Model reduction and uncertainty quantification of multiscale diffusions with parameter uncertainties using nonlinear expectations
MMODEL REDUCTION AND UNCERTAINTY QUANTIFICATIONOF MULTISCALE DIFFUSIONS WITH PARAMETERUNCERTAINTIES USING NONLINEAR EXPECTATIONS
HAFIDA BOUANANI, CARSTEN HARTMANN, AND OMAR KEBIRI
Abstract.
In this paper we study model reduction of linear and bilinear qua-dratic stochastic control problems with parameter uncertainties. Specifically,we consider slow-fast systems with unknown diffusion coefficient and study theconvergence of the slow process in the limit of infinite scale separation. Theaim of our work is two-fold: Firstly, we want to propose a general frameworkfor averaging and homogenisation of multiscale systems with parametric un-certainties in the drift or in the diffusion coefficient. Secondly, we want to usethis framework to quantify the uncertainty in the reduced system by derivinga limit equation that represents a worst-case scenario for any given (possiblypath-dependent) quantity of interest. We do so by reformulating the slow-fastsystem as an optimal control problem in which the unknown parameter playsthe role of a control variable that can take values in a closed bounded set. Forsystems with unknown diffusion coefficient, the underlying stochastic controlproblem admits an interpretation in terms of a stochastic differential equationdriven by a G-Brownian motion. We prove convergence of the slow process withrespect to the nonlinear expectation on the probability space induced by theG-Brownian motion. The idea here is to formulate the nonlinear dynamic pro-gramming equation of the underlying control problem as a forward-backwardstochastic differential equation in the G-Brownian motion framework (in brief:G-FBSDE), for which convergence can be proved by standard means. We illus-trate the theoretical findings with two simple numerical examples, exploitingthe connection between fully nonlinear dynamic programming equations andsecond-order BSDE (2BSDE): a linear quadratic Gaussian regulator problemand a bilinear multiplicative triad that is a standard benchmark system inturbulence and climate modelling. Introduction
Modelling of real-world processes often involves high-dimensional stochastic dif-ferential equations with multiple time and length scales. In many cases the relevantbehaviour is given by the largest scales in the system (e.g., phase transitions), thedirect numerical simulation of which is difficult. Moreover, the system variablesmay be only be partially observable, or the model may involve unknown param-eters, which makes the numerical or analytical treatment of the high-dimensionalequations by multiscale techniques as well as the uncertainty quantification (UQ)of derived quantities difficult. Recently, data-driven approaches have been devel-oped in order to account for model uncertainties or partial observability [8]. Theidea there is to postulate a reduced-order model for some given quantities of interest
Key words and phrases.
Slow-fast system, parametric systems, unknown diffusion, G-Brownianmotion, G-FBSDE, fully nonlinear Hamilton-Jacobi-Bellman equation, second-order BSDE, opti-mal control, linear and bilinear stochastic regulator, multiplicative triad. a r X i v : . [ m a t h . O C ] F e b HAFIDA BOUANANI, CARSTEN HARTMANN, AND OMAR KEBIRI (QoI), based on either physical principles [35] or classical multiscale techniques suchas averaging and homogenisation [33], and then parameterise the resulting modelequations using dense observations of the resolved variables or the QoI [15]. Formultiscale systems with more than two scales, the parameter estimation is known tobe a difficult task, for the standard estimators may be strongly biased and requiresophisticated subsampling strategies to reduce the bias [36].
A sublinear expectation framework for multiscale diffusions.
We studyslow-fast stochastic differential equations (SDE) with unobservable fast variableswhere the latter are driven by a G-Brownian motion (G-SDE). The approach pur-sued in this paper is different from the aforementioned methods in that it em-ploys analytical techniques for the elimination of fast variables, like averaging orhomogenisation that normally require that the model is fully specified (with all pa-rameters being available), and yet takes the inherent uncertainty of the unresolvedor partially resolved (“underresolved”) fast variables into account. Specifically, theparameter uncertainty of the underresolved fast variables is taken into account bymodelling them as a stationary G-SDE that then generates a parametric family ofprobability distributions that, consequently, give rise to a family of reduced-ordermodels for the slow variables or QoI. More specifically, we will study averagingof G-SDE and forward-backward G-SDE (G-FBSDE) and discuss the relation totraditional SDE and FBSDE multiscale methods (e.g. [36, 14]).The idea is to consider path functionals of the resolved variables and to maxi-mize the expected difference between the original and the limit dynamics over theuncertain parameters. Exploiting the specific properties of the driving G-Brownianmotion and related nonlinear expectation concepts [39], we show that the maxi-mum difference goes to zero in the limit of infinite scale separation. Depending onthe specific situation at hand (i.e. whether drift or diffusion coefficients are uncer-tain), the nonlinear expectation boils down to a g-expectation [40] and an averagingproblem for a standard (uncoupled) FBSDE, or to a G-expectation [39] and an av-eraging problem for a G-FBSDE. For finite scale separation, the maximiser of thefunctional generates a worst-case scenario for the deviation between the multiscaleand the limit dynamics, and thus the maximiser quantifies the uncertainty in theQoI. For finite scale separation, the maximiser of the functional generates a worst-case scenario for the deviation between the multiscale and the limit dynamics, andthus the maximiser quantifies the uncertainty in the QoI.
Existing work.
Backward stochastic differential equations (BSDE) as a proba-bilistic representation of semilinear partial differential equations (PDE) have re-ceived a lot of attention, starting with the work of Peng and Pardoux [37]. Afterthat, the theory of forward-backward stochastic differential equations (FBSDE) andtheir applications to stochastic control problems developed quickly, following thework of Antonili [1]; see also [11, 30]. Recently, Redjil et al. [42] proved the exis-tence of an optimal control of a controlled forward SDE driven by a G-Brownianmotion (G-SDE), allowing to model situations in which the randomness in a modelcoming from, e.g., measurements or model uncertainties does not satisfy the usuali.i.d. assumption, so that the classical limit theorems like the law of large numbersor the central limit theorem do not apply. The theory and the stochastic calculusfor G-SDE have been developed by Peng and co-workers [39, 12]. Relevant prelimi-nary work on existence and uniqueness of fully coupled FBSDE, G-FBSDE and the
ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 3 corresponding dynamic programming (Hamilton-Jacobi Bellman or HJB) equationsis due to Redjil & Choutri [42] and Kebiri et al. [4, 3, 2, 26], showing the existenceof a relaxed control based on results of El-Karoui et al. [25].Systematic model reduction and uncertainty quantification methods for multi-scale parametric systems are still at their infancies, notwithstanding recent advancesin the field; see [9, 31, 34] and the references therein. Related work on model re-duction of controlled multiscale diffusions using duality arguments and Fleming’stechnique of logarithmic transformations (see [17, Ch. VI]) has been carried by oneof the authors [7, 22, 20, 21]. Despite recent progress on the theoretical foundationsof G-(F)BSDE and G-Brownian motion, there have been relatively few practicallyoriented works in the context of uncertainty quantification (see e.g. [23, 38, 41])and even fewer on numerical methods for G-Brownian motions (see e.g. [43]).
Outline of the article.
The idea of using the G-Brownian motion frameworkto do uncertainty quantification for multiscale systems is explained in Section 2.The key theoretical result of this paper, the convergence of the value function andits derivative, is formulated and proved in Section 3. To illustrate the theoreticalfindings, we discuss two numerical examples with unceetain diffusions in Section4: a linear quadratic Gaussian regulator with uncertain diffusion and an uncon-trolled bilinear benchmark system from turbulence modelling. We summarise thekey observations and main results in Section 5. The article contains two appen-dices: Appendix A records basic definitions related to Peng’s nonlinear expectationand inequalities for G-Brownian motion that will be used throughout the article.Appendix B records basic definitions and identities related to the stochastic repre-sentations of fully nonlinear partial differential equations in terms of second-orderBSDE that are used to carry out the numerical simulations in Section 42.
Slow-fast system
Let x = ( r, u ) ∈ R n = R n s × R n f and (cid:15) > dR (cid:15)t = (cid:18) f ( R (cid:15)t , U (cid:15)t ) + 1 √ (cid:15) f ( R (cid:15)t , U (cid:15)t ) (cid:19) dt + α ( R (cid:15)t , U (cid:15)t ) dV t (1a) dU (cid:15)t = 1 (cid:15) g ( R (cid:15)t , U (cid:15)t ; θ ) dt + 1 √ (cid:15) β ( R (cid:15)t , U (cid:15)t ; θ ) dW t , (1b)where all coefficients are assumed to be such that the SDE has a unique strongsolution for all times. We call R (cid:15)t the resolved (slow) variable and U (cid:15)t the unresolved(fast) variable that is not fully accessible and depends on an unknown parameter θ ∈ Θ ⊂ R p , where for convenience we suppress the dependence on θ .The aim is to derive a closed equation for R (cid:15) for (cid:15) → (cid:15) is sufficiently small. Since the fast process dependson an unknown parameter, the answer to the question what the best approximation is remains ambigous. HAFIDA BOUANANI, CARSTEN HARTMANN, AND OMAR KEBIRI -1 -0.5 0 0.5 1-505 =0=1/3=1
Figure 1.
Limiting vector field F ( · , θ ) for θ ∈ { , / , } . Thevalue θ = 1 / Goal-oriented uncertainty quantification.
To illustrate the ambiguity inthe reduced dynamics, let us consider the degenerate diffusion dR t = ( R t − U t ) dt , R = r (2a) dU t = 1 (cid:15) ( R (cid:15)t − U t ) dt + (cid:114) θ(cid:15) dW t , U = u . (2b)for θ ∈ [0 ,
1] where, for simplicity, we use the shorthand (
R, U ) = ( R (cid:15) , U (cid:15) ) ∈ R × R and suppress the dependence on the small parameter (cid:15) .When (cid:15) (cid:28)
1, the fast dynamics becomes “slaved” by the slow dynamics andrandomly fluctuates around R t . The unique limiting invariant measure of the fastvariables conditional on R t = r is given by µ r = N ( r, θ ) when θ ∈ (0 , µ r = δ r for θ = 0. As (cid:15) → R = R (cid:15) converges pathwise to a limitprocess that is the solution of the (here: deterministic) initial value problem(3) drdt = F ( r ; θ ) , r (0) = r , where(4) F ( r, θ ) = − r + r (1 − θ ) , θ ∈ [0 , . Figure 1 shows the vector field F ( · , θ ) for three different values of θ and illustratesthat the limit dynamics undergoes a supercritical pitchfork bifurcation at θ = 1 / F ( r, · ) is continuous at θ = 0, nevertheless,depending on value of θ , the qualitative properties of the limit dynamics changedrastically as θ varies. It therefore makes sense to modify the best approximationquestion slightly and instead ask for a worst-case scenario in terms of the unknownparameter for a given quantity of interest (QoI).Let ϕ : C ([0 , T ]) → R be a suitable test function. The objects of interest arepath functionals of the form φ (cid:15) = ϕ ( R (cid:15) ), with R (cid:15) = ( R (cid:15),θt ) t ∈ [0 ,T ] . To this end let r = ( r θt ) t ∈ [0 ,T ] denote the candidate limit process as (cid:15) → φ = ϕ ( r ). ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 5
A worst-case scenario for the convergence of R (cid:15) to the limiting process r can beexpressed by the G-expectation using the representation formula (46):(5) ˆ E ( | φ (cid:15) − φ | ) = sup θ ∈ Θ E θ ( | φ (cid:15) − φ | ) . For example the worst-case approximation for the variance (or the second moment)may be different from the approximation of the slow process itself, in that theycorrespond to different values of the unknown parameter θ .If the linear expectation on the right hand side of (5) converges for every fixed θ ∈ Θ, stability results (e.g. [44, Thm. 3.1]) for G-BSDE imply that(6) lim (cid:15) → ˆ E ( | φ (cid:15) − φ | ) = 0 . If φ (cid:15) is regarded as data , then the G-expectation defines some kind of trackingproblem for the limit dynamics, with θ playing the role of the control variable.(There may be an additional control variable in the equations though.) An equiv-alent statement is that the value function, i.e. the unique viscosity solution of theunderlying dynamic programming equation converges as (cid:15) → ϕ via the optimal parameter θ ∗ . In general, by the dynamic pro-gramming principle, θ ∗ = θ ∗ ( t ) will be time dependent or a feedback law, thereforethe limit equations are not simply obtained by setting θ equal to some appropriatevalue. They are moreover goal-oriented, in that they depend on the QoI.3. Convergence of the quantity of interest
In this section we study the convergence of the slow component of a slow-fastsystem driven by a G-Brownian motion. Specicifally, we prove convergence ofthe corresponding value function that is associated with the QoI. For the sake ofsimplicity, the proof will be given for a linear controlled G-SDE only, but we stressthat the proof carries over to the case of a nonlinear G-SDE with or without controland under standard Lipschitz conditions, using essentially the same techniques.
Controlled linear-quadratic slow-fast system and related QoI.
We considerthe following controlled stochastic differential equation(7) dX (cid:15)s = ( A (cid:15) X (cid:15)s + B (cid:15) α s ) ds + C (cid:15) dW s ; X (cid:15)t = x, with X (cid:15)s = ( U s , R s ) taking values in R n s × R n f where n s + n f = n , and x =( r, u ) denotes the decomposition of the state vector x into slow (resolved) and fast(unresolved) components. We will suppress the dependence of R and U on (cid:15) , untilfurther notice. Here W = ( W t ) t ≥ is a standard R m -valued Brownian motion on(Ω , F , P ) that is endowed with its own filtration ( F t ) t ≥ , and α = ( α t ) t ≥ denotesan adapted control variable with values in R k . Let A (cid:15) = (cid:18) A (cid:15) − / A (cid:15) − / A (cid:15) − A (cid:19) ∈ R n × n , with the natural partitioning into A ∈ R n s × n s , etc. where assume that thematrix A ∈ R n f × n f is Hurwitz, i.e. all of its eigenvalues are lying in the open lefthalf-plane. The control and the noise coefficients are partitioned as follows: B (cid:15) = (cid:18) B (cid:15) − / B (cid:19) ∈ R n × k , C (cid:15) = (cid:18) C (cid:15) − / C (cid:19) ∈ R n × m . HAFIDA BOUANANI, CARSTEN HARTMANN, AND OMAR KEBIRI
We assume that, for all (cid:15) >
0, the columns of B (cid:15) lie in the column space of thematrix C (cid:15) , i.e. ran( B (cid:15) ) ⊂ ran( C (cid:15) ) or, equivalently, the column space of B (cid:15) isorthogonal to the kernel of ( C (cid:15) ) T , so that the equation(8) C (cid:15) ξ = B (cid:15) c has a (not necessarily unique) solution for every c ∈ R k . We seek a control α thatminimises the following quadratic cost functional(9) J ( α ; t, x ) = E t,x (cid:20) (cid:90) τt R Ts Q R s + | α s | ds + 12 R Tτ Q R τ (cid:21) , where τ is a bounded stopping time given by τ = inf { s ∈ [ t, T ] : X t / ∈ S } where S is a bounded subset of R n s × R n f which containe the initial state x , and where Q , Q ∈ R n s × n s are any given symmetric positive semi-definite matrices. Notethat even though the cost depends only on the slow process, the expected costdepends on the initial conditions of both r and u . We call(10) q = r T Q r , q = r T Q r . The corresponding value function is our QoI, it is given by(11) V (cid:15) ( t, x ) = inf α ∈A J ( α ; t, x ) . where A is the space of all admissible controls α , such that (7) has a unique strongsolution. (Likewise we may consider q , q or J to be our quantities of interest.)Assuming that all coefficients are known, the averaging principle for linear-quadratic control systems of the form (7)–(9) implies that, under mild conditionson the system matrices, the value function V (cid:15) converges uniformly on any com-pact subset of [0 , T ] × R n to a value function v = v ( t, r ); see e.g. [27]. The latteris the value function of the following linear-quadratic stochastic control problem:minimise the reduced cost functional(12) ¯ J ( α ; t, r ) = E t,r (cid:20) (cid:90) τt q ( ¯ R s ) + | α s | ds + 12 q ( ¯ R τ ) (cid:21) , subject to(13) d ¯ R s = ( ¯ A ¯ R s + ¯ Bα s ) ds + ¯ CdW s , where the coefficients of the reduced system are given by(14) ¯ A = A − A A − A , ¯ B = B − A A − B , ¯ C = C − A A − C . Multiscale system with unknown diffusion coefficient.
We suppose that thenoise coefficients C and/or C are unknown. This situation is common in manyapplications, since especially the diffusion coefficient of the unresolved variables isdifficult to estimate. Very often, however, an educated guess can be made as towhich set or interval the unknown coefficient lies in. Specifically, we suppose that( C , C ) T ∈ A Θ , ∞ which is the collection of all Θ-valued adapted process on [0 , ∞ )where Θ is a given bounded and closed subset in R ( n s + n f ) × m .Following the work by Denis and co-workers [12, 13] we exploit the link betweenthe G − expectation framework and diffusion controlled processes and define D (cid:15) ˜ W t = (cid:90) t CdW s , ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 7 for each C (cid:15) ∈ A Θ , ∞ , such that C (cid:15) = D (cid:15) (cid:18) C C (cid:19) , D (cid:15) = (cid:18) I n s (cid:15) − / I n f (cid:19) so that ( ˜ W s ) s ≥ is a d − dimensional G-Brownian motion. As the main result, wewill show below that the value function converges uniformly on any compact subsetof [0 , T ] × R n . The result does not rely on any compactness or periodicity assump-tions of the fast variables with unknown diffusion; the key idea is to recast the fullynonlinear dynamic programming (or: G-Hamilton-Jacobi-Bellman) equation of thefull G-stochastic optimal control problem as a G-FBSDE and then study conver-gence to the limiting G-FBSDE, which implies convergence of the correspondingdynamic programming equation. Nonlinear dynamic programming equation.
By the dynamic programmingprinciple for controlled G-SDE [16], the G-Hamilton-Jacobi-Bellman (G-HJB) equa-tion associated with our uncertain stochastic control problem (7)–(9) reads(15) − ∂v (cid:15) ∂t = inf c { G ( DD T : ∇ v (cid:15) ) + (cid:104)∇ v (cid:15) , Ax + Bc (cid:105) + 12 q + 12 | c | ) } , with terminal condition(16) v (cid:15) ( τ, · ) = 12 q . Note that we v (cid:15) is different from the value function V (cid:15) in (11), since the diffusioncoefficient in (11) is assumed constant, whereas, here, it is part of the nonlineargenerator that involves a maximisation over the coefficient. Further note that wehave dropped the (cid:15) in A = A (cid:15) , B = B (cid:15) and D = D (cid:15) . We can get rid of the outerinfimum since the diffusion part is independent of the control variable, andinf c (cid:26) (cid:104)∇ V (cid:15) , B (cid:15) c (cid:105) + 12 | c | (cid:27) = − | c | BB T , where | c | BB T = (cid:104) c, BB T c (cid:105) . This implies that (15) is equivalent to(17) ∂v∂t + G ( DD T : ∇ v ) + (cid:104)∇ v, Ax (cid:105) − | c | BB T + 12 q = 0 , with the associated G-FBSDE system given by(18) dX (cid:15)s = AX (cid:15)s ds + D d ˜ W s , X (cid:15)t = xY (cid:15)t = 12 q ( R τ ) − (cid:90) τt q ( R s ) ds + 12 (cid:90) τt | B T ( D T ) (cid:93) Z (cid:15)s | ds − (cid:90) τt Z (cid:15)s d ˜ W s − ( K τ − K t )Here(19) Y (cid:15)s = v (cid:15) ( s, X (cid:15)s ) , Z (cid:15)s = ∇ v (cid:15) ( s, X (cid:15)s ) , t ≤ s ≤ τ , and (cid:93) denotes the Moore-Penrose pseudo inverse of a matrix. The process K is adecreasing G-martingale with K = 0 that is a consequence of the G-martingalerepresentation theorem [39]. HAFIDA BOUANANI, CARSTEN HARTMANN, AND OMAR KEBIRI
Strong convergence of the quantity of interest.
Since the G-FBSDE is de-coupled and running and terminal cost q , q depend only on the resolved variables,we can infer the candidate for the limiting process:(20) d ¯ R s = ( ¯ A ¯ R s + ¯ Bα s ) ds + ¯ D d ˜ W s with ¯ D d ˜ W given by ¯ CdW s = ( C − A A − C ) dW s = C dW s − A A − C dW s = d ˜ W s − A A − d ˜ W s =: ¯ D d ˜ W . in other words, ¯ D = ( I n s , − A A − ). The associated limiting G-FBSDE reads(21) d ¯ R s = ¯ A ¯ R s ds − ¯ D d ˜ W s , ¯ R t = r ¯ Y s = 12 q ( ¯ R τ ) − (cid:90) τt q ( ¯ R s ) ds + 12 (cid:90) τt | ¯ B T ( ¯ D T ) (cid:93) ¯ Z s | ds − (cid:90) τt ¯ Z s d ˜ W s − ( ¯ K τ − ¯ K t )The corresponding limit G-HJB equation is then given by(22) ∂ ¯ v∂t + G ( ¯ D ¯ D T : ∇ ¯ v ) + (cid:104)∇ ¯ v, ¯ Ar (cid:105) − |∇ ¯ v | B ¯ B T + 12 q = 0 . with the natural terminal condition(23) ¯ v ( τ, · ) = 12 q . Theorem 1.
Let v (cid:15) be the classical solution of the dynamic programming equation17 and ¯ v be the solution of 22, then, as (cid:15) → v (cid:15) → ¯ v , ∇ v (cid:15) → ∇ ¯ v where the convergence of v (cid:15) is uniform on any compact subset of [0 , T ] × R n s andpointwise for ∇ v (cid:15) for all ( t, x ) ∈ [0 , T ] × R n s .Proof. Subtracting the G-BSDE part of (21) from (18) yields(24) Y (cid:15)t − ¯ Y t = 12 q ( R τ ) − q ( ¯ R τ ) − (cid:90) τt q ( R s ) ds + 12 (cid:90) τt q ( ¯ R s ) ds + 12 (cid:90) τt | B T ( D T ) (cid:93) Z (cid:15)s | ds − (cid:90) τt | ¯ B T ( ¯ D T ) (cid:93) ¯ Z s | ds − (cid:90) τt Z (cid:15)s d ˜ W s + (cid:90) τt ¯ Z s d ˜ W s − ( K τ − K t ) + ( ¯ K τ − ¯ K t )Let γ > y t = Y (cid:15)t − ¯ Y t , M t = K t − ¯ K t , we can apply Itˆo’sformula to | y t | e γt for 0 ≤ t < τ ≤ T , which yields ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 9 (25) | y t | e γt + (cid:90) τt | Z (cid:15)s − ¯ Z s | d (cid:104) ˜ W (cid:105) s + (cid:90) τt γ | y s | e γs ds = (cid:12)(cid:12)(cid:12)(cid:12) q ( R τ ) − q ( ¯ R τ ) (cid:12)(cid:12)(cid:12)(cid:12) e γτ − (cid:90) τt y s e γs (cid:0) q ( R s ) − q ( ¯ R s ) (cid:1) ds + (cid:90) τt y s e γs (cid:0) | B T ( D T ) (cid:93) Z (cid:15)s | − | ¯ B T ( ¯ D T ) (cid:93) ¯ Z s | (cid:1) ds − ( ¯ M τ − ¯ M t ) , where ¯ M τ − ¯ M t = 2 (cid:90) τt y s e γs dM s + 2 (cid:90) τt y s e γs (cid:0) Z (cid:15)s − ¯ Z s (cid:1) d ˜ W s . It is convenient to write e γs on the right hand side as e γs/ e γs/ . Now droppingthe quadratic variation term on the left and using Young’s inequality (cf. Lemma9) gives after rearranging terms(26) | y t | e γt + γ (cid:90) τt | y s | e γs ds + ( ¯ M τ − ¯ M t ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) q ( R τ ) − q ( ¯ R τ ) (cid:12)(cid:12)(cid:12)(cid:12) e γτ + (cid:90) τt (cid:18) λ | y s | e γs + e γs λ (cid:0) q ( ¯ R s ) − q ( R s ) (cid:1) (cid:19) ds + (cid:90) τt (cid:18) λ | y s | e γs + e γs λ (cid:0) | B T ( D T ) (cid:93) Z (cid:15)s | − | ¯ B T ( ¯ D T ) (cid:93) ¯ Z s | (cid:1) (cid:19) ds, where we have defined λ , λ by γ = λ / λ /
2. As a consequence,(27) | y t | e γt + ( ¯ M τ − ¯ M t ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) q ( R τ ) − q ( ¯ R τ ) (cid:12)(cid:12)(cid:12)(cid:12) e γτ + (cid:90) τt e γs λ (cid:0) q ( ¯ R s ) − q ( R s ) (cid:1) ds + (cid:90) τt e γs λ (cid:0) | B T ( D T ) (cid:93) Z (cid:15)s | − | ¯ B T ( ¯ D T ) (cid:93) ¯ Z s | (cid:1) ds. Using the shorthands N = ( B , B ) T and k s = (cid:0) N Z (cid:15)s + N ¯ Z s (cid:1) , with (cid:0) ( B T (( D (cid:15) ) T ) (cid:93) Z (cid:15)s ) − ( ¯ B T ( D T ) (cid:93) ¯ Z s ) (cid:1) = (cid:0) N Z (cid:15)s − N ¯ Z s (cid:1) , the pathwise convergence E (cid:34) sup t ∈ [0 ,T ] | R t − ¯ R t | (cid:35) = O ( (cid:15) )as (cid:15) → | y t | e γt + ( ¯ M τ − ¯ M t ) ≤ l(cid:15) e γτ (cid:90) τt l(cid:15) e γs λ ds + (cid:90) τt | k s | (cid:107) N N T (cid:107) F λ | Z (cid:15)s − ¯ Z s | e γs ds for some generic constant l ∈ (0 , ∞ ) that may change from equation to equation.Taking the supremum and the using the fact that ¯ M is a symmetric G-martingale,it follows again by Young’s inequality that (29) ˆ E (cid:32) sup s ∈ [ t,τ ] | y s | e γs (cid:33) ≤ l(cid:15) e γτ l(cid:15) e γτ γλ + l λ ˆ E (cid:18)(cid:90) τt | Z (cid:15)s − ¯ Z s | e γs ds (cid:19) . Now using (25) again, together with the BDG-type inequalities (47)–(48) for thequadratic variation and Young’s inequality for the integrals involving y s e γs on theright hand side, we obtain after dropping the quadratic terms in y :(30) σ ˆ E (cid:18)(cid:90) τt | Z (cid:15)s − ¯ Z s | e γs ds (cid:19) ≤ | l(cid:15) | e γτ (cid:90) τt | l(cid:15) | e γs α ds + ( k ) N N T α ˆ E (cid:18)(cid:90) τt | Z (cid:15)s − ¯ Z s | e γs ds (cid:19) . where α , α are defined by γ = α / α /
2. Hence(31) ˆ E (cid:18)(cid:90) τt | Z (cid:15)s − ¯ Z s | e γs ds (cid:19) ≤ | l(cid:15) | e γτ σ + (cid:90) τt | l(cid:15) | e γs α σ ds + ( k ) N N T σ α ˆ E (cid:18)(cid:90) τt | Z (cid:15)s − ¯ Z s | e γs ds (cid:19) , which can be rearranged to give(32) (cid:18) − ( k ) N N T lσ α (cid:19) ˆ E (cid:18)(cid:90) τt | Z (cid:15)s − ¯ Z s | e γs ds (cid:19) ≤ | l(cid:15) | e γτ lσ + | l (cid:15) | ( e γτ − e γt )2 γα lσ . The last inequality can be combined with (29), so that we obtain(33)ˆ E (cid:32) sup s ∈ [ t,τ ] | y s | e γs (cid:33) + (cid:18) − ( k ) N N T lσ α − ( k ) N N T λ (cid:19) ˆ E (cid:18)(cid:90) τt | Z (cid:15)s − ¯ Z s | e γs ds (cid:19) ≤ | l(cid:15) | e γτ (cid:90) τt | l (cid:15) | ( e γτ − e γt )2 γλ + | l(cid:15) | e γτ lσ + | l (cid:15) | ( e γτ − e γt )2 γα lσ . As a consequence, (cid:107) Y (cid:15) − ¯ Y (cid:107) γ := ˆ E (cid:32) sup s ∈ [ t,τ ] | y s | e γs (cid:33) , (cid:107) Z (cid:15) − ¯ Z (cid:107) γ := ˆ E (cid:18)(cid:90) τt | Z (cid:15)s − ¯ Z s | e γs ds (cid:19) go to zeros as (cid:15) → (cid:15) . Since (cid:107) · (cid:107) γ and (cid:107) · (cid:107) γ =0 are equivalent, it followsthat Y (cid:15)t → ¯ Y t uniformly for t ∈ [0 , T ], and therefore, as (cid:15) → v (cid:15) ( · , x ) = Y (cid:15) → ¯ Y = ¯ v ( · , x )uniformly on any compact subset of [0 , T ] × R n s . Likewise, ∇ v (cid:15) ( t, x ) = Z (cid:15)t → ¯ Z t = ∇ ¯ v ( t, x ) , ( t, x ) ∈ [0 , T ] × R n s as (cid:15) →
0, which implies the convergence of the optimal control in (7)–(9). (cid:3)
Remark 2.
The theorem also holds if the underlying G-SDE is nonlinear, as longas the averaging principle applies (e.g. when the drift is uniformly Lipschitz).
Remark 3.
When B = D in (15) then the corresponding G-BSDE and the limitG-BSDE simplify to (34) Y (cid:15)t = 12 q ( R τ ) − (cid:90) τt q ( R s ) ds + 12 (cid:90) τt | Z (cid:15)s | ds − (cid:90) τt Z (cid:15)s d ˜ W s − ( K τ − K t ) ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 11 and (35) ¯ Y s = 12 q ( ¯ R τ ) − (cid:90) τt q ( ¯ R s ) ds + 12 (cid:90) τt | ¯ Z s | ds − (cid:90) τt ¯ Z s d ˜ W s − ( ¯ K τ − ¯ K t ) . Numerical illustration
In this section we present two numerical examples to verify that the value func-tion of the original system (17) converges to the solution of the reduced system(22) as (cid:15) →
0. The corresponding fully nonlinear HJB equations (17) and (22) arenumerically solved by exploiting the link between fully nonlinear PDE and second-order BSDE (2BSDE); see e.g. [10]. The numerical algorithm for solving 2BSDEis based on the deep 2BSDE solver introduced by Beck et al. [6].4.1.
Linear quadratic Gaussian regulator.
The first example is a 2-dimensionallinear quadratic regulator problem given by the SDE(36) dX (cid:15)t = ( A (cid:15) X (cid:15)t + B (cid:15) u (cid:15)t ) dt + √ σB (cid:15) dW t , X (cid:15) = x , with unknown diffusion coefficient σ ∈ [ σ, σ ] and the cost functional(37) J ( u ; t, x ) = 12 E (cid:34)(cid:90) Tt (( X (cid:15)s ) T Q X (cid:15)s + | u (cid:15)s | ) ds + ( X (cid:15)T ) T Q X (cid:15)T (cid:35) . Here x = ( r, u ) ∈ R and the coefficients are given by A (cid:15) = (cid:18) − − /(cid:15) /(cid:15) − /(cid:15) (cid:19) , B (cid:15) = (cid:18) . /(cid:15) (cid:19) , Q = 0 , Q = (cid:18) (cid:19) , and we define the value function as v (cid:15) ( t, x ) = inf u J ( u ; t, x ). The G-PDE corre-sponding to the Gaussian regulator problem (37)–(36) is then given by(38) ∂v (cid:15) ∂t + G ( a (cid:15) : ∇ v (cid:15) ) + (cid:104)∇ v (cid:15) , A (cid:15) x (cid:105) − (cid:104) B (cid:15) , z (cid:105) = 0 , v (cid:15) ( T, x ) = x where we have used the shorthand a (cid:15) = σB (cid:15) ( B (cid:15) ) T . Calling a = σ ¯ B ¯ B T , the G-PDEof the limiting value function ¯ v = lim (cid:15) → v (cid:15) then reads(39) ∂ ¯ v∂t + G ( a : ∇ ¯ v ) + (cid:104)∇ ¯ v, ¯ A ¯ x (cid:105) − (cid:104)∇ ¯ v, ¯ B (cid:105) = 0 , ¯ v ( T, r ) = r .
The function G : R → R is defined by: G ( x ) = x (cid:26) ¯ σ if x ≥ σ if x < . Numerical results.
We consider the two value functions in the time interval[0 , .
1] with fixed initial condition x = ( r, u ) = (1 , . σ ∈ [0 . , f of the 2BSDE corresponding to theG-PDE (38) of the original system is f ( t, x, y, z, S ) = G ( a (cid:15) : S ) + 12 (cid:104) B (cid:15) , z (cid:105) + 12 (cid:104) A (cid:15) x, z (cid:105) , whereas the 2BSDE corresponding to the limiting G-PDE (39) has the driver¯ f ( t, x, y, z, ¯ S ) = G ( a : ¯ S ) + 12 (cid:10) ¯ B, z (cid:11) + 12 (cid:10) ¯ Ax, z (cid:11) . We compare v (cid:15) (0 , x ) and ¯ v (0 , r ) and call δ v ( (cid:15) ) = | v (cid:15) (0 , x ) − ¯ v (0 , r ) | , -1.5 -1 -0.5 0 0.5 1 1.5 r -1.5-1-0.500.511.5 r =1.0=2.0 Figure 2.
Vector field f of the limit triad system for A = A = 1and A = − λ .Denoting by u (cid:15) and u the corresponding optimal controls for any given noise coef-ficient σ (that can expressed in terms of the value function for fixed σ ), we have v (cid:15) (0 , x ) = ˆ E (cid:34)(cid:90) T | u (cid:15)s | ds (cid:35) , ¯ v (0 , r ) = ˆ E (cid:34)(cid:90) T | u s | ds (cid:35) . The simulation results are shown in the following table: (cid:15) δ v Triad system for climate prediction.
We consider a stochastic climatemodel which can be represented as a bilinear system with additive noise [32](40) dX (cid:15) ( t ) = 1 (cid:15) L ( X (cid:15) ( t )) dt + 1 (cid:15) B ( X (cid:15) ( t ) , X (cid:15) ( t )) dt + 1 (cid:15) Σ dW t , X (cid:15) (0) = x, where X (cid:15) ( t ) = ( R (cid:15) ( t ) , R (cid:15) ( t ) , U (cid:15) ( t )) ∈ R and L ( x ) = − u , B ( x, x ) = A r uA r uA r r , Σ = λ , where 0 < (cid:15) (cid:28)
1, and A , A , A are real numbers such that A + A + A = 0 , and λ ∈ [ σ, σ ]is the unknown diffusion coefficient. Equation (40), which is a time rescaled versionof (1a)–(1b), is a simplified stochastic turbulence model that comprises triad waveinteractions between two climate variables r , r and a single stochastic variable u .The noise level λ cannot be accurately estimated, nevertheless it may have a hugeimpact on the dynamics, even though there are no bifurcations for λ >
0. Equation(40) can thus be considered an SDE driven by a G-Brownian motion. It is shown
ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 13 in [32] that, for any finite value λ >
0, the first two components R (cid:15) = ( R (cid:15) , R (cid:15) )converge strongly in L p for p = 1 , , T ] to thesolution of the nonlinear SDE with multiplicative noise(41) dR ( t ) = f ( R ( t )) dt + σ ( R ( t )) dW t , R (0) = r, where R ( t ) = ( R ( t ) , R ( t )) and f ( r ) = (cid:32) A r ( A r + λ A ) A r ( A r + λ A ) (cid:33) , σ ( x ) = λγ (cid:18) A r A r (cid:19) . The pathwise convergence R (cid:15) → R together with the stability result of Zhang andChen [44, Thm. 3.1] implies thatˆ E ( sup t ∈ [0 ,T ] | R (cid:15) ( t ) − R ( t ) | ) → (cid:15) → I ( r , r ) = A r − A r is a conserved quantity for both the reduced and the original system. We considerthe case A , A > A <
0, in which case the level sets of I are hyperbola,and the origin is an unstable hyperbolic equilibrium. The rays that connect theorigin with any of the four equilibria r ∗± , ± = (cid:32) ± σ (cid:115) A | A | , ± σ (cid:115) A | A | (cid:33) , A , A > . are (locally hyperbolically unstable) invariant sets. Figure 2 shows representativevector fields f of the limit system for different noise coefficients λ = 1 . λ = 2 . A = A >
0. It can be seen that the repulsive and attractive regions on theinvariant diagonals change as the coefficient λ varies.For illustration, Figure 3 shows three representative samples of R (0 .
5) for A =0 . A = 0 .
25 and A = − .
0, with λ = 1 . λ = 1 . λ = 1 .
0, all startingfrom the same initial value R (0) = (1 , − λ in a non-trivial fashion. Goal-oriented uncertainty quantification.
We now compare the full triad sys-tem (40) and the limit system (41) for a specific quantity of interest (QoI) usingthe G-BSDE framework. To this end, we consider the QoI mean (42) v (cid:15) ( t, x ) = E t,x ( X (cid:15) ( T )) , v ( t, r ) = E t,r ( R ( T ))as a function of the initial data ( t, x ) and ( t, r ) where x = ( r, u ) = ( r , r , u ) and T > v (cid:15) and v solve the followingnonlinear dynamic programming (HJB-type) equations(43) ∂v (cid:15) ∂t + G ( a (cid:15) : ∇ v (cid:15) ) + (cid:104)∇ v (cid:15) , b (cid:15) (cid:105) = 0 , v (cid:15) ( T, x ) = x and(44) ∂v∂t + G ( a : ∇ v + (cid:104)∇ v, f (cid:105) ) + (cid:104)∇ v, f (cid:105) = 0 , v ( T, r ) = r , with the shorthands b (cid:15) = 1 (cid:15) L + 1 (cid:15) B , a (cid:15) = 1 (cid:15) ΣΣ T , a = σσ T , f = λ A A r , f = f − f . -1 0 1 2-2.4-2.3-2.2-2.1-2-1.9 invariant manifold=1.0=1.5=2.0 Figure 3.
Independent realisations of the limit triad system for A = A = 1 and A = − λ ∈ [1 , T = 0 .
5. For every parameter value, we have gener-ated 100 independent realisations, all starting from the same initialvalue r = (1 , − λ , nevertheless thedynamics on the invariant manifolds are different.The nonlinearity G in (43) and (44) is defined by G ( x ) = x (cid:26) σ if x ≥ σ if x < G as the nonlinear generator of the parameter-dependent partof the corresponding G-SDE.) We solve the fully nonlinear HJB equations by ex-ploiting the aforementioned relation to second-order BSDE (2BSDE) and using thedeep learning approximation developed by Beck et al. [6]. Numerical results.
As a first example, we consider the triad system and its ho-mogenisation limit, with the parameters A = A = 1 , A = − λ ∈ [0 . , . T = 0 . x = ( r, u ) = (1 , − , − T the 2BSDE solution for (cid:15) = 0 . v (cid:15) (0 , x ) = 0 . v (0 , r ) = 0 . | v (cid:15) (0 , x ) − v (0 , r ) | v (0 , r ) = 0 . (cid:15) = 0 .
2, but with the different set of parameters A = 1 , A = 2 , A = − λ ∈ [0 . , .
2] and T = 0 .
5, and found v (cid:15) (0 , x ) = 1 . v (0 , r ) = 1 . | v (cid:15) (0 , x ) − v (0 , r ) | v (0 , r ) = 0 . . ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 15 t * Figure 4.
The plot shows the parameter σ ∗ that maximises thenonlinear part G of the generator in (45) for fixed initial conditionover the noise coefficient λ ∈ [1 ,
2] .It is illustrative to consider the parameter for which the maximum in the nonlin-ear part G of the generator is attained. For example, for the original triad system,(45) G ( a (cid:15) : ∇ v (cid:15) ) = max λ ∈ [ σ,σ ] a (cid:15) ( λ ) : ∇ v (cid:15) = 1 (cid:15) max λ ∈ [ σ,σ ] λ ∂ v (cid:15) ∂u , which is identically equal to σ if v (cid:15) is strictly concave in its third argument, u , andequal to σ if it is strictly convex in u . For a G-PDE of the form (43) that containsno running cost, one can show that the value function is strictly convex or concave ifthe terminal condition is strictly convex or concave (since the solution of the forwardSDE is a strictly increasing function of the initial value). In general, however, itis not the convexity that determines, for which parameter value the maximum isattained, as the limit G-PDE (44) shows. In fact, the optimal parameter will be afeedback function that depends on ( t, x ) or ( t, r ).Figure 4 shows the maximiser in (45) as function of t for a fixed value of x . Itcan be seen that the optimal parameter value is time-dependent, which underpinsthe fact that the optimal parameter depends on the QoI (here also through theinitial data) in a nontrivial way; cf. Figure 3.As a final numerical test, we consider the triad system with A = 0 . , A =0 . , A = − λ ∈ [1 , T = 0 . (cid:15) = 0 . v (cid:15) (0 , x ) = 0 . v (0 , r ) = 0 . | v (cid:15) (0 , x ) − v (0 , r ) | v (0 , r ) = 0 . . Conclusions
We have sketched a general framework for goal-oriented uncertainty quantifica-tion and model reduction of parametric multiscale diffusions. The framework isbased on the notion of sublinear G-expectations and the related G-Brownian mo-tion. The sublinear expectation framework allows to define worst-case scenariosfor any given, possibly path-dependent quantity of interest (QoI), and we haveproved pathwise convergence of the corresponding G-BSDE for the case when themultiscale system depends on a small parameter that can be sent to zero.Given the rather restrictive assumptions in this paper, it can only serve as astarting point for further studies. For example, we have assumed that the unknownparameters are from a compact set, and it would be desirable to allow for unboundedparameters. Since the nonlinear generator of the underlying G-Brownian motionmay not be unambigously defined then, this calls for a suitable regularisation that islikely to have a Bayesian interpretation that may open up new algorithmic possibil-ities to quantify the uncertainty in the reduced system. Another obvious extensionof this study is a formulation of the G-BSDE of the reduced system in a purelydata-driven fashion using simulation data from the original model (rather than itsvalue function). This will lead to a tracking-type functional for the QoI that entersthe G-BSDE that determines the worst-case parameter(s) for the reduced model.Finally, in the combination with controlled systems, questions regarding the com-muntativity of the control optimisation with the nonlinear expectation remain tobe addressed. All this questions will be adressed in forthcoming papers.
Acknowledgement
This work has been partially supported by the Collaborative Research Center
Scaling Cascades in Complex Systems (DFG-SFB 1114) through project A05 andby the MATH+ Cluster of Excellence (DFG-EXC 2046) through the projects EP4-4and EF4-6. Hafida Bouanani gratefully acknowledges funding by ATRST (Algeria).
Appendix A. Nonlinear expectation
To begin with, we fix the notation and review the fundamentals of G-Brownianmotion and the related nonlinear expectation [19].Let Ω = C ([0 , ∞ )) denote the the space of real valued continuous functions( ω t ) t ≥ with the property ω = 0. We denote by C b , lip ( R d ) the space of boundedand Lipschitz continuous functions on R d and, for every T >
0, we define L ip (Ω T ) = (cid:8) ϕ ( B t , . . . , B t n ) : n ≥ , t , . . . , t n ∈ [0 , T ] , ϕ ∈ C b , lip ( R d × n ) (cid:9) and L ip (Ω) = ∞ (cid:91) T =0 L ip (Ω T ) . We call the corresponding Banach space L (Ω) = ( L ip (Ω) , (cid:107) · (cid:107) ∞ )the Lipschitz space on Ω and define L p (Ω) = { X ∈ L (Ω) : | X | p ∈ L (Ω) } . Follow-ing Peng [39], a sublinear expectation (also called: G-expectation) is a functionalˆ E ( · ) : L ip (Ω) → R with the following properties:(1) ˆ E ( X ) ≥ ˆ E ( Y ) if X ≥ Y ( monotonicity ) ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 17 (2) ˆ E ( l ) = l for every l ∈ R ( preservation of constant )(3) ˆ E ( X + Y ) ≤ ˆ E ( X ) + ˆ E ( Y ) ( sub-additivity )(4) ˆ E ( λX ) = λ ˆ E ( X ) for all λ ≥ positive homogeneity ).The triple (Ω , L ip (Ω) , ˆ E ) is called a sublinear expectation space. The sublinearexpectation admits a variational representation in terms of a family { E P : P ∈ P} of linear expectations where P is a family of probability measures on (Ω , B (Ω)):(46) ˆ E ( X ) = max P ∈P E P ( X ) , X ∈ L ip (Ω)The corresponding canonical process ( B t ) t ≥ on the sublinear expectation space(Ω , L ip (Ω) , ˆ E ) is called a G-Brownian motion and is characterized as follows:
Definition 4. A d -dimensional process ( B t ) t ≥ is called a G-Brownian motionunder the sublinear expectation ˆ E if the following properties hold: (1) B ( ω ) = 0 . (2) For every t, s ≥ and any n ∈ N , the increment B t + s − B t is independentof of the collection ( B t , B t , . . . , B t n ) of random variables, ≤ t ≤ . . . ≤ t n ≤ t where two random vectors X, Y are independent if for all boundedand Lipschitz continuous test functions ϕ ˆ E ( ϕ ( X, Y )) = ˆ E (ˆ E ( ϕ ( x, Y ) | x = X )(3) For every t, s ≥ , the increment B t + s − B t is normally distributed withmean 0 and covariance s Σ where Σ ∈ R d × d is a symmetric and positivesemidefinite matrix that is independent of s or t . An interesting property of the G-Brownian motion is that its quadratic variation (cid:104) B (cid:105) has almost the same properties as the G-Brownian motion itself: (cid:104) B (cid:105) = 0, theincrement (cid:104) B (cid:105) t + s − (cid:104) B (cid:105) t is independent of the collection( (cid:104) B (cid:105) t , (cid:104) B (cid:105) t , . . . , (cid:104) B (cid:105) t n ) , ≤ t ≤ . . . ≤ t n ≤ t of quadratic variations, and (cid:104) B (cid:105) t + s − (cid:104) B (cid:105) t d = (cid:104) B (cid:105) s .A.1. Path properties of G-Brownian motion.
We will need the following in-equalities in the G-framework, all of which have straighforward interpretations interms of the standard Brownian motion under the linear expectation. We define M p ([0 , T ]) = (cid:40) η t = N − (cid:88) i =0 ξ i { t i ,t i +1 } : 0 = t < . . . < t N = T, ξ i ∈ L p (Ω t i ) (cid:41) to be the space of simple processes on [0 , T ] and call M pG ([0 , T ]) the completion of M p ([0 , T ]) under the norm (cid:107) η (cid:107) M,p = (cid:32) ˆ E (cid:34)(cid:90) T | η s | p ds (cid:35)(cid:33) p and H pG ([0 , T ]) the completion of M p ([0 , T ]) under the norm (cid:107) η (cid:107) H,p = ˆ E (cid:32)(cid:90) T | η s | ds (cid:33) p p . In the following, let B be a G-Brownian motion, with l = − ˆ E ( −| B | ) ≤ ˆ E ( | B | ) = ¯ l . Proposition 5. (Itˆo isometry inequality, [39, Prop. 6.4])
Let β ∈ M pG ([0 , T ]) forsome p ≥ . Then (cid:90) T β t dB t ∈ L p (Ω T ) and (47) ˆ E (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T β t dB t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:33) ≤ C p ˆ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T β t d (cid:104) B (cid:105) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p , for some C p > . Proposition 6. (Burkholder-Davis-Gundy inequality, [24, Prop. 2.6])
For each η ∈ H αG ([0 , T ]) for some α ≥ and p ∈ (0 , α ] , we have (48) l p c p ˆ E (cid:32)(cid:90) T η s ds (cid:33) p ≤ ˆ E (cid:34) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t η s dB s (cid:12)(cid:12)(cid:12)(cid:12) p (cid:35) ≤ ¯ l p C p ˆ E (cid:32)(cid:90) T η s ds (cid:33) p where < c p < C p < ∞ are constants. Proposition 7. (Isometry inequality, [5, Lemma 2.19])
Let p ≥ , η ∈ M pG ([0 , T ]) and ≤ s ≤ t ≤ T . Then (49) ˆ E (cid:18) sup s ≤ u ≤ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) us η r d (cid:104) B (cid:105) r (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) ≤ (cid:18) l + ¯ l (cid:19) p ( t − s ) p − ˆ E (cid:18)(cid:90) ts | η u | p du (cid:19) . The previous inequalities in Propositions (47), (48) and (49) hold true also forintervals in the form [ t, τ ] . Now let S ([0 , T ]) := (cid:8) h ( t, B t ∧ t , . . . , B t n ∧ t ) : t , . . . , t n ∈ [0 , T ] , h ∈ C b , lip ( R n +1 ) (cid:9) , with S pG ([0 , T ]) denoting the completion of S ([0 , T ]) under the norm (cid:107) η (cid:107) S,p = (cid:32) ˆ E (cid:34) sup s ∈ [0 ,T ] | η s | p (cid:35)(cid:33) p . Corollary 8.
For θ ∈ S G , we have ˆ E (cid:34)(cid:90) T | θ s | d (cid:104) B (cid:105) s (cid:35) ≤ T ¯ l ˆ E (cid:34) sup s ∈ [0 ,T ] | θ s | (cid:35) . Furthermore, for any η ∈ H G ([0 , T ]) , the process (cid:18)(cid:90) t η s θ s dB s (cid:19) t ∈ [0 ,T ] is a uniformly integrable martingale, with ˆ E (cid:34)(cid:90) Tt η s θ s dB (cid:35) = 0 . ODEL REDUCTION OF PARAMETRIC MULTISCALE DIFFUSIONS 19
A.2.
Some inqualities.
We have
Lemma 9.
For r > and < q, p < ∞ , with p + q = 1 , we have (50) | a + b | r ≤ max { , r − } ( | a | r + | b | r ) for a, b ∈ R (51) | ab | ≤ | a | p p + | b | q q . Using the sublinearity of ˆ E , the following is a straight consequence (see [39]): Proposition 10.
For any
X, Y so that the moments below exist, we have (52) ˆ E ( | X + Y | r ) ≤ r − (cid:16) ˆ E ( | X | r ) + ˆ E ( | Y | r ) (cid:17) (53) ˆ E ( XY ) ≤ (cid:16) ˆ E ( | X | p ) p + ˆ E ( | Y | q ) q (cid:17) (54) (cid:16) ˆ E ( | X + Y | p ) (cid:17) p ≤ (cid:16) ˆ E ( | X | p ) (cid:17) p + (cid:16) ˆ E ( | Y | p ) (cid:17) p , where r ≥ and < p, q < ∞ , with p + q = 1 . In particular, for ≤ p < p (cid:48) , (cid:16) ˆ E ( | X | p ) (cid:17) p ≤ (cid:16) ˆ E ( | X | p (cid:48) ) (cid:17) p (cid:48) . Appendix B. We give the formal definition of a 2BSDE. For details we refer to [10].
Definition 11 (2BSDE) . Let ( t, x ) ∈ [0 , T ) × R d , ( X t,xs ) s ∈ [ t,T ] a diffusion processand ( Y s , Z s , Γ s , A s ) s ∈ [ t,T ] a quadruple of F t,T -progressively measurable processestaking values in R , R d , S d and R d , respectively. Then we say that the quadru-ple ( Y, Z, Γ , A ) is a solution to the second order backward stochastic differentialequation (2BSDE) corresponding to ( X t,x , f, g ) if dY s = f ( s, X t,xs , Y s , Z s , Γ s ) ds + Z (cid:48) s ◦ dX t,xs , s ∈ [ t, T ) , (55) dZ s = A s ds + Γ s dX t,xs , s ∈ [ t, T ) , (56) Y T = g (cid:0) X t,xT (cid:1) , (57) where Z (cid:48) s ◦ dX t,xs denotes Fisk–Stratonovich integration, which is related to Itˆo in-tegration by Z (cid:48) s ◦ dX t,xs = Z (cid:48) s dX t,xs + 12 d (cid:10) Z, X t,xs (cid:11) = Z (cid:48) s dX t,xs + 12 Tr[Γ s σ ( X t,xs ) σ ( X t,xs ) (cid:48) ] ds . Now we present the relation between the 2BSDE (55)-(57) and fully non-linearparabolic PDEs: Let f : [0 , T ) × R d × R × R d × S d → R and g : R d → R arecontinuous functions, and assume v : [0 , T ] × R d → R is a C , function such that v t , Dv, D v, L Dv ∈ C ([0 , T ) × R d )and v solves the PDE(58) − v t ( t, x ) + f (cid:0) t, x, v ( t, x ) , Dv ( t, x ) , D v ( t, x ) (cid:1) = 0 on [0 , T ) × R d , with terminal condition(59) v ( T, x ) = g ( x ) , x ∈ R d in the classical sense. Then it follows directly from Itˆo’s formula that for each pair( t, x ) ∈ [0 , T ) × R d , the processes Y s = v (cid:0) s, X t,xs (cid:1) , s ∈ [ t, T ] ,Z s = Dv (cid:0) s, X t,xs (cid:1) , s ∈ [ t, T ] , Γ s = D v (cid:0) s, X t,xs (cid:1) , s ∈ [ t, T ] ,A s = L Dv (cid:0) s, X t,xs (cid:1) , s ∈ [ t, T ] , solve the 2BSDE corresponding to ( X t,x , f, g ).The converse is also true: The first component of the solution of the 2BSDE(55) at the initial time is a solution of the fully nonlinear PDE (58). We use this2BSDE representation in Section 4 to solve fully nonlinear dynamic programmingequations associated with a G-(B)SDE control problem. References [1] F. Antonelli. Backward forward stochastic differential equations.
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Laboratory of Stochastic Models, Statistics and Applications, University of SaidaDr Moulay Tahar, Algeria
Email address : [email protected] Institute of Mathematics, Brandenburgische Technische Universit¨at Cottbus-Senftenberg,03046 Cottbus, Germany
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