eta\|_{F,\infty}^2}{K_{\kappa,\epsilon}}(1-e^{-K_{\kappa,\epsilon}t})\,. \end{align}\tag{22}\]

While the \(n=1\) case 22 explicitly enters into our sensitivity bounds, to justify the use of Girsanov’s theorem in Section 4.2.2 via a Novikov-type condition we will also require sufficient control on \(\mathbb{E}[\|Z_t\|^{2n}]\) for all \(n\in\mathbb{Z}^+\). To that end, we take the expected value of 20 and use the martingale property to compute \[\begin{align} &e^{Kt}\mathbb{E}[\|Z_t\|^{2n}]\leq\mathbb{E}[\|Z_0\|^{2n}]- \left( 2n(\kappa+C_G-L_{\tilde{G}} )-K\right)\int_0^t e^{Ks} \mathbb{E}[\|Z_s\|^{2n}]ds\\ &+\int_0^t 2n(n-1)e^{Ks}\mathbb{E}[\|Z_s\|^{2(n-2)} \|(\sigma(s,\tilde{X}_s)-\eta(s,Y_s))^T Z_s\|^2]ds\notag\\ &+\int_0^t ne^{Ks}\mathbb{E}[\|Z_s\|^{2(n-1)}\|\sigma(s,\tilde{X}_s)-\eta(s,Y_s)\|_F^2]ds\notag\\ \leq &\mathbb{E}[\|Z_0\|^{2n}]-(2n(\kappa+C_G -L_{\tilde{G}}) -K)\int_0^t e^{Ks} \mathbb{E}[\|Z_s\|^{2n}]ds\notag\\ &+n^2(2\sup_{s,x,y}\|\sigma(s,x)-\eta(s,y)\|_2^2 +\sup_{s,x,y}\|\sigma(s,x)-\eta(s,y)\|_F^2)\!\int_0^t e^{Ks}\mathbb{E}[\|Z_s\|^{2(n-1)}]ds\,.\notag \end{align}\] Note that here a much rougher bound on the diffusion terms will suffice. Letting \(K= 2n(\kappa+C_G-L_{\tilde{G}})\) we obtain \(f_n(t)\leq b_n+a_n\int_0^t e^{\tau s}f_{n-1}(s)ds\) for all \(n\in\mathbb{Z}^+\), \(t\geq0\), where \[\begin{align} &f_n(t)\mathrel{\vcenter{:}}= e^{n\tau t}\mathbb{E}[\|Z_t\|^{2n}]\,,\,\,\,\,\tau\mathrel{\vcenter{:}}= 2(\kappa+C_G-L_{\tilde{G}})\,,\,\,\,\,b_n\mathrel{\vcenter{:}}=\mathbb{E}[\|Z_0\|^{2n}]\,,\\ &a_n\mathrel{\vcenter{:}}= n^2(2\sup_{s,x,y}\|\sigma(s,x)-\eta(s,y)\|_2^2 +\sup_{s,x,y}\|\sigma(s,x)-\eta(s,y)\|_F^2)\mathrel{\vcenter{:}}= n^2 D_{\sigma,\eta}\,. \end{align}\] Functions that satisfy such a recursive sequence of inequalities can be bounded as follows.

Applying Lemma 1 together with Assumption 1 . [assump:ICS] yields \[\begin{align} e^{n\tau t}\mathbb{E}[\|Z_t\|^{2n}]\leq& \sum_{j=0}^n \frac{\prod_{k=j+1}^n a_k}{(n-j)! \tau^{n-j}} b_j e^{(n-j)\tau t} \\ \leq & \alpha n!\sum_{j=0}^n \frac{n!}{j!(n-j)!} \beta^j (e^{\tau t} D_{\sigma,\eta}/\tau)^{n-j} = \alpha n!(\beta+e^{\tau t} D_{\sigma,\eta}/\tau)^n\,,\notag \end{align}\] where we used Assumption 1 . [assump:ICS]. Therefore we can conclude \[\begin{align} \label{eq:E95Z952n95final95bound} \mathbb{E}[\|Z_t\|^{2n}]\leq& \alpha n!\left(\beta e^{-2(\kappa+C_G-L_{\tilde{G}})t}+ \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)^n\,\,\,\, \text{ for all n\in\mathbb{Z}^+, t\geq 0.} \end{align}\tag{23}\]

In addition to bounds on \(Z_t\), we will require the following bounds on the moments of \(\tilde{X}_t\). \[\begin{align} \label{eq:E95X95295bound} &\mathbb{E}[\|\tilde{X}_t\|^{2}] \leq \mathbb{E}[\| X_0\|^{2}] e^{-\tilde{K}_{\kappa,\delta}t}\\ & +\left(2A_G+ \|\sigma\|_{F,\infty}^2 +\frac{\kappa}{\delta } \left(\mathbb{E}[\|Z_0\|^{2}] +\frac{(1+1/\epsilon)\|\sigma-\eta\|_{F,\infty}^2}{K_{\kappa,\epsilon}} \right) \right)\frac{1-e^{-\tilde{K}_{\kappa,\delta}t}}{\tilde{K}_{\kappa,\delta}}\notag \end{align}\tag{24}\] and, for all \(t>0\), \(n\in\mathbb{Z}^+\), \[\begin{align} \label{eq:f95n95bound95X95tilde95final} \mathbb{E}[\|\tilde{X}_t\|^{2n}] \leq& n! \left(\tilde{\alpha} + \alpha /2 \right) \left(\frac{D_{\sigma,G} }{ B_G}+\max\left\{\tilde{\beta} , (2\kappa/B_G)^{2 } \left(\beta + \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)\right\}\right)^n\notag\,. \end{align}\tag{25}\] The proofs are extremely similar to the above calculations; see Appendix 6.2 for details.

4.1.1 Observables Under \(\tilde{X}_t\) Compared to \(Y_t\)↩︎

We are now prepared to show that observables of \(\tilde{X}_t\) and \(Y_t\) are close to one another in several cases. First we consider time-averaged and discounted observables, defined in terms of a function \(h(t,x)\) satisfying the following.

In the time-averaged case, the above computations imply the following.

Next we consider the case of discounted observables; for simplicity, and because we view it to be the primary case of interest in such applications, we assume the initial conditions are equal. To avoid making additional integrability assumptions (regarding the limit as \(T\to\infty\)), we phrase the result as follows.

Finally, we consider the observables that are related to the linear parabolic PDE 5 .

4.2 Information-Theoretic UQ Bound Comparing \(\tilde{X}_t\) and \(X_t\)↩︎

We now proceed to show that observables of \(\tilde{X}_t\) and \(X_t\) are close to one another by combining information-theoretic UQ bounds with functional inequalities, an approach originally developed in [8]. This approach requires that the two SDEs being compared share the same diffusion term. Hence the first stage of our method provides the key to transforming the problem into one that is within reach of such techniques. In this section we generalize the method from [8] to cover time-dependent observables by using the bounds on the Feynman-Kac semigroup with time-dependent potential derived in [35].

Start by applying the information-theoretic UQ bound given in Section 2.2 of [3] to an observable (measurable function) on path-space, \(g:C([0,T],\mathbb{R}^d)\to\mathbb{R}\), and the distributions of \(\{\tilde{X}_t\}_{t\in[0,T]}\) and \(\{X_t\}_{t\in[0,T]}\); we denote the corresponding distributions by \(P_{\tilde{X}|_{[0,T]}}\) and \(P_{{X}|_{[0,T]}}\) respectively and expectations with respect to these will be denoted by \(E_{\tilde{X}|_{[0,T]}}\) and \(E_{{X}|_{[0,T]}}\). With this notation, the UQ bound is given by the following.

In ?? , \(\mathrm{KL}\) denotes the KL-divergence, i.e., relative entropy, defined by \(\mathrm{KL}(Q\|P)\mathrel{\vcenter{:}}= E_{Q}[\log(dQ/dP)]\) if \(Q\ll P\) and \(\mathrm{KL}(Q\|P)\mathrel{\vcenter{:}}=\infty\) otherwise.

The objective of the minimization on the right-hand sides of ?? consists of two terms:

(a) The first involves the centered MGF of the observable, \(g\). For this term we will employ standard tail-bounds, i.e., sub-Gaussian and sub-exponential. The case of time-averaged observables requires additional care to ensure the bounds behave well as \(T\to\infty\). We address this using \(\log\)-Sobolev inequalities, as discussed in Section 4.2.1. This technique also leads to sharper sensitivity bounds for parabolic PDEs.

(b) The second term involves the relative entropy between the distributions on path-space of the solutions to the SDEs; in particular, it does not depend on the choice of observable. We discuss this term further in Section 4.2.2.

4.2.1 MGF and UQ Bound for Time-Averaged Observables via \(\log\)-Sobolev Inequalities↩︎

Here we derive a bound on the centered MGF of time-averaged observables which is well-behaved in the limit as \(T\to\infty\). We will work under the following additional assumptions.

To bound the MGF term in ?? for the observable \(g(\gamma)=\int_0^T h(t,\gamma_t)dt\) we utilize the approach developed in [35] for bounding the Feynman-Kac semigroup with a time-dependent potential. Specifically, we require a variant of Theorem 1 from [35]. The setting under consideration here allows for a streamlined proof; for completeness, we provide the details in Appendix 6.3. The following two lemmas further specialize this result to two important cases; sub-Gaussian and sub-exponential observables, though other forms of tail behavior could similarly be analyzed.

First we consider sub-Gaussian observables; given \(\sigma_{\mathrm{SG}}\in(0,\infty)\), recall that a random variable \(g\) is \(\sigma_{\mathrm{SG}}\)-sub-Gaussian with respect to a distribution \(Q\) if \[\begin{align} \label{eq:sub95Gauss95def} E_Q\left[ \exp\left(\lambda (g-E_Q[g])\right)\right]\leq \exp\left(\lambda^2\sigma_{\mathrm{SG}}^2/2\right) \, \text{ for all }\,\lambda\in\mathbb{R}\,. \end{align}\tag{26}\] In particular, this case applies to bounded \(h\); see, e.g., Chapter 2 in [36] for background.

Next we consider sub-exponential observables. More specifically, we consider those satisfying a Bernstein-type MGF bound. Given \(\sigma_B,b\in(0,\infty)\), we say that a random variable \(g\) satisfies a \((\sigma_B,b)\)-Bernstein MGF bound with respect to a distribution \(Q\) if \[\begin{align} \label{eq:Bernstein95def} E_Q\left[ \exp\left(\lambda (g-E_Q[g])\right)\right]\leq \exp\left(\frac{\lambda^2\sigma_B^2}{2(1-b|\lambda|)}\right) \, \text{ for all }\,|\lambda|<1/b\,. \end{align}\tag{27}\] Such bounds can be proven via the Bernstein condition; see Proposition 2.10 in [38].

4.2.2 Relative Entropy Bound via Girsanov’s Theorem↩︎

To bound the relative entropy term in ?? , we utilize Girsanov’s theorem together with the moment bounds derived in Section 4.1. We will require the following additional assumptions on the diffusion for \(X_t\).

5 Conclusion↩︎

In this work we have used a combination of stochastic calculus estimates, functional inequalities, and information-theoretic UQ bounds to derive novel sensitivity bounds on the solutions to SDEs, allowing for perturbation of both the diffusion and the drift. To the best of the author’s knowledge, these results are the first of this type that allow for perturbation of the diffusion and which produce results that are well-behaved at large times; in particular, previous information-theoretic sensitivity bounds only apply to perturbations to the drift. Several applications of our new method were studied: 1) sensitivity bounds on time-averaged and exponentially discounted observables, 2) bounds on the \(1\)-Wasserstein distance between invariant distributions of the baseline and perturbed SDEs, 3) sensitivity bounds on solutions to linear parabolic PDEs. By examining invariant distributions of OU processes, we showed that our method produces bounds that scale optimally in the size of the perturbation to the diffusion and to the drift. Currently, the method is restricted to the case of a uniformly-elliptic baseline SDE; the investigation of more general cases (e.g., underdamped Langevin dynamics) is left for future work.

6 Additional Proofs↩︎

Below we provide additional proof details regarding several results discussed in the main text.

6.1 Theorem 5: Bernstein MGF Bound Case↩︎

In this section we provide details regarding the proof of ?? from Theorem 5 under Bernstein MGF bound assumptions.

Starting from the UQ bound 12 , we apply the Bernstein MGF bound from Lemma 7 to obtain

\[\begin{align} &\left|\mathbb{E}\left[\frac{1}{T}\int_0^T h(t,\tilde{X}_t)dt\right]- \frac{1}{T}\int_0^T E_{\mu_X^*}[h_t] dt \right|\\ \leq& \inf_{c\in(0,1/(C_*b)}\left\{\frac{1}{T} \frac{ c C_*}{2(1-c C_*b)} \int_0^T \sigma_{h_t}^2dt+\frac{1}{cT}\mathrm{KL}\left(P_{\tilde{X}|_{[0,T]}}\|P_{{X}|_{[0,T]}}\right)\right\}\,.\notag \end{align}\] The optimization over \(c\) can be computed exactly to yield \[\begin{align} &\left|\mathbb{E}\left[\frac{1}{T}\int_0^T h(t,\tilde{X}_t)dt\right]- \frac{1}{T}\int_0^T E_{\mu_X^*}[h_t] dt \right|\\ \leq& \left(2 C_* \frac{1}{T}\int_0^T \sigma_{h_t}^2dt \frac{1}{T} \mathrm{KL}\left(P_{\tilde{X}|_{[0,T]}}\|P_{{X}|_{[0,T]}}\right)\right)^{1/2} +\frac{C_*b}{T}\mathrm{KL}\left(P_{\tilde{X}|_{[0,T]}}\|P_{{X}|_{[0,T]}}\right)\,.\notag \end{align}\] Combining this with 11 and 13 completes the proof.

6.2 Bounds on \(\tilde{X}_t\)↩︎

In addition to the bounds on the moments of \(Z_t=\tilde{X}_t-Y_t\) that were derived in Section 4.1, we will require bounds on the moments of \(\tilde{X}_t\); we continue to work under Assumption 1. Similarly to the computations in Section 4.1, for \(\tilde{K}\in\mathbb{R}\), \(n\in\mathbb{Z}^+\) we can use Itô’s formula along with Assumption 1 . [assump:G95confining] to compute \[\begin{align} \label{eq:Ito95tilde95X95app} &e^{\tilde{K}t}\|\tilde{X}_t\|^{2n}\\ =&\|\tilde{X}_0\|^{2n}+\int_0^t \tilde{K}e^{\tilde{K}s} \|\tilde{X}_s\|^{2n}ds +2n\int_0^te^{\tilde{K}s}\|\tilde{X}_s\|^{2(n-1)}\tilde{X}_s\cdot\sigma(s,\tilde{X}_s)dW_s\notag\\ &+2n\int_0^te^{\tilde{K}s}\|\tilde{X}_s\|^{2(n-1)}\tilde{X}_s\cdot (G(s,\tilde{X}_s)-\kappa(\tilde{X}_s-Y_s))ds\notag\\ &+\frac{1}{2}\sum_{i,j,k}\int_0^t e^{\tilde{K}s}\left(2n(n-1)\|\tilde{X}_s\|^{2(n-2)}2 \tilde{X}_s^j\tilde{X}_s^i+2n\|\tilde{X}_s\|^{2(n-1)}\delta_{ij}\right) \sigma_k^i(s,\tilde{X}_s)\sigma_k^j(s,\tilde{X}_s) ds\notag\\ \leq&\|\tilde{X}_0\|^{2n}+\int_0^t \tilde{K}e^{\tilde{K}s} \|\tilde{X}_s\|^{2n}ds+2n\int_0^te^{\tilde{K}s}\|\tilde{X}_s\|^{2(n-1)}\tilde{X}_s\cdot\sigma(s,\tilde{X}_s)dW_s\notag\\ &+2n\int_0^te^{\tilde{K}s}\|\tilde{X}_s\|^{2(n-1)}( A_G-B_G\|\tilde{X}_s\|^2+\kappa \|\tilde{X}_s\|\|Z_s\|)ds\notag\\ &+\frac{1}{2}\sum_{i,j,k}\int_0^t e^{\tilde{K}s}\left(2n(n-1)\|\tilde{X}_s\|^{2(n-2)}2 \tilde{X}_s^j\tilde{X}_s^i+2n\|\tilde{X}_s\|^{2(n-1)}\delta_{ij}\right) \sigma_k^i(s,\tilde{X}_s)\sigma_k^j(s,\tilde{X}_s) ds\notag\,. \end{align}\tag{31}\] Moreover, \[\begin{align} \mathbb{E}\left[\int_0^t\left(e^{\tilde{K}s}\|\tilde{X}_s\|^{2(n-1)}\|\sigma^T(s,\tilde{X}_s)\tilde{X}_s\|\right)^2ds\right]<\infty\,, \end{align}\] for all \(t\), which implies \(\int_0^te^{\tilde{K}s}\|\tilde{X}_s\|^{2(n-1)}\tilde{X}_s\cdot\sigma(s,\tilde{X}_s)dW_s\) is a martingale.

Specializing 20 to \(n=1\), taking the expectation of both sides and using the martingale property, we obtain \[\begin{align} e^{\tilde{K}t}\mathbb{E}[\|\tilde{X}_t\|^{2}] \leq &\mathbb{E}[\|\tilde{X}_0\|^{2}]+\int_0^t \tilde{K}e^{\tilde{K}s} \mathbb{E}[\|\tilde{X}_s\|^{2}]ds+\int_0^t e^{\tilde{K}s}\mathbb{E}\left[\|\sigma(s,\tilde{X}_s)\|_F^2 \right]ds\notag\\ &+2\int_0^te^{\tilde{K}s}(A_G-B_G\mathbb{E}[\|\tilde{X}_s\|^2])ds+2\kappa\int_0^te^{\tilde{K}s}\mathbb{E}[\|\tilde{X}_s\|\|Z_s\|]ds \notag\,. \end{align}\] With \(\delta\) chosen according to Assumption 1 . [assump:delta] and letting \(\tilde{K}=\tilde{K}_{\kappa,\delta}\) we can compute \[\begin{align} & e^{\tilde{K}_{\kappa,\delta}t}\mathbb{E}[\|\tilde{X}_t\|^{2}]\\ \leq &\mathbb{E}[\|\tilde{X}_0\|^{2}]+(\tilde{K}_{\kappa,\delta}-2B_G + \kappa\delta)\int_0^t e^{\tilde{K}_{\kappa,\delta}s} \mathbb{E}[\|\tilde{X}_s\|^{2}]ds\notag\\ &+\int_0^t e^{\tilde{K}_{\kappa,\delta}s}\mathbb{E}\left[\|\sigma(s,\tilde{X}_s)\|_F^2 \right]ds+2A_G\int_0^te^{\tilde{K}_{\kappa,\delta}s} ds+\kappa\delta^{-1}\int_0^te^{\tilde{K}_{\kappa,\delta}s}\mathbb{E}[\|Z_s\|^2]ds \notag\\ \leq &\mathbb{E}[\|\tilde{X}_0\|^{2}]+(2A_G+ \|\sigma\|_{F,\infty}^2 )\frac{e^{\tilde{K}_{\kappa,\delta}t}-1}{\tilde{K}_{\kappa,\delta}}+\kappa\delta^{-1}\int_0^te^{\tilde{K}_{\kappa,\delta}s}\mathbb{E}[\|Z_s\|^2]ds \,,\notag \end{align}\] and hence \[\begin{align} \mathbb{E}[\|\tilde{X}_t\|^{2}] \leq &\mathbb{E}[\|\tilde{X}_0\|^{2}] e^{-\tilde{K}_{\kappa,\delta}t} +(2A_G+ \|\sigma\|_{F,\infty}^2 )\frac{1-e^{-\tilde{K}_{\kappa,\delta}t}}{\tilde{K}_{\kappa,\delta}}\\ &+\kappa\delta^{-1}e^{-\tilde{K}_{\kappa,\delta}t}\int_0^te^{\tilde{K}_{\kappa,\delta}s}\mathbb{E}[\|Z_s\|^2]ds \,.\notag \end{align}\] Using 22 and Assumption 1 . [assump:ICS] we obtain \[\begin{align} \label{eq:E95X95295bound95app} &\mathbb{E}[\|\tilde{X}_t\|^{2}]\\ \leq &\mathbb{E}[\| X_0\|^{2}] e^{-\tilde{K}_{\kappa,\delta}t} +\left(2A_G+ \|\sigma\|_{F,\infty}^2 +\frac{\kappa}{\delta } \left(\mathbb{E}[\|Z_0\|^{2}] +\frac{(1+1/\epsilon)\|\sigma-\eta\|_{F,\infty}^2}{K_{\kappa,\epsilon}} \right) \right)\frac{1-e^{-\tilde{K}_{\kappa,\delta}t}}{\tilde{K}_{\kappa,\delta}}\,.\notag \end{align}\tag{32}\]

As was the case for \(Z_t\), we will also require control of higher moments in order to justify our use of Girsanov’s theorem in Section 4.2.2. By taking the expectation of 31 and using the martingale property we can compute \[\begin{align} &e^{\tilde{K}t}\mathbb{E}[\|\tilde{X}_t\|^{2n}]\\ \leq&\mathbb{E}[\|\tilde{X}_0\|^{2n}]+\int_0^t \tilde{K}e^{\tilde{K}s} \mathbb{E}[\|\tilde{X}_s\|^{2n}]ds \notag\\ &+2n\int_0^te^{\tilde{K}s}\mathbb{E}[\|\tilde{X}_s\|^{2(n-1)}( A_G-B_G\|\tilde{X}_s\|^2+\kappa \|\tilde{X}_s\|\|Z_s\|)]ds\notag\\ &+\frac{1}{2}\sum_{i,j,k}\int_0^t e^{\tilde{K}s}\mathbb{E}[\left(2n(n-1)\|\tilde{X}_s\|^{2(n-2)}2 \tilde{X}_s^j\tilde{X}_s^i+2n\|\tilde{X}_s\|^{2(n-1)}\delta_{ij}\right) \sigma_k^i(s,\tilde{X}_s)\sigma_k^j(s,\tilde{X}_s)]ds\notag\\ \leq&\mathbb{E}[\|\tilde{X}_0\|^{2n}]+\int_0^t \tilde{K}e^{ \tilde{K}s} \mathbb{E}[\|\tilde{X}_s\|^{2n}]ds\notag\\ &+2n\int_0^te^{ \tilde{K}s} ( A_G\mathbb{E}[\|\tilde{X}_s\|^{2(n-1)}]-B_G\mathbb{E}[\|\tilde{X}_s\|^{2n}]+\kappa\mathbb{E}[\|\tilde{X}_s\|^{2n-1}\|Z_s\|])ds\notag\\ &+\left(2n(n-1)\|\sigma\|^2_{2,\infty} +n\|\sigma\|_{F,\infty}^2 \right)\int_0^t e^{ \tilde{K}s} \mathbb{E}\left[\|\tilde{X}_s\|^{2(n-1)}\right] ds\,.\notag \end{align}\] Using Young’s inequality with \(p=2n\), \(1/q=1-1/2n=(2n-1)/2n\), for all \(\epsilon>0\) we can further compute \[\begin{align} e^{ \tilde{K}t}\mathbb{E}[\|\tilde{X}_t\|^{2n}] \leq&\mathbb{E}[\|\tilde{X}_0\|^{2n}]+\kappa\epsilon^{-(2n-1)}\int_0^te^{ \tilde{K}s}\mathbb{E}[\|Z_s\|^{2n}]ds\\ &-(2nB_G+\kappa\epsilon-2n\kappa\epsilon- \tilde{K})\int_0^te^{ \tilde{K}s} \mathbb{E}[\|\tilde{X}_s\|^{2n}]ds\notag\\ &+\left(2n(n-1)\|\sigma\|^2_{2,\infty} +n\|\sigma\|_{F,\infty}^2+2n A_G \right)\int_0^t e^{ \tilde{K}s} \mathbb{E}\left[\|\tilde{X}_s\|^{2(n-1)}\right] ds\,.\notag \end{align}\] Let \(\epsilon=B_G/(2\kappa)\) and \(\tilde{K}=nB_G\). Fixing \(T>0\), for \(t\in[0,T]\) the above implies \[\begin{align} \label{eq:tilde95X952n95bound95recursive} e^{nB_Gt}\mathbb{E}[\|\tilde{X}_t\|^{2n}] \leq&\mathbb{E}[\|\tilde{X}_0\|^{2n}]+\kappa(2\kappa/B_G)^{2n-1}\int_0^Te^{nB_Gs}\mathbb{E}[\|Z_s\|^{2n}]ds\\ &+n^2D_{\sigma,G}\int_0^t e^{nB_Gs} \mathbb{E}\left[\|\tilde{X}_s\|^{2(n-1)}\right] ds\,,\notag \end{align}\tag{33}\] where \(D_{\sigma,G}\mathrel{\vcenter{:}}= 2\|\sigma\|^2_{2,\infty}+\|\sigma\|_{F,\infty}^2+2A_G\). The bound 33 has the form \[\begin{align} &f_n(t)\leq b_n+a_n\int_0^t e^{\tau s} f_{n-1}(s) ds\,,\,\,t\in[0,T]\,. \end{align}\] where \[\begin{align} f_n(t)\mathrel{\vcenter{:}}= &e^{n\tau t}\mathbb{E}[\|\tilde{X}_t\|^{2n}]\,,\,\,\, \tau\mathrel{\vcenter{:}}= B_G\,,\\ b_n\mathrel{\vcenter{:}}=&\mathbb{E}[\|\tilde{X}_0\|^{2n}]+\kappa(2\kappa/B_G)^{2n-1}\int_0^Te^{nB_Gs}\mathbb{E}[\|Z_s\|^{2n}]ds\,,\\ a_n\mathrel{\vcenter{:}}= &n^2D_{\sigma,G}\,. \end{align}\] Therefore we can apply Lemma 1, which yields the bound \[\begin{align} \label{eq:f95n95bound95X95tilde} e^{nB_G t}\mathbb{E}[\|\tilde{X}_t\|^{2n}]\leq& \sum_{j=0}^n \frac{\prod_{k=j+1}^n a_k}{(n-j)! \tau^{n-j}} b_j e^{(n-j)\tau t} \\ =& \sum_{j=0}^n \frac{(n!/j!)^2}{(n-j)!} (D_{\sigma,G} e^{ B_G t}/ B_G)^{n-j}b_j \notag \end{align}\tag{34}\] for all \(t\in[0,T]\), where \(b_0\mathrel{\vcenter{:}}= 1\). Using Assumption 1 . [assump:ICS] along with 23 , for \(n\in\mathbb{Z}^+\) we can compute \[\begin{align} b_n\leq & \tilde{\alpha}\tilde{\beta}^n n!+\kappa \alpha n! (2\kappa/B_G)^{2n-1} \left(\beta + \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)^n\int_0^Te^{nB_Gs}ds \\ \leq &n!\left(\tilde{\alpha}\tilde{\beta}^n + \frac{1}{2}\alpha \left( (2\kappa/B_G)^{2 }e^{ B_GT} \left(\beta + \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)\right)^n \right) \notag\\ \leq &n!\left(\tilde{\alpha} + \alpha /2 \right)\max\left\{\tilde{\beta}, (2\kappa/B_G)^{2 }e^{ B_GT} \left(\beta + \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)\right\}^n\notag \end{align}\] (note that the final line also bounds \(b_0\) when \(n=0\)). Combining this with 34 we obtain \[\begin{align} &e^{nB_G t}\mathbb{E}[\|\tilde{X}_t\|^{2n}]\\ \leq& n! \left(\tilde{\alpha} + \alpha /2 \right) \sum_{j=0}^n \frac{n! }{j!(n-j)!} \left(\frac{D_{\sigma,G} e^{ B_G t}}{ B_G}\right)^{n-j}\notag\\ &\qquad\qquad\qquad\times \max\left\{\tilde{\beta}, (2\kappa/B_G)^{2 }e^{ B_GT} \left(\beta + \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)\right\}^j \notag\\ =&n! \left(\tilde{\alpha} + \alpha /2 \right) \left(\frac{D_{\sigma,G} e^{ B_G t}}{ B_G}+\max\left\{\tilde{\beta}, (2\kappa/B_G)^{2 }e^{ B_GT} \left(\beta + \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)\right\}\right)^n\notag \end{align}\] for all \(t\in[0,T]\). Dividing by \(e^{nB_G t}\), letting \(t=T\), and recalling that \(T>0\) was arbitrary, we find \[\begin{align} \label{eq:f95n95bound95X95tilde95final95app} \mathbb{E}[\|\tilde{X}_t\|^{2n}] \leq& n! \left(\tilde{\alpha} + \alpha /2 \right) \left(\frac{D_{\sigma,G} }{ B_G}+\max\left\{\tilde{\beta} e^{-B_G t}, (2\kappa/B_G)^{2 } \left(\beta + \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)\right\}\right)^n\\ \leq& n! \left(\tilde{\alpha} + \alpha /2 \right) \left(\frac{D_{\sigma,G} }{ B_G}+\max\left\{\tilde{\beta} , (2\kappa/B_G)^{2 } \left(\beta + \frac{D_{\sigma,\eta}}{2(\kappa+C_G-L_{\tilde{G}})} \right)\right\}\right)^n\notag \end{align}\tag{35}\] for all \(t>0\), \(n\in\mathbb{Z}^+\).

6.3 Bound on the Feynman-Kac Semigroup with a Time-Dependent Potential↩︎

In this section we derive a bound on the Feynman-Kac semigroup with a time-dependent potential via a streamlined version of the argument used in [35] to study concentration inequalities for time-inhomogeneous systems; see Theorem 1 therein. Specifically, we use the Feynman-Kac formula, i.e., that (under appropriate assumptions) \[\begin{align} u^f_T(t,x)= \mathbb{E}\left[ f(X^{t,x}_T)\exp\left(\int_t^T h(s, X_s^{t,x})ds\right)\right] \end{align}\] solves the PDE \[\begin{align} &\partial_t u^f_T(t,x)=-A_{X}[u^f_T(t,\cdot)](t,x)-h(t,x) u^f_T(t,x)\,,\,\,\,\,u^f_T(T,x)=f(x)\,, \end{align}\] together with a \(\log\)-Sobolev inequality for the generator, \(A_X\). We note that related techniques have also previously been used to derive concentration inequalities for Markov processes for time-independent \(h(x)\); see [10], [32]–[34].

References↩︎