Continuous-time perpetuities and time reversal of diffusions
aa r X i v : . [ m a t h . P R ] J a n CONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OFDIFFUSIONS
CONSTANTINOS KARDARAS AND SCOTT ROBERTSON
Abstract.
We consider the problem of estimating the joint distribution of a continuous-timeperpetuity and the underlying factors which govern the cash flow rate, in an ergodic Markovianmodel. Two approaches are used to obtain the distribution. The first identifies a partial differentialequation for the conditional cumulative distribution function of the perpetuity given the initialfactor value, which under certain conditions ensures the existence of a density for the perpetuity.The second (and more general) approach, using techniques of time reversal, identifies the joint lawas the stationary distribution of an ergodic multi-dimensional diffusion. This later approach allowsfor efficient use of Monte-Carlo simulation, as the distribution is obtained by sampling a singlepath of the reversed process.
Introduction
Discussion.
In this article, we consider a continuous-time perpetuity given by the random variable(0.1) X := Z ∞ D t f ( Z t )d t. Above, Z = ( Z t ) t ∈ R + represents the value of an economic factor that determines a cash flow rate( f ( Z t )) t ∈ R + . Cash flows are discounted according to D = ( D t ) t ∈ R + ; therefore, X represents thewhole payment in units of account at time zero. Our main concern is the identification of anefficient means to obtain the joint distribution of ( Z , X ), as naive estimation of the distributionby simulating sample paths of Z and approximating X through numerical integration may beprohibitively slow. As Z is typically observable, the joint distribution of ( Z , X ) also allows usto obtain the conditional distribution of X given Z .In order to make the problem tractable, we work in a diffusive, Markovian environment where Z and D are solutions to the respective stochastic differential equations (written in integrated form) (0.2) Z = Z + Z · m ( Z t )d t + Z · σ ( Z t )d W t , (0.3) D = 1 − Z · D t (cid:0) a ( Z t )d t + θ ( Z t ) ′ σ ( Z t )d W t + η ( Z t ) ′ d B t (cid:1) . Date : July 26, 2018. Throughout the text, the prime symbol ( ′ ) denotes transposition. In the above equations, W and B are independent Brownian motions of dimension d and k respec-tively, while m , σ , a , θ and η are given functions. (Precise assumptions on all the model coefficientsare given in Section 1.) We assume Z is stationary and ergodic with invariant density p . Equation(0.3) includes in particular the case when D is smooth; in other words D = exp (cid:0) − R · a ( Z t )d t (cid:1) ,where a represents a short-rate function. However, the more general form of (0.3) is considered toaccommodate a broader range of situations. For example: • when payment streams are sometimes denominated in units of different account (for ex-ample, another currency, or financial assets), in which case discounting has to take intoaccount the “exchange rate”. • when, for pricing purposes, the payment stream, though denominated in domestic currency,must incorporate both traditional discounting and the density of the pricing kernel.The two main results of the paper—Theorem 2.1 and Theorem 2.4—identify the distribution of( Z , X ) in different ways. First, in the case where η in (0.3) is non-degenerate and f in (0.1) issufficiently regular, the conditional cumulative distribution function of X given Z is shown tocoincide with the explosion probability of an associated locally elliptic diffusion and, hence, throughthe Feynman-Kac formula satisfies a partial differential equation (PDE): see Theorem 2.1. Second,for general η and f , using methods of diffusion time-reversal, we identify an “ergodic” process( ζ, χ ) whose invariant distribution coincides with the joint distribution of ( Z , X ). In particular,for any fixed starting point x > χ , the (random) empirical time-average law of ( ζ, χ ) on [0 , T ]almost surely converges to the joint distribution of ( Z , X ) in the weak topology: see Theorem2.4. The time-reversal result has the advantage of leading to an efficient method for obtaining thedistribution via simulation, as the ergodic theorem enables estimation of the entire distributionbased upon a single realization of ( ζ, χ ); a numerical example in Section 3 dramatically reinforcesthis point. However, it must be noted that the invariant distribution p for Z appears in the reverseddynamics, and hence must be known to perform simulation. When Z is one-dimensional, or moregenerally, reversing, p is given in explicit form with respect to the model parameters. In the generalmulti-dimensional setup, lack of knowledge of p could pose an issue; however, we provide a potentialway to amend the situation in the discussion after Theorem 2.4. Note also that in the PDE resultin Theorem 2.1, explicit knowledge of p is not necessary. Existing literature and connections.
Obtaining the distribution of the perpetuity X is ofgreat importance in the areas of finance and actuarial science; for this reason, perpetuities with aform similar to X have been extensively studied. For example, [12] deals with the case where X = Z ∞ e − σB t − νt d t, establishing that X has an inverse gamma distribution. This fits into the set-up of (0.2), (0.3) bytaking a = ν − σ / f = 1, θ = 0 and η = σ . Note that here Z plays no role. In a similar manner, ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 3 [32, 10, 11] consider the case X = Z ∞ e − R t Z u d u d t ; d Z t = κ ( θ − Z t )d t + ξ p Z t d W t ; E = (0 , ∞ ) , and obtain the first moment, along with bounds for other moments, of X . In [17], the perpetuitytakes the form(0.4) X = Z ∞ e − Q t dP t , with P and Q being independent L´evy processes . Under certain conditions on P and Q , the distribution of X is implicitly calculated by identifyingthe characteristic function and/or Laplace transform for X . In fact, the results of [17] are pre-dated (for highly particular P and Q ), in [25, 22]. The Laplace transform method is also used in[27, 26] to treat (0.4) when P t = t and Q is a diffusion. In addition to identifying a degenerateelliptic partial differential equation for the Laplace transform, they propose a candidate recurrentMarkov chain whose invariant distribution has the law of X . Lastly, the setup of [17] is significantlyextended in [7] where, under minimal assumptions on P and Q , the distribution of X is shown tocoincide with the unique invariant measure for a certain generalized Ornstein-Uhlenbeck process,a relationship that is confirmed in our current setting in Proposition 8.2.The use of time-reversal to identify the distribution of a discrete-time perpetuity is well known,dating at least back to [13], where X takes the form X = ∞ X n =1 n Y i =1 D i ! f n , where the discount factors ( D n ) n ∈ N and cash flows ( f n ) n ∈ N are two independent sequences ofindependent, identically distributed (iid) random variables. To provide insight, the time-reversalargument in [13] is briefly presented here. With X ( N )0 := P Nn =1 ( Q ni =1 D i ) f n it is clear by theiid property that X ( N )0 has the same distribution as e X N := D N f N + D N D N − f N − + .... + (cid:16)Q Nj =1 D j (cid:17) f . Straightforward calculations show that the reversed process ( e X n ) n ∈ N satisfies therecursive equation e X n = D n (cid:0) e X n − + f n (cid:1) . Thus, assuming that ( e X n ) n ∈ N converges to a randomvariable e X in distribution, e X must solve the distributional equation e X = D ( e X + f ), where D , f and e X are independent, D has the same law as D and f has the same law as f . In [31] solutionsto the aforementioned distributional equation are obtained based upon the expectation of log( | D | )and log + ( | Df | ). The tails of e X , as well as convergence of iterative schemes, are studied in [15];furthermore, [18] gives “almost” if and only if conditions for the convergence of iterative schemes.In a continuous time setting, we employ an argument similar in spirit, but rather different in exe-cution, to [13]. Specifically, we extend X to a whole “forward” process X := (1 /D ) R ∞· D t f ( Z t ) dt and then, for each T > ζ T , χ T ) on [0 , T ] by ζ Tt := Z T − t , χ Tt := X T − t :see (2.7), (2.8). Using results on time reversal of diffusions from [20] (alternatively, see [24, 3, 8, 14]), CONSTANTINOS KARDARAS AND SCOTT ROBERTSON as well as additional elementary calculations, we obtain the dynamics for ( ζ T , χ T ). In fact, Propo-sition 7.5 shows the generator of ( ζ T , χ T ) does not depend upon T and ergodicity can be studiedfor the process ( ζ, χ ) with the given generator. When | η | > E and f is sufficiently regular, thisgenerator is locally elliptic and the associated process ( ζ, χ ) is ergodic with invariant distributionequalling that of ( Z , X ): see Proposition 8.2. In the general case a slightly weaker (but stillsufficient) form of ergodicity still holds: starting ζ off its invariant distribution p and χ off anystarting point x >
0, the (random) empirical time-average laws of ( ζ, χ ) converge almost surely inthe weak topology to the distribution of ( Z , X ). Structure.
This paper is organized as follows: in Section 1 we precisely state the given assump-tions on the processes Z and D , as well as the function f , paying particular attention to derivingsharp conditions under which X is almost surely finite or infinite. The main results are thenpresented in Section 2. First, when | η | > E and f is sufficiently regular, the conditionalcumulative distribution function of X given Z = z is shown to satisfy a certain partial differentialequation. Then, using the method of time reversal, we construct a probability space and diffusion( ζ, χ ) such that with probability one its empirical time-average laws weakly converge to the jointdistribution of ( Z , X ) for all starting points of χ . Section 2 concludes with a brief discussion howthe distribution may be estimated via simulation, in particular proposing a method for obtainingthe desired distribution when the invariant density p for Z is not explicitly known. Section 3provides a numerical example in a specific case where the joint distribution of ( Z , X ) is explicitlyidentifiable. Here, we compare the performance of the reversal method versus the direct methodfor obtaining the distribution of X . In particular we show that for a given desired level of accuracy(see Section 3 for a more precise definition), the method of time reversal is approximately 175 to 300times faster than the direct method. The remaining sections contain the proofs: Section 5 provesthe statements regarding the finiteness of X ; Section 6 proves the partial differential equationresult; Section 7 obtains the dynamics for the time-reversed process ( ζ, χ ); Section 8 proves the(weak) ergodicity with the correct invariant distribution. Finally, a number of technical supportingresults are included in the appendix. 1. Problem Setup
Well-posedness and ergodicity.
The first order of business is to specify precise coefficientassumptions so that Z in (0.2) and D in (0.3), are well-defined. As for Z , we work in the standardlocally elliptic set-up for diffusions: for more information, see [28]. Let E ⊆ R d be an open,connected region. We assume the existence of γ ∈ (0 ,
1] such that:(A1) there exists a sequence of regions ( E n ) n ∈ N such that E = S ∞ n =1 E n , each E n being open,connected, bounded, with ∂E n being C ,γ and satisfying ¯ E n ⊂ E n +1 for all n ∈ N . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 5 (A2) m ∈ C ,γ ( E ; R d ) and c ∈ C ,γ ( E ; S d ++ ), where S d ++ is the space of symmetric and strictlypositive definite ( d × d )-dimensional matrices.With the provisos in (A1) and (A2), define L Z as the generator associated to ( m, c ):(1.1) L Z := 12 d X i,j =1 c ij ∂ ij + d X i =1 m i ∂ i . Under (A1) and (A2), one can infer the existence of a solution to the martingale problem for L Z on E , with the possibility of explosion to the boundary of E : see [28] We wish for something stronger;namely, to construct a filtered probability space (Ω , F , P ) on which there is a strong, stationary,ergodic solution to the SDE in (0.2) with invariant density p . In (0.2), W is a d -dimensionalBrownian Motion and σ = √ c , the unique positive definite symmetric matrix such that σ = c . Inorder to achieve this, we ask that(A3) The martingale problem for L Z on E is well posed and the corresponding solution is re-current. Furthermore, there exists a strictly positive p ∈ C ,γ ( E, R ) with R E p ( z )d z = 1satisfying ˜ L Z p = 0, where ˜ L Z is the formal adjoint of L Z given by(1.2) ˜ L Z := 12 c ij ∂ ij − (cid:0) m i − ∂ j c ij (cid:1) ∂ i − (cid:18) ∂ i m i − ∂ ij c ij (cid:19) . We summarize the situation in the following result: the extra Brownian motion B in its statementwill be used to define the process D via (0.3) later on. Theorem 1.1.
Under assumptions (A1), (A2) and (A3), there exists a filtered probability space (Ω , F , P ) satisfying the usual conditions supporting two independent Brownian motions W and B , d -dimensional and k -dimensional respectively, such that Z satisfies (0.2) and is stationary andergodic with invariant density p .Remark . According to [28, Corollary 5.1.11], in the one-dimensional case, where E = ( α, β ) for −∞ ≤ α < β ≤ ∞ , the above assumption (A3) is true if and only if for some z ∈ E Z z α exp (cid:18) − Z zz m ( s ) c ( s ) d s (cid:19) d z = ∞ , Z βz exp (cid:18) − Z zz m ( s ) c ( s ) d s (cid:19) d z = ∞ , Z βα c ( z ) exp (cid:18) Z zz m ( s ) c ( s ) d s (cid:19) d z < ∞ . In this case, it holds that p ( z ) = Kc − ( z ) exp (cid:18) Z zz m ( s ) c ( s ) d s (cid:19) , z ∈ ( α, β ) , In the sequel the summands will be omitted using Einstein’s convention; therefore, L Z is written as L Z =(1 / c ij ∂ ij + m i ∂ i . CONSTANTINOS KARDARAS AND SCOTT ROBERTSON where
K > H : E R with the propertythat c − (2 m − div ( c )) = ∇ H , where div ( c ) is the (matrix) divergence defined by div ( c ) i = ∂ j c ij , i = 1 , ..., d . Then, Z is a reversing Markov process. Furthermore Assumption (A3) follows ifit can be shown that Z does not explode to the boundary of E and K := R E exp( H ( z ))d z < ∞ .Indeed, by construction p = e H /K satisfies ˜ L Z p = 0, R E p ( z )d z = 1. Thus, if Z does not explode,it follows from [28, Theorem 2.8.1, Corollary 4.9.4] that Z is recurrent. In fact, Z is ergodic, asshown in [28, Theorems 4.3.3, 4.9.5]. Absent the reversing case, there are many known techniquesfor checking ergodicity—see [6, 28]. For example, if there exist a smooth function u : E R , aninteger N and constants ε > C > L Z u ≤ − ε and u ≥ − C on E \ E N , then (A3)holds.In order to ensure that D in (0.3) is well defined, we assume that(A4) a ∈ C ,γ ( E ; R + ), η ∈ C ,γ ( E ; R k ), and θ ∈ C ,γ ( E ; R d ).Given (A4) and all previous assumptions, it follows that (0.3) possesses a strong solution on(Ω , F , P ) of Theorem 1.1; in fact, defining R := − log( D ), it holds that(1.3) R = Z · (cid:18) a + 12 (cid:0) θ ′ cθ + | η | (cid:1)(cid:19) ( Z t )d t + Z · θ ( Z t ) ′ σ ( Z t )d W t + Z · η ( Z t ) ′ d B t . Finiteness of X . Having the set-up for the existence of Z and D , we proceed to X . Forthe time being, we shall just assume that the function f : E R + is in L ( E, p ) . For the PDEresults of Theorem 2.1 below we require a slightly stronger regularity assumption on f , though thetime-reversal results of Theorem 2.4 make no additional assumptions. Now, for f not necessarilyin L ( E, p ), it is entirely possible that X takes infinite values with positive probability. In thissection, conditions are given under which P [ X < ∞ ] = 1 or, conversely, when P [ X < ∞ ] = 0. Lemma 1.3.
Let (A1), (A2), (A3) and (A4) hold. For the invariant density p of Z , assume thereexists ε > such that (1.4) (cid:18) a + 1 − ε θ ′ cθ + η ′ η ) (cid:19) − ∈ L ( E, p ) , and Z E (cid:18) a + 1 − ε θ ′ cθ + η ′ η ) (cid:19) ( z ) p ( z )d z > . Then, the following hold:i) There exists κ > such that for all z ∈ E , P (cid:2) lim t →∞ e κt D t = 0 | Z = z (cid:3) = 1 . In particular, lim t →∞ e κt D t = 0 P a.s..ii) For any f ∈ L ( E, p ) , it holds that P [ X < ∞ ] = 1 . This definition is equivalent to the standard definition of divergence for matrices, where the divergence operator isapplied to the columns, by the symmetry of c . Also, to differentiate the matrix divergence from its vector counterpart,we will write div ( A ) for symmetric matrices A and ∇ · v for vector valued functions v . We define L ( E, p ) to be those Borel measurable functions g on E so that R D | g ( z ) | p ( z ) dz < ∞ . Thus, Borelmeasurability is implicitly assumed throughout. ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 7
Remark . Note that (1.4) holds if a > E . The more complicated form in (1.4) allows a to take (unbounded) negative values. Furthermore, in the case where ( θ ′ cθ + η ′ η ) ∈ L ( E, p ) thenequation (1.4) is equivalent to:(1.5) (cid:18) a + 12 ( θ ′ cθ + η ′ η ) (cid:19) − ∈ L ( E, p ) , and Z E (cid:18) a + 12 ( θ ′ cθ + η ′ η ) (cid:19) ( z ) p ( z )d z > . As a partial converse to Lemma 1.3 we have
Lemma 1.5.
Let (A1), (A2), (A3) and (A4) hold. For the invariant density p of Z , assume thereexists ε > such that (1.6) (cid:18) a + 1 + ε θ ′ cθ + η ′ η ) (cid:19) + ∈ L ( E, p ) , and Z E (cid:18) a + 1 + ε θ ′ cθ + η ′ η ) (cid:19) ( z ) p ( z )d z ≤ . (If θ ′ cθ + η ′ η ≡ , then assume that a + ∈ L ( E, p ) and R E a ( z ) p ( z )d z < .) If f is such that R E f ( z ) p ( z )d z > , then P [ X < ∞ ] = 0 .Remark . Let (A1), (A2), (A3) and (A4) hold, and assume that a is non-negative. A combinationof Lemma 1.3 and Lemma 1.5 yield sharp conditions for the finiteness of X that do not requireknowledge of p , at least for bounded f . • If a + (1 / θ ′ cθ + η ′ η )
0, then P [ X < ∞ ] = 1 holds if f ∈ L ( E, p ). • If a + (1 / θ ′ cθ + η ′ η ) ≡ P [ X < ∞ ] = 0 holds if R E f ( z ) p ( z )d z > f ∈ L ( E, p ), R E f ( z ) p ( z )d z >
0, and there exists ε > (cid:18) a + 1 − ε θ ′ cθ + η ′ η ) (cid:19) − ∈ L ( E, p ) , and Z E (cid:18) a + 1 − ε θ ′ cθ + η ′ η ) (cid:19) ( z ) p ( z )d z > . To recapitulate, for the remainder of the article the following is assumed:
Assumption 1.7.
We enforce throughout all above assumptions (A1), (A2), (A3), (A4) and (A5).2.
Main Results
The distribution of X via a partial differential equation. Define the cumulative dis-tribution function g of X given Z by(2.1) g ( z, x ) := P [ X ≤ x | Z = z ] , ( z, x ) ∈ F := E × (0 , ∞ ) . Next, recall that Assumption 1.7 implies that Z has a density p , and define the joint distribution π of ( Z , X ) by(2.2) π ( A ) := Z Z A p ( z ) g ( z, d x )d z ; A ∈ B ( F ) . Under Assumption 1.7, as well as an additional smoothness requirement on f and non-degeneracyrequirement on η , the first main result (Theorem 2.1 below) shows g solves a certain PDE on the CONSTANTINOS KARDARAS AND SCOTT ROBERTSON state space F . This will imply that the joint distribution of ( Z , X ) has a density (still labeled π )and the law of X charges all of (0 , ∞ ).To motivate the result, as well as to fix notation, for each x ∈ (0 , ∞ ), consider the process(2.3) Y x := 1 D (cid:18) x − Z · D t f ( Z t )d t (cid:19) . Since Assumption 1.7 implies P [lim t →∞ D t = 0 | Z = z ] = 1 for all z ∈ E , it is clear that given Z = z , on { X < x } the process Y x tends to ∞ . Alternatively, on { X > x } , Y x will hit 0 atsome finite time. What happens on { X = x } is not immediately clear but it will be shown underthe given assumptions there is no probability of this occurring. For fixed ( z, x ) ∈ F , it followsthat 1 − g ( z, x ) equals the probability that Y x hits zero, given Z = z . According the Feynman-Kac formula, such probabilities “should” solve a PDE. To identify the PDE, note that the jointequations governing Z and Y x are Z = Z + Z · m ( Z t )d t + Z · σ ( Z t )d W t ,Y x = x + Z · (cid:0) − f ( Z t ) + Y xt (cid:0) a ( Z t ) + θ ′ cθ ( Z t ) + η ′ η ( Z t ) (cid:1)(cid:1) d t + Z · Y xt (cid:0) θ ′ σ ( Z t )d W u + η ( Z t ) ′ d B t (cid:1) . Define b : F R d +1 and A : F S d +1++ by(2.4) b ( z, x ) := m ( z ) − f ( z ) + x ( a + θ ′ cθ + η ′ η ) ( z ) ! ; A ( z, x ) := c ( z ) xcθ ( z ) xθ ′ c ( z ) x ( θ ′ cθ + η ′ η ) ( z ) ! , for all ( z, x ) ∈ F . Note that if, in addition to Assumptions 1.7, | η | ( z ) > , z ∈ E then A is locallyelliptic. Let L be the second order differential operator associated to ( A, b ), i.e.,(2.5) L := 12 A ij ∂ ij + b i ∂ i . Note that Lφ = L Z φ for functions φ of z ∈ E alone. With the previous notation, the first mainresult now follows. Theorem 2.1.
Let Assumptions 1.7 hold, and suppose further that a) f ∈ C ,γ ( E ; R + ) and b) | η ( z ) | > for all z ∈ E . Then, g ∈ C ,γ ( F ) satisfies Lg = 0 with the following “locally uniform”boundary conditions (2.6) lim n →∞ sup x ≤ n − ,z ∈ E k g ( z, x ) = 0; lim n →∞ inf x ≥ n,z ∈ E k g ( z, x ) = 1 , ∀ k ∈ N . Furthermore, g is unique within the class of solutions to Lg = 0 taking values in [0 , with theabove boundary conditions. ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 9
Remark . The non-degeneracy assumption on η is essential for the existence of a density; if η ≡ X has an atom. Indeed, take f ≡ a ≡ η ≡ θ ≡ X = R ∞ e − t d t = 1 with probability one. Remark . Theorem 2.1 implies the law of X charges all of (0 , ∞ ), even for those functions f which are bounded from above. Theorem 2.1 also implies that X has a density without imposingHormander’s condition [23, Chapter 2] on the coefficients in (2.4). Rather, the infinite horizoncombined with the presence of the independent Brownian motion B “smooth out” the distributionof X .Theorem 2.1 is certainly important from a theoretical viewpoint. However, it appears to be oflimited practical use. Even under the force of the extra non-degeneracy condition | η | >
0, it isunclear how to numerically solve the PDE Lg = 0 with the given boundary conditions (2.6), asthere are no natural auxiliary boundary conditions in the spatial domain of z ∈ E . In Subsection2.2 that follows we provide an alternative, more useful method for estimating numerically the lawof ( Z , X ).2.2. The distribution of ( Z , X ) via diffusion time-reversal. The goal here is to show thatthe distribution of ( Z , X ) coincides with the invariant distribution of a positive recurrent process( ζ, χ ). In order to see the connection, extend X to a whole process ( X t ) t ∈ R + defined via(2.7) X := 1 D Z ∞· D t f ( Z t )d t, and note that ( Z t , X t ) t ∈ R + is a stationary process under P . Fix T >
0, and define the process( ζ Tt , χ Tt ) t ∈ [0 ,T ] via time-reversal:(2.8) ζ Tt := Z T − t ; χ Tt := X T − t ; t ∈ [0 , T ] . It still follows that ( ζ T , χ T ) is stationary under P , with the same one-dimensional marginal distri-bution as ( Z , X ). Furthermore, stationarity of ( Z, X ) clearly implies that the law of the process( ζ T , χ T ) does not depend on T (except for its time-domain of definition). Therefore, one may cre-ate a new process ( ζ t , χ t ) t ∈ R + such that the law of ( ζ T , χ T ) is the same as the law of ( ζ t , χ t ) t ∈ [0 ,T ] for all t ∈ T . If one can establish that ( ζ, χ ) is ergodic, then the distribution of ( Z , X ) may beefficiently estimated via the ergodic theorem.Towards this end, one needs to understand the behavior of ( ζ, χ ). Standard results (e.g. [20])in the theory of time-reversal imply that ζ is a diffusion in its own filtration, and identify thecorresponding coefficients. In order to deal with χ , we return to the definition of χ T and define yetone more process (∆ Tt ) t ∈ [0 ,T ] via(2.9) ∆ Tt = D T D T − t , t ∈ [0 , T ] . Using all previous definitions, we obtain that χ Tt = X T − t = 1 D T − t Z ∞ T − t D u f ( Z u )d u = D T D T − t (cid:18) X T + Z TT − t D u D T f ( Z u )d u (cid:19) = ∆ Tt (cid:18) χ T + Z t Tu f ( ζ Tu )d u (cid:19) , t ∈ [0 , T ] . (2.10)As it turns out, one can describe the joint dynamics of ( ζ T , ∆ T ) in appropriate filtrations (and thesedynamics do not depend on T , as expected). To ease the presentation, recall from Section 1 thatfor any S d ++ valued smooth function A on E the (matrix) divergence is defined by div ( A ) i = ∂ j A ij for i = 1 , ..., d . It is then shown in Section 7 that ( ζ T , ∆ T ) is such that ζ T = ζ T + Z · (cid:18) c ∇ pp + div ( c ) − m (cid:19) ( ζ Tt )d t + Z · σ ( ζ Tt )d W Tt , ∆ T = 1 + Z · ∆ Tt (cid:18) θ ′ c ∇ pp + ∇ · ( cθ ) − a (cid:19) ( ζ Tt )d t + Z · ∆ Tt (cid:0) η ( ζ Tt ) ′ d B Tt + θ ′ σ ( ζ Tt )d W Tt (cid:1) = 1 + Z · ∆ Tt (cid:0) θ ′ ( m − div ( c )) + ∇ · ( cθ ) − a (cid:1) ( ζ Tt )d t + Z · ∆ Tt (cid:0) η ( ζ Tt ) ′ d B Tt + θ ( ζ Tt ) ′ d ζ Tt (cid:1) for independent Brownian motions ( W T , B T ) in an appropriate filtration.From the joint dynamics of ( ζ T , ∆ T ) one obtains the joint dynamics of ( ζ T , χ T ), which again donot depend on T . In particular, since ∆ T is a semimartingale, (2.10) yields that ζ T = ζ T + Z · (cid:18) c ∇ pp + div ( c ) − m (cid:19) ( ζ Tt )d t + Z · σ ( ζ Tt )d W Tt χ T = χ T + Z · (cid:18) f ( ζ Tt ) − χ Tt (cid:18) a − θ ′ c ∇ pp − ∇ · ( cθ ) (cid:19) ( ζ Tt ) (cid:19) d t + Z · χ Tt (cid:0) η ( ζ Tt ) ′ d B Tt + θ ′ c ( ζ Tt ) ′ d W Tt (cid:1) . (2.11)For a generic version ( ζ, χ ) with the same generator (which does not depend upon time) as( ζ T , χ T ) above, ergodicity of Z implies ergodicity of ζ (see Proposition 7.1 later on in the text).Furthermore, χ is “mean reverting” as can easily be seen when θ ≡
0, and a >
0, and continuesto be true in the general case. Thus, one expects the empirical laws of ( ζ, χ ) to satisfy a certainstrong law of large numbers, an intuition that is made precise in the following result.
Theorem 2.4.
Let Assumption 1.7 hold. Then, there exists a probability space (Ω , F , Q ) supportingindependent d and k dimensional Brownian motions W and B , as well as process ζ satisfying ζ = ζ + Z · (cid:18) c ∇ pp + div ( c ) − m (cid:19) ( ζ t )d t + Z · σ ( ζ t )d W t , where ζ is an F -measurable random variable with density p . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 11
Define the process ∆ as the solution to the linear differential equation (2.12) ∆ = 1 + Z · ∆ t (cid:0) θ ′ ( m − div ( c )) + ∇ · ( cθ ) − a (cid:1) ( ζ t )d t + Z · ∆ t (cid:0) η ( ζ t ) ′ d B t + θ ( ζ t ) ′ d ζ t (cid:1) , and then, for any x ∈ (0 , ∞ ) , define χ x as the solution to the linear differential equation (2.13) χ x = x + Z · χ xt d∆ t ∆ t + Z · f ( ζ t )d t. Lastly, let x ∈ (0 , ∞ ) , T ∈ (0 , ∞ ) and set F = E × (0 , ∞ ) as in (2.1) . Define the (random)empirical measure b π xT on B ( F ) , the Borel subsets of F by (2.14) b π xT [ A ] := 1 T Z T I A ( ζ t , χ xt )d t, A ∈ B ( F ) . With the above notation, there exists a set Ω ∈ F ∞ with Q [Ω ] = 1 such that (2.15) lim T →∞ b π xT ( ω ) = π weakly, for all x ∈ (0 , ∞ ) and ω ∈ Ω , where π is the joint distribution of ( Z , X ) under P given in (2.2) .Remark . In the context of Theorem 2.4, note that the processes ∆ and χ x can be given inclosed form in terms of ζ ; indeed,∆ = exp (cid:18)Z · (cid:0) θ ′ ( m − div ( c )) + ∇ · ( cθ ) − a (cid:1) ( ζ t )d t (cid:19) E (cid:18)Z · (cid:0) η ( ζ t ) ′ d B t + θ ( ζ t ) ′ d ζ t (cid:1)(cid:19) · ,χ x = ∆ (cid:18) x + Z · t f ( ζ t )d t (cid:19) , x ∈ (0 , ∞ ) . Theorem 2.4 provides a way to efficiently estimate the joint distribution of ( Z , X ) efficientlythrough Monte-Carlo simulation. Indeed, one need only obtain a single path of the reversed process( ζ, χ x ) to recover the distribution π . However, the applicability of the result above depends heavilyon whether or not the distribution p for Z is known, as it (together with its gradient) appearsin the dynamics of ζ . In the case where Z is one-dimensional, or more generally, reversing, p can be expressed in closed form from the model coefficients m and c in the dynamics for Z .Furthermore, there are certain cases of non-reversing, multi-dimensional diffusions, where p can be(semi-)explicitly computed, as the next example shows. Example . Assume that Z is a multi-dimensional Ornstein-Uhlenbeck process with dynamics dZ t = − γ ( Z t − Θ) dt + σdW t , t ∈ R + , where γ ∈ R d × d , Θ ∈ R d , and σ ∈ R d × d . Here, E = R d and (A1) clearly holds. Furthermore (A2)is satisfied when c = σσ ′ is (strictly) positive definite; in fact, we take σ as the unique positivedefinite square root of c . The process Z need not be reversing, as can clearly be seen when σ is the identity matrix, Θ = 0 and γ is not symmetric. However, as will be argued below, theergodic assumption (A3) holds when all eigenvalues of γ have strictly positive real part, and one may identify the invariant density “almost” explicitly. To see this, a direct calculation shows thatif a symmetric matrix J satisfies the Riccati equation(2.16) J J = σγ ′ σ − J + J σ − γσ, then the function p ( z ) = exp (cid:18) −
12 ( z − Θ) ′ σ − J σ − ( z − Θ) (cid:19) , z ∈ R d , satisfies ˜ L Z p = 0 where ˜ L Z is as in (1.2). If J is additionally positive definite then, up to anormalizing constant, p is the density for a normal random variable with mean Θ and covariancematrix Σ = σJ − σ . Thus, p is integrable on R d and (A3) follows from [28, Corollary 4.9.4] whichproves recurrence for Z .It thus remains to construct a symmetric, positive definite solution to (2.16). From [1, Lemma2.4.1, Theorem 2.4.25] such a solution (called the “stabilizing solution” therein) exits if a) the pair( σ − γσ, d ) is stabilizable , in that there exists a matrix F such that σ − γσ − F has eigenvalueswith strictly negative real part and b) the eigenvalues of σ − γσ have strictly positive real part.In the present case, each of these statements readily follows: for the first statement, one can take F = σ − γσ + 1 d ; for the second statement, note that the eigenvalues of σ − γσ coincide with thoseof γ , which by assumption have strictly positive real part. Therefore, even in this non-reversingcase one may still identify p .The previous interesting Example 2.6 notwithstanding, for non-reversing, multi-dimensional dif-fusions, even after verifying the ergodicity of Z (and hence the existence of p ) one does not typicallyknow p explicitly. In such cases, the following simulation method is proposed: fix a large enough T and first simulate ( Z t ) t ∈ [0 , T ] via (0.2), starting from any point Z (since the invariant density isunknown). If the choice of T is large enough, the process ( Z t ) t ∈ [ T, T ] will behave as the stationaryversion in (0.2), since Z T will have approximately density p . In that case, defining ( ζ t ) t ∈ [0 ,T ] via ζ t = Z T − t for t ∈ [0 , T ], ζ should behave as it should in the dynamics (7.7), even with ζ having(approximate) density p . Now, given ζ , χ x may be defined via the formulas of Remark 2.5; there-fore, for large enough T , the empirical measure b π xT should approximate in the weak sense the jointlaw π .Note finally that when p is known and | η | >
0, and under certain mixing conditions (see [30, 29]),one can also obtain uniform estimates for the speed at which the above convergence takes place.
Remark . In the case when θ = η ≡ f ∈ C ,γ ( E ; R + ), it is possible to explicitly identifythe support of π . Such an identification follows from more general ergodic results on “stochasticdifferential systems” obtained in [5, 4]. To identify the support, note that when θ = η ≡
0, itfollows that ∆ t = exp (cid:16) − R T a ( ζ u ) du (cid:17) . A direct calculation using Remark 2.5 shows that χ x has ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 13 dynamics d χ xt = ( f ( ζ t ) − χ xt a ( ζ t )) d t. (2.17)Hence, the paths of χ x are of bounded variation. Now, define(2.18) ˆ u := inf (cid:26) x | sup z ∈ E ( f ( z ) − xa ( z )) ≤ (cid:27) ; ˆ l := sup (cid:26) x | inf z ∈ E ( f ( z ) − xa ( z )) ≥ (cid:27) . Assumption 1.7 implies a ( z ) > z ∈ E and thus 0 ≤ ˆ l ≤ ˆ u ≤ ∞ with ˆ l = ˆ u if and onlyif for some constant c , f ( z ) = ca ( z ) for all z ∈ E . In this case, X = c P z almost surely for all z ∈ E . With this notation, [5] proves: Proposition . ( [5, Section III] ) Let Assumptions 1.7 hold. Assume that f ∈ C ,γ ( E ; R + ) and η, θ ≡ . Then the support of π is ¯ E × [ˆ l, ˆ u ] ( [ˆ l, ∞ ) if ˆ u = ∞ ). A Numerical Example
We now provide an example which highlights the superiority (in terms of computational effi-ciency) of the time-reversal method over the naive method for obtaining the distribution of X .Consider the case E = R , and(3.1) dZ t = − γZ t d t + d W t ; X = Z ∞ Z t e − at d t ; γ, a > . Note that the function R ∋ z f ( z ) = z fails to be non-negative. However, as argued below,the results of Theorem 2.4 still hold. As Z is a mean-reverting Ornstein Uhlenbeck process, it isstraight-forward to verify Assumption (A3) with p ( z ) = p γ/πe − γz , so that Z ∼ N (0 , / (2 γ )).We claim that ( Z , X ) is normally distributed with mean vector (0 ,
0) and covariance matrixΣ = γ γ ( a + γ )12 γ ( a + γ ) 12 γa ( a + γ ) ! . Indeed, integration by parts shows that for
T > Z T e − at Z t d t = Z a + γ + 1 a + γ Z T e − at d W t − a + γ e − aT Z T . The ergodicity of Z implies lim T →∞ ( Z T /T ) = − γ R R zp ( z ) dz = 0 almost surely; therefore, it followsthat lim T →∞ e − aT Z T = 0 holds almost surely. Next, note that Y T := R T e − at d W t is independentof Z and normally distributed with mean 0 and variance (1 − e − aT ) / (2 a ). Lastly, as a process, Y = ( Y T ) T ≥ is an L -bounded martingale and hence Y ∞ := lim T →∞ Y T almost surely exists,where Y ∞ is independent of Z , and normally distributed with mean 0 and variance 1 / (2 a ). Thus X = lim T ↑∞ R T e − at Z t dt exists almost surely and X = Z a + γ + Y ∞ a + γ ; Z ⊥⊥ Y ∞ ; Z ∼ N (cid:18) , γ (cid:19) , Y ∞ ∼ N (cid:18) , a (cid:19) , Figure 1.
Kolmogorov-Smirnov distances between the empirical and true distri-bution for X . The solid line is for the reversal method starting ζ ∼ p and thedashed line for the reversal method running Z to 2 T and setting ζ t = Z T − t .Here, T = 10 , δ = 1 / γ = 2 and a = 1. Computations were per-formed using Mathematica and the code can be found on the author’s website .from which the joint distribution follows. Now, even though f ( z ) = z can take negative values, thetime reversal dynamics in (2.17) still hold, taking the formd ζ t = − γζ t d t + d W t ; d χ t = ( a − ζ t χ t ) d t. Lastly, even though Theorem 2.4 no longer directly applies, it is shown in [5, Theorem 3.3, Section3.D, Proposition 3.15] that ( ζ, χ ) is still ergodic , in that (2.15) holds.For these dynamics, we performed the following test: for a fixed terminal time T and mesh size δ , we estimated the distribution of X in two ways. First, (“Method A”) by sampling ζ ∼ p andsetting χ = 1, and second (“Method B”) by running the forward process Z until 2 T then setting ζ t = Z T − t , χ = 1. For each simulation we computed the empirical distribution along a singlepath and then estimated the Kolmogorov-Smirnov distance ( d KS ( F, G ) = sup x | F ( x ) − G ( x ) | , fordistribution functions F, G ) between the empirical and true distribution for X . The parametervalues were γ = 2, a = 1, T = 10 ,
000 and δ = 1 / ζ to its invariant distribution [9]. Table 1 provides summarystatistics regarding the median distances and simulation times, as well as the standard deviationand tail data. The tightness condition in Proposition 3.15 is straightforward to verify.
ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 15
Method A Method BMedian Distance 0.00887 0.00882Standard Deviation 0.00405 0.0041399 th Percentile 0.02168 0.022551 st Percentile 0.00405 0.00290Median Time (seconds) 2.694 8.766
Table 1.
Statistics on Kolmogorov-Smirnov distances between the empirical andtrue distribution for X using methods A and B. Figure 2.
Histogram for the number N of paths necessary so that, using thenaive simulation for X , the Kolmogorov-Smirnov distance between the empiricaldistribution and true distribution for X fell below the median distance d usingMethod A from Table 1. The integral was computed using T = 100 with mesh sizeof δ = 1 /
24; furthermore, the values γ = 2 and a = 1 we used. Computations wereperformed using Mathematica and the code can be found on the author’s website .Having obtained Kolmogorov-Smirnov distances using reversal methods, we next compared ourresults to a naive simulation of X , obtained by sampling Z ∼ p and computing X via (3.1)directly. Here, for the median distance d using Method A from Table 1, we sampled X stoppingat the first instance N so that the Kolmogorov-Smirnov distance between the empirical and truedistribution for X fell below d . As can be seen from Figure 2 and the summary statistics inTable 2, the naive simulation performs significantly worse: at the median it took 7 ,
002 paths anda simulation time of 8 .
66 minutes to achieve the same level of accuracy as 1 path (2 .
94 seconds)of the reversed process. Further, the histogram shows the presence of a significant number oftrials where significantly more than the median number of paths were needed to achieve the givenaccuracy.
Summary for the Forward SimulationMedian Number of Paths 7,002Mean Number of Paths 11,446Standard Deviation 10,165Minimum Number of Paths 1,846Maximum Number of Paths 45,004Median Simulation Time (minutes) 8.66Mean Simulation Time (minutes) 14.34
Table 2.
Summary statistics using the naive forward simulation method.4.
Conclusion
In this work, using the method of time reversal, an efficient method for simulating the jointdistribution of ( Z , X ) for perpetuities of the form (0.1) is obtained. The joint distribution may beobtained by sampling a single path of the reversed process, as opposed to sampling numerous pathsof X using the naive method. However, the effectiveness of the proposed method depends on beingto obtain analytic representations for the distribution p of Z : an undertaking that, though alwayspossible in the one-dimensional case, is often not possible for non-reversing multi-dimensionaldiffusions. Furthermore, results are presented for perpetuities with non-negative underlying cashflow rates. As such, more research is needed to identify an effective time reversal method forperpetuities of the form X = Z ∞ D t dF t for general Markovian processes F (i.e., not just dF t = f ( Z t ) dt ) containing both jumps and diffusiveterms. Additionally, the performance of the method where Z is run until a large time 2 T and thensetting ζ t = Z T − t must be thoroughly analyzed: in particular, how fast does the distribution of Z T converge to p given a fixed starting point? To answer these questions, one must first analyzethe resultant backwards dynamics and associated PDEs for the invariant density, obtaining ratesof convergence. 5. Proofs from Section 1.2
We present here the proofs of Lemma 1.3 and Lemma 1.5.
Proof of Lemma 1.3.
Let ε > θ ′ cθ + η ′ η ≡
0. Then R = R · a ( Z t )d t and (1.4) reads a − ∈ L ( E, p ) and R E a ( z ) p ( z )d z >
0. Set κ := (1 / R E a ( z ) p ( z )d z >
0. Fix z ∈ E and denote by P z the probability obtained by conditioning upon Z = z . The positive recurrenceof Z implies ([28, Theorem 4.9.5]), there exists a P z -a.s. finite random variable T ( z ) such that t ≥ T ( z ) implies that R t = R t a ( Z u )d u ≥ κt and hence the first conclusion of Lemma 1.3 holds. ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 17
Furthermore, since Z is stationary, ergodic under P , the ergodic theorem implies there is a P a.s.finite random variable T such that t ≥ T implies R t ≥ κt . Now, let n ∈ N be such that n > / (2 κ ).We have sup t ≥ ( t/n − R t ) ≤ sup t ≤ T ( t/n − R t ) < ∞ , where the last inequality follows by the regularity of a and the non-explositivity of Z . Thus X = Z ∞ e − R t f ( Z t )d t ≤ e sup t ≤ T ( t/n − R t ) Z ∞ e − t/n f ( Z t )d t. By the stationarity of Z : E (cid:20)Z ∞ e − t/n f ( Z t )d t (cid:21) = Z ∞ e − t/n E [ f ( Z t )] d t = n Z E f ( z ) p ( z ) dz < ∞ , hence P (cid:2)R ∞ e − t/n f ( Z t ) dt < ∞ (cid:3) = 1, which in turn implies that P [ X < ∞ ] = 1.Assume now that θ ′ cθ + η ′ η
0, which by continuity of all involved functions implies that R E ( θ ′ cθ + η ′ η ) ( z ) p ( z )d z >
0. Fix z ∈ E . Positive recurrence of Z gives that lim t →∞ R t ( θ ′ cθ + η ′ η )( Z u )d u = ∞ with P z probability one. On the event (cid:8) R t ( θ ′ cθ + η ′ η )( Z u )d u > (cid:9) , note that − R t = − Z t a ( Z u )d u + Z t ( θ ′ cθ + η ′ η )( Z u )d u − − R t θ ′ σ ( Z u )d W u + η ( Z u )d B u R t ( θ ′ cθ + η ′ η )( Z u )d u ! . By the Dambis, Dubins and Schwarz theorem and the strong law of large numbers for Brownianmotion, it follows that there exists a P z -a.s. finite random variable T ( z ) such that t ≥ T ( z ) = ⇒ − R t θ ′ σ ( Z u )d W u + η ( Z u )d B u R t ( θ ′ cθ + η ′ η )( Z u )d u ≤ ε t ≥ T ( z ) = ⇒ − R t ≤ − Z t (cid:18) a + 1 − ε θ ′ cθ + η ′ η ) (cid:19) ( Z u )d u. With κ := (1 / R E ( a + (1 − ε )( θ ′ cθ + η ′ η ) / z ) p ( z )d z >
0, and increasing T ( z ) if necessary (stillkeeping it P z -a.s. finite), it follows that t ≥ T ( z ) implies − R t ≤ − κt . Hence the first part ofLemma 1.3 holds true again. Additionally, the ergodic theorem applied with P gives a P -a.s. finiterandom variable T such that t ≥ T implies − R t ≤ − κt . Again, for n ∈ N such that n > / (2 κ )we have X = Z ∞ e − R t f ( Z t )d t ≤ e sup t ≤ T ( t/n − R t ) Z ∞ e − t/n f ( Z t )d t. from which P [ X < ∞ ] = 1 follows by the same line of reasoning as above. (cid:3) Proof of Lemma 1.5.
The proof is nearly identical that if Lemma 1.3. Namely, in each of the cases θ ′ cθ + η ′ η ≡ θ ′ cθ + η ′ η
0, under the given hypothesis there is a constant κ ≥ P -a.s.finite random variable T such that − R t ≥ κt holds for t ≥ T . This gives that(5.1) X ≥ Z ∞ T e κt f ( Z t )d t ≥ Z ∞ T e κt ( f ∧ N )( Z t ) dt, where N is large enough so that R E ( f ( z ) ∧ N ) p ( z )d z >
0. We thus have X ≥ Z ∞ e κt ( f ∧ N )( Z t ) dt − Nκ ( e κT − . Ergodicity of Z implies that P almost surelylim u →∞ u Z u ( f ∧ N )( Z t )d t = Z E ( f ( z ) ∧ N ) p ( z )d z > , so that lim u →∞ R u e κt ( f ∧ N )( Z t )d t = ∞ , proving the result. (cid:3) Proof of Theorem 2.1
Under the given assumptions there exists a unique solution, ( P z,x ) ( z,x ) ∈ F to the generalizedmartingale problem for L on F , where L is from (2.5). Here, the measure space is ( e Ω , e F ), where e Ω = ( C [0 , ∞ ); ˆ F ), with ˆ F being the one-point compactification of F . The filtration e F is theright-continuous enlargement of the filtration generated by the coordinate process ( e Z, e Y ) on e Ω.Let ( F n ) n ∈ N be an increasing sequence smooth, bounded, open, connected domains of F suchthat F = ∪ n F n . Note that F n can be obtained by smoothing out the boundary of E n × (1 /n, n ).By uniqueness of solutions to the generalized martingale problem, for each n , the law of of ( e Z, e Y )is the same as the law of ( Z, Y x ) under P [ · | Z = z ] (where the latter will always denote a versionof the conditional probability) up until the first exit time of F n . Furthermore, since the process Z is recurrent, with ( P z ) z ∈ E being the restriction of ( P z,x ) ( z,x ) ∈ F to the first d coordinates, for z ∈ E , the law of e Z under P z is the same as the law of Z under P [ · | Z = z ]. For these reasons,and in order to ease the reading, we abuse notation and still use ( Z, Y ) instead of ( e Z, e Y ) for thecoordinate process on e Ω. The underlying space we are working on will be clear from the context.Denote by τ n the first exit time of ( Z, Y ) from F n . Assumption 1.7 implies Z does not ex-plode under P z,x and Y cannot explode to infinity since D is strictly positive almost surely under P [ · | Z = z ] for all z ∈ E . Therefore, the explosion time τ := lim n →∞ τ n for ( Z, Y ) is the firsthitting time of Y to 0 and the law of τ under P z,x is the same as the law of the first hitting of Y x to 0 under P [ · | Z = z ].Note that Y xt = D − t (cid:0) x − X + R ∞ t D u f ( Z u )d u (cid:1) . Assumption 1.7 implies (6.1) P (cid:20)Z ∞ t D u f ( Z u )d u > (cid:12)(cid:12)(cid:12) Z = z (cid:21) = 1 , z ∈ E, t ≥ . Therefore, g ( z, x ) = P [ X ≤ x | Z = z ] = P z,x [ Y xt > , ∀ t ≥
0] = P z,x [ τ = ∞ ] . Define(6.2) h ( z, x ) := P z,x h lim t →∞ Y t = ∞ i , ( z, x ) ∈ F This follows by the ergodic theorem since (cid:8)R ∞ t f ( Z u ) D u d u = 0 (cid:9) ⊂ n lim k →∞ (1 /k ) R t + kt f ( Z u )d u = 0 o . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 19
Fix ( z, x ) ∈ F and let 0 < ε < x . Note that Y xt = Y x − εt + ε/D t . Since lim t →∞ D t = 0 holds P [ · | Z = z ]-a.s., it follows that P z,x − ε [ τ = ∞ ] = P (cid:2) Y x − εt > ∀ t ≥ | Z = z (cid:3) ≤ P [ Y xt ≥ ε/D t , ∀ t ≥ | Z = z ] ≤ P h lim t →∞ Y xt = ∞ | Z = z i = P z,x h lim t →∞ Y t = ∞ i ≤ P z,x [ τ = ∞ ] . (6.3)Therefore, g ( z, x − ε ) ≤ h ( z, x ) ≤ g ( z, x ). By definition, g ( z, x ) is right-continuous in x for a fixed z and so g ( z, x ) ≤ lim inf ε → h ( z, x + ε ) ≤ lim sup ε → h ( z, x + ε ) ≤ lim sup ε → g ( z, x + ε ) = g ( z, x ) . Therefore, if h ( z, x ) is continuous it follows that h ( z, x ) = g ( z, x ). It is now shown that in fact h is in C ,γ ( F ) and satisfies Lh = 0. This gives the desired result for g since g = h .Let ψ : (0 , ∞ ) (0 ,
1) be a smooth function such that lim x → ψ ( x ) = 0, lim x →∞ ψ ( x ) = 1. Bythe classical Feynman-Kac formula u n ( z, x ) := E P z,x [ ψ ( Y τ n )] , satisfies Lu n = 0 in F n with u n ( z, x ) = ψ ( x ) on ∂F n . Since P [ X < ∞ | Z = z ] = 1 there exists apair ( z , x ) ∈ F so that P [ X < x | Z = z ] >
0. Using (6.3) this gives(6.4) h ( z , x ) ≥ P [ X < x | Z = z ] > . Therefore, ( P z,x ) ( z,x ) ∈ F is transient [28, Chapter 2] and, since ( P z ) z ∈ E is positive recurrent, thisimplies that for all ( z, x ), with P z,x -probability one, either lim t → τ Y t = 0 or lim t → τ Y t = ∞ , wherein the latter case, τ = ∞ since Y cannot explode to ∞ . This in turn yields that Y τ n → Y τ n → ∞ with P z,x -probability one and hence by the dominated convergence theorem(6.5) lim n →∞ u n ( z, x ) = P z,x h lim t → τ Y t = ∞ i = P z,x h lim t →∞ Y t = ∞ i = h ( z, x ) . For ( z , x ) from (6.4), g ( z , x ) ≥ h ( z , x ) > g ( z, x ) > z, x ) ∈ F [28,Theorem 1.15.1]. But this implies h ( z, x ) ≥ g ( z, x/ >
0, and so from (6.5) the u n are convergingpoint-wise to a strictly positive function. Thus, by the interior Schauder estimates and Harnack’sinequality, it follows by “the standard compactness” argument ([28, Page 147]) that there exists a C ,γ ( F ), strictly positive, function u such that u n converges to u in the C ,γ ( D ) Holder space forall compact D ⊂ F . Clearly, this function u satisfies Lu = 0 in F . In fact, since u n converges to h pointwise, h = u and hence Lh = 0. The boundary conditions for g are now considered. Let the integer k be given. It suffices toshow for each ε > n ( ε ) such that(6.6) sup x ≤ n ( ε ) − ,z ∈ E k g ( z, x ) ≤ ε ; inf x ≥ n ( ε ) ,z ∈ E k g ( z, x ) ≥ − ε The condition near x = 0 is handled first. By way of contradiction, assume there exists some ε > n there exists z n ∈ E k , x n ≤ /n such that g ( z n , x n ) > ε . Since the z n are all contained within E k there is a sub-sequence (still labeled n ) such that z n → z for z ∈ ¯ E k .Let δ > N δ such that n ≥ N δ implies n − ≤ δ . Since g is increasing in x , ε < g ( z n , δ ). Since g is continuous, ε ≤ g ( z, δ ). Since this is true for all δ >
0, lim x → g ( z, x ) ≥ ε .But, this is a contradiction : lim x → g ( z, x ) = 0 for each z ∈ E . To see this, let δ > β > P [ X ≥ β | Z = z ] ≥ − δ . This is possible in view of (6.1). Thus, for x < β , g ( z, x ) ≤ P [ X < β | Z = z ] ≤ δ and hence lim sup x → g ( z, x ) ≤ δ . Taking δ → x → ∞ is very similar. Assume by contradiction that there is some ε > n there exist z n ∈ E k , x n ≥ n such that g ( z n , x n ) < − ε . Again, by takingsub-sequences, it is possible to assume z n → z ∈ ¯ E k . Fix M >
0. For n ≥ M , since g is increasingin x , g ( z n , M ) < − ε . Since g is continuous, g ( z, M ) ≤ − ε . Since this holds for all M ,lim x →∞ g ( z, x ) ≤ − ε . But, this violates the condition that under P [ · | Z = z ], X < ∞ almostsurely.The uniqueness claim is now proved. Let ˜ g be a C ( F ) solution of L ˜ g = 0 such that 0 ≤ ˜ g ≤ σ k := inf { t ≥ Z t E k } ; ρ k := inf { t ≥ Y t = k } . By Ito’s formula, for any k, n, m ˜ g ( z, x ) = E P z,x h g ( Z σ k ∧ ρ /n ∧ ρ m , Y σ k ∧ ρ /n ∧ ρ m ) (cid:16) ρ /n <σ k ∧ ρ m + 1 ρ /n ≥ σ k ∧ ρ m (1 τ< ∞ + 1 τ = ∞ ) (cid:17)i . Since P z,x almost surely lim m →∞ ρ m = ∞ , taking m → ∞ yields˜ g ( z, x ) = ˆ E P z,x h g ( Z σ k ∧ ρ /n , Y σ k ∧ ρ /n ) (cid:16) ρ /n <σ k + 1 ρ /n ≥ σ k (1 τ< ∞ + 1 τ = ∞ ) (cid:17)i . On (cid:8) ρ /n < σ k (cid:9) , Z ρ /n ∈ E k , Y ρ /n ≤ /n and hence by 0 ≤ ˜ g ≤ ε > n ( ε ) such that for n ≥ n ( ε )˜ g ( z, x ) ≤ ε + P z,x (cid:2) ρ /n ≥ σ k , τ < ∞ (cid:3) + P z,x (cid:2) ρ /n ≥ σ k , τ = ∞ (cid:3) . Taking n → ∞ thus gives˜ g ( z, x ) ≤ ε + P z,x [ τ ≥ σ k , τ < ∞ ] + P z,x [ τ = ∞ ] . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 21
Taking k → ∞ gives ˜ g ( z, x ) ≤ ε + P z,x [ τ = ∞ ] . and hence taking ε → g ( z, x ) ≤ P z,x [ τ = ∞ ] = g ( z, x ). Similarly, for k, n, m ˜ g ( z, x ) = E P z,x h g ( Z σ k ∧ ρ /n ∧ ρ m , Y σ k ∧ ρ /n ∧ ρ m ) (cid:16) ρ m <σ k ∧ ρ /n + 1 ρ m ≥ σ k ∧ ρ /n (cid:17)i , ≥ (1 − ε )ˆ P z,x h ρ m < σ k ∧ ρ /n , lim t →∞ Y t = ∞ i , for all ε > m ≥ m ( ε ) for some m ( ε ). Note that the set (cid:8) ρ m < σ k ∧ ρ /n (cid:9) is restricted toinclude { lim t →∞ Y t = ∞} but this is fine since lower bounds are considered. Now, on the event { lim t →∞ Y t = ∞} it holds that ρ /n → ∞ . Thus, taking n → ∞ ˜ g ( z, x ) ≥ (1 − ε ) P z,x h ρ m < σ k , lim t →∞ Y t = ∞ i . Taking k → ∞ gives ˜ g ( z, x ) ≥ (1 − ε ) P z,x h ρ m < ∞ , lim t →∞ Y t = ∞ i . Taking m → ∞ and noting that for m large enough ρ m < ∞ on lim t →∞ Y t = ∞ it holds that˜ g ( z, x ) ≥ (1 − ε ) P z,x h lim t →∞ Y t = ∞ i = (1 − ε ) h ( z, x ) . where the last equality follows by the definition of h in (6.2). Now, in proving Lg = 0 it was shownthat g = h and hence ˜ g ( z, x ) ≥ (1 − ε ) g ( z, x ). Taking ε → g ( z, x ) ≥ g ( z, x ), finishingthe proof. 7. Dynamics for the Time-Reversed Process
The goal of the next two sections is to prove Theorem 2.4. We keep all notation from Subsection2.2. We first identify the dynamics for ζ T . Proposition 7.1.
Let Assumptions 1.7 hold. Then, for each
T > , the law of ζ T under P solvesthe martingale problem on E (for t ≤ T ) for the operator L ζ := (1 / c ij ∂ ij + µ i ∂ i where (7.1) µ := c ∇ pp + div ( c ) − m. The operator L ζ does not depend upon T . Thus, if ( Q z ) z ∈ E denotes the solution of the generalizedmartingale problem for L ζ on E , then in fact ( Q z ) ζ ∈ E solves the martingale problem for L ζ on E and is positive recurrent.Remark . If Z is reversing then p satisfies m = (1 /
2) ( c ∇ p/p + div ( c )). Thus, in this instance, µ = m and, as the name suggests, ζ T has the same dynamics as Z . Proof.
The first statement regarding the martingale problem is based off the argument in [20].Since Z is positive recurrent with invariant measure p and Z has initial distribution p under P , Z is stationary with distribution p . Since ˜ L Z p = 0, equation (2 .
5) in [20] holds noting that p doesnot depend upon t .For a given s ≤ t ≤ t and g ∈ C ∞ c ( E ) define the function v ( s, z ) := E (cid:2) g ( X t ) (cid:12)(cid:12) Z s = z (cid:3) . TheFeynman-Kac formula implies v satisfies v s + L z v = 0 on 0 < s < t, z ∈ E with v ( t, z ) = g ( z ) : see[21, 19] for an extension of the classical Feynman-Kac formula to the current setup. Therefore, thecondition in equation (2 .
7) of [20] holds as well. Thus, the formal argument on page 1191 of [20]is rigorous and the law of ζ T under P solves the martingale problem for L ζ .Turning to the statement regarding ( Q z ) z ∈ E , set ˜ L ζ as the formal adjoint to L ζ . ˜ L ζ is given by(1.2) with µ replacing m therein. Using the formula for µ in (7.1) and for ˜ L Z in (1.2) calculationshows that ˜ L ζ f = ˜ L Z f − ∇ · (cid:18) fp (cid:18)
12 ( c ∇ p + p div ( c )) − pm (cid:19)(cid:19) . Since(7.2) 0 = ˜ L Z p = ∇ · (cid:18)
12 ( c ∇ p + p div ( c )) − pm (cid:19) , it follows by considering f = p above that ˜ L ζ p = 0. Therefore, p is an invariant density for L ζ if an only if the diffusion corresponding to the operator ˜ L ζ,p does not explode, where ˜ L ζ,p is theh-transform of ˜ L ζ [28, Theorem 4.8.5]. But, by definition of the h-transform [28, pp. 126] and (1.2)with µ replacing m :˜ L ζ,p f := 1 p ˜ L ζ ( f p ) = 12 c ij ∂ ij f − µ i − div ( c ) i − (cid:18) c ∇ pp (cid:19) i ! ∂ i f + fp ˜ L ζ p, = 12 c ij ∂ ij f + m i ∂ i f = L Z f, where the third equality follows from (7.1). Thus, Assumption 1.7 (specifically the fact that Z is ergodic and R E p ( z ) dz = 1) implies the diffusion for ˜ L ζ,p not only does not explode but also ispositive recurrent, finishing the proof. (cid:3) In preparation for the proof of the main result of this Section, which is Proposition 7.5, it is firstneeded to define a certain “backwards” filtration G T and to present two Lemmas. Fix T ∈ (0 , ∞ )and t ∈ [0 , T ] and let e G Tt be the σ -field generated by X T , ( Z T − u ) u ∈ [0 ,t ] , ( W T − W T − u ) u ∈ [0 ,t ] and( B T − B T − u ) u ∈ [0 ,t ] . Then, let G T := ( G Tt ) t ∈ [0 ,T ] be the usual augmentation of ( e G Tt ) t ∈ [0 ,T ] . Itis easy to check that ( χ T , ζ T ) is G T -adapted for all T ∈ R + , as well as that the process B T defined via B Tt := B T − t − B T is a k dimensional Brownian motion on (Ω , G T , P ), independent of( χ T , ζ T ) = ( X T , Z T ). However, the G T -adapted process ( W T − t − W T ) t ∈ [0 ,T ] is not necessarily aBrownian motion on (Ω , G T , P ).With this notation, the following two Lemmas are essential for proving Proposition 7.5. ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 23
Lemma 7.3.
Let Assumptions 1.7 hold. For locally bounded Borel function η : E R and ≤ s ≤ t ≤ T , it holds that (7.3) − Z T − sT − t η ( Z u ) ′ d B u = Z ts η ( ζ Tu ) ′ d B Tu . Furthermore, if θ : E R d is continuously differentiable, then (7.4) − Z T − sT − t θ ′ ( Z u )d Z u = Z ts θ ′ ( ζ Tu )d ζ Tu + Z ts (cid:0) ∇ · ( cθ ) − θ ′ div ( c ) (cid:1) ( ζ Tu )d u. Proof.
Fix 0 ≤ s ≤ t ≤ T . For each n ∈ N and i ∈ { , . . . , n } , let(7.5) u ni := T − t + i ( t − s ) /n. First, assume that η is twice continuously differentiable. The standard convergence theorem forstochastic integrals implies that (the following limit is to be understood in measure P ): Z ts η ( ζ Tu ) ′ d B Tu + Z T − sT − t η ( Z u ) ′ d B u = − lim n →∞ n X i =1 (cid:16) η ( Z u ni ) − η ( Z u ni − ) (cid:17) ′ (cid:16) B u ni − B u ni − (cid:17)! . Since B and Z are independent, by Ito’s formula the last quadratic covariation is zero. Therefore,(7.3) holds for twice continuously differentiable η . The fact that (7.3) holds whenever η is locallybounded follows from a monotone class argument.In a similar manner, assume that θ is twice continuously differentiable. The standard convergencetheorem for stochastic integrals implies that Z ts θ ′ ( ζ Tu )d ζ Tu + Z T − sT − t θ ′ ( Z u )d Z u = − lim n →∞ n X i =1 (cid:16) θ ( Z u ni ) − θ ( Z u ni − ) (cid:17) ′ ( Z u ni − Z u ni − ) ! . The last quadratic covariation process (without the minus sign) is equal to Z T − sT − t ˜ F ( c, θ )( Z u )d u = Z ts ˜ F ( c, θ )( ζ Tu )d u, where ˜ F ( c, θ ) : E R is given by˜ F ( c, θ ) = d X i,j =1 c ij ∂ z i θ j = d X i,j =1 (cid:0) ∂ z i ( c ij θ j ) − θ j ∂ z i (( c ′ ) ji ) (cid:1) = ∇ · ( cθ ) − θ ′ div ( c ) , since c ′ = c . Thus, (7.4) is established in the case where θ is twice continuously differentiable.The fact that (7.4) holds whenever θ is continuously differentiable follows form a density argu-ment, noting that there exists a sequence ( θ n ) n ∈ N of polynomials such that lim n →∞ θ n = θ andlim n →∞ ∇ θ n = ∇ θ both hold, where the convergence is uniform on compact subsets of E . (cid:3) Lemma 7.4.
Let Assumptions 1.7 hold. For each T ∈ R + , define the G T -adapted continuous-path ∆ T as in (2.9) . Then ∆ T is a semimartingale on (Ω , G T , P ) . More precisely, for t ∈ [0 , T ]∆ Tt = 1 + Z t ∆ Tu (cid:18) θ ′ c ∇ pp + ∇ · ( cθ ) − a (cid:19) ( ζ Tu )d u + Z t ∆ Tu (cid:0) η ( ζ Tu ) ′ d B Tu + θ ′ σ ( ζ Tu )d W Tu (cid:1) . (7.6) Proof.
Define ( ρ Tt ) t ∈ [0 ,T ] by ρ Tt := R T − R T − t , for t ∈ [0 , T ]. In view of (0.2), (1.3), (7.1) andLemma 7.3, ρ T = Z TT −· (cid:18) a + 12 ( θ ′ cθ + η ′ η ) (cid:19) ( Z t )d t + Z TT −· (cid:0) η ( Z t ) ′ d B t + θ ′ σ ( Z t )d W t (cid:1) = Z TT −· (cid:18) a − θ ′ m + 12 ( θ ′ cθ + η ′ η ) (cid:19) ( Z t )d t + Z TT −· (cid:0) η ( Z t ) ′ d B t + θ ′ ( Z t )d Z t (cid:1) = Z · (cid:18) a − θ ′ m + θ ′ div ( c ) − ∇ · ( cθ ) + 12 ( θ ′ cθ + η ′ η ) (cid:19) ( ζ Tt )d t − Z · (cid:0) η ( ζ Tt ) ′ d B Tt + θ ′ ( ζ Tt )d ζ Tt (cid:1) , = Z · (cid:18) a − θ ′ c ∇ pp − ∇ · ( cθ ) + 12 ( θ ′ cθ + η ′ η ) (cid:19) ( ζ Tt )d t − Z · (cid:0) η ( ζ Tt ) ′ d B Tt + θ ′ σ ( ζ Tt )d W Tt (cid:1) . The fact that D = exp( − R ) gives ∆ T = exp( − ρ T ). Then, the dynamics for ∆ T follow from thedynamics of ρ T . (cid:3) Proposition 7.5.
Let Assumptions 1.7 hold. Then, for each
T > there is a filtration G T satisfying the usual conditions and d and k dimensional independent ( P , G T ) Brownian motions W T , B T on [0 , T ] so that the pair ( ζ T , χ T ) have dynamics ζ Tt = ζ T + Z T (cid:18) c ∇ pp + div ( c ) − m (cid:19) ( ζ Tu )d u + Z T σ ( ζ Tu )d W Tu ,χ Tt = χ T + Z T (cid:18) f ( ζ Tu ) − χ Tu (cid:18) a − θ ′ c ∇ pp − ∇ · ( cθ ) (cid:19) ( ζ Tu ) (cid:19) d u + Z T χ Tu (cid:0) θ ′ σ ( ζ Tu )d W Tu + η ( ζ Tu ) ′ d B Tu (cid:1) . (7.7) Proof of Proposition 7.5.
Proposition 7.1 immediately implies that under P , ζ T has dynamics: ζ Tt = ζ T + Z t (cid:18) c ∇ pp + div ( c ) − m (cid:19) ( ζ Tu ) du + Z t σ ( ζ Tu ) dW Tu ; t ∈ [0 , T ] , (7.8)where ( W Tt ) t ∈ [0 ,T ] is a Brownian motion on (Ω , G T , P ). In order to specify the dynamics for χ T ,recall the definition of ∆ T from (2.9). Observe that X T − t = 1 D T − t Z ∞ T − t D u f ( Z u )d u = D T D T − t (cid:18) X T + Z TT − t D u D T f ( Z u )d u (cid:19) ; t ∈ [0 , T ] . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 25
Then, using the definitions of χ T , ζ T and ∆ T , the above is rewritten as(7.9) χ Tt = ∆ Tt (cid:18) χ T + Z t Tu f ( ζ Tu )d u (cid:19) ; t ∈ [0 , T ] . Lemma 7.4 implies ∆ T is a semimartingale, and hence (7.9) yields χ Tt = χ T + Z t χ Tu d∆ Tu ∆ Tu + Z t f ( ζ Tu )d u ; t ∈ [0 , T ] . The result now follows by plugging in for d∆ Tu / ∆ Tu from (7.6). (cid:3) Proof of Theorem 2.4
Preliminaries.
We first prove two technical results. The first asserts the existence of aprobability space and stationary processes ( ζ, χ ) consistent with ( ζ, χ x ) in Theorem 2.4 in thatgiven χ = x , it holds that χ t = χ xt , t ≥
0. The second proposition shows that under the non-degeneracy assumption | η | ( z ) > , z ∈ E and regularity assumption f ∈ C ( E ; R + ) it follows that( ζ, χ ) is ergodic. Lemma 8.1.
Let Assumption 1.7 hold. Then, there is a filtered probability space (Ω , F , Q ) , sup-porting independent d and k dimensional Brownian motions W and B , F measurable randomvariables ζ , χ with joint distribution π , as well as a stationary process ζ with dynamics (8.1) ζ = ζ + Z · (cid:18) c ∇ pp + div ( c ) − m (cid:19) ( ζ t )d t + Z · σ ( ζ t )d W t . Furthermore, with ∆ , χ x defined as in (2.12) , (2.13) , if the process χ is defined by χ t := χ χ t (seeRemark 2.5) then ( ζ, χ ) are stationary with invariant measure π and joint dynamics d ζ t = (cid:18) c ∇ pp + div ( c ) − m (cid:19) ( ζ t )d t + σ ( ζ t )d W t , t ∈ R + , d χ t = (cid:18) f ( ζ t ) − χ t (cid:18) a − θ ′ c ∇ pp − ∇ · ( cθ ) (cid:19) ( ζ t ) (cid:19) d t + χ t (cid:0) θ ′ σ ( ζ t )d W t + η ( ζ t ) ′ d B t (cid:1) , t ∈ R + . (8.2) Proof.
This result follows from Proposition 7.1. Indeed, one can start with a probability space(Ω , F , Q ) supporting independent d and k dimensional Brownian motions W and B respectively,as well as a F measurable random variable ( ζ , χ ) ∼ π (hence independent of W and B ). Underthe given regularity assumptions, Proposition 7.1 yields a strong, stationary solution ζ satisfying(8.1). Then, defining ∆ as in (2.9) and, for x > χ x as in (2.13), it follows that ( ζ, χ x ) andhence ( ζ, χ ) satisfy the SDE in (8.2). Under the given regularity assumptions the law under P of( ζ T , χ T ) given ζ T = z, χ T = x coincides with the law under Q of ( ζ, χ x ) given that ζ = z . Sinceby construction, π is an invariant measure for ( ζ T , χ T ), it follows from the Markov property that π is invariant for ( ζ, χ ) under Q and hence ( ζ, χ ) is stationary with invariant measure π . (cid:3) Define the measures Q z,x for ( z, x ) ∈ F via(8.3) Q z,x [ A ] = Q [ A | ζ = z, χ = x ] ; A ∈ F ∞ We now consider when | η | > E and f ∈ C ( E ; R + ). According to Theorem 2.1, g ∈ C ,γ ( F )and hence π possesses a density satisfying(8.4) π ( z, x ) = p ( z ) ∂ x g ( z, x ); ( z, x ) ∈ F. Additionally, we have the following Proposition:
Proposition 8.2.
Let Assumption 1.7 hold, and additionally suppose that | η | ( z ) > for z ∈ E and f ∈ C ( E ; R + ) . Then the process ( ζ, χ ) from Lemma 8.1 is ergodic. Thus, for all boundedmeasurable functions h on F and all ( z, x ) ∈ F (8.5) lim T →∞ T Z T h ( ζ t , χ t ) dt = Z F hdπ ; Q z,x a.s. . Proof of Proposition 8.2.
Recall A from (2.4) and define b R : F R d +1 by b R ( z, x ) := ( c ( ∇ p/p ) + div ( c ) − m ) ( z ) f ( z ) − x ( a − θ ′ c ( ∇ p/p ) − ∇ · ( cθ )) ( z ) ! . (8.6)From (8.2) it is clear that the generator for ( ζ, χ ) is L R := (1 / A ij ∂ ij + ( b R ) i ∂ i . As an abuse ofnotation, let ( Q z,x ) ( z,x ) ∈ F also denote the solution to the generalized martingale problem for L R on F . Using Theorem 2.1, and the fact that under the given coefficient regularity assumptions, g ∈ C ( F ) (see [16, Ch. 6]) a lengthy calculation performed in Lemma A.1 below shows that thedensity π from (8.4) solves ˜ L R π = 0 where ˜ L R is the formal adjoint to L . Since by construction, RR F π ( z, x ) dzdx = 1, positive recurrence will follow once it is shown that ( Q z,x ) ( z,x ) ∈ F is recurrent.By Proposition 7.1, the restriction of Q z,x to the first d coordinates (i.e. the part for ζ ) is positiverecurrent. Since by (2.13) it is evident that χ does not hit 0 in finite time, it follows that that χ does not explode under Q z,x . Thus, [28, Corollary 4.9.4] shows that ( ζ, χ ) is recurrent. Now, that(8.5) holds follows from [28, Theorem 4.9.5]. (cid:3) Proof of Theorem 2.4.
The proof of Theorem 2.4 uses a number of approximations argu-ments. To make these arguments precise, we first enlarge the original probability space (Ω , F , P )so that it contains a one dimensional Brownian motion ˆ B which is independent of Z , W and B .Let D be as in (0.3), and for ε >
0, define D ε := D E ( √ ε ˆ B ). Similarly to (0.1) define(8.7) X ε := Z ∞ D εt f ( Z t ) dt. Note that D ε takes the form (0.3) for η ε ( z ) = ( η ( z ) , √ ε ) and when the Brownian motion B thereinis the k + 1 dimensional Brownian motion ( B, ˆ B ). Note that | η ε | = | η | + ε >
0. Denote by π ε ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 27 the joint distribution of ( Z , X ε ) under P and by g ε the conditional cdf of X ε given Z = z . ByTheorem 2.1 it follows that g ε ∈ C ,γ ( F ) and hence π ε admits a density.In a similar manner, by enlarging the probability space (Ω , F , Q ) of Lemma 8.1 to include aBrownian motion (still labeled ˆ B ) which is independent of ζ , χ , W and B and defining thefamily of processes (∆ ε ) ε> and ( χ ε,x ) ε> for x > εt := ∆ t E ( √ ε ˆ B ) t ; t ≥ χ ε,xt := ∆ εt (cid:18) x + Z t εu f ( ζ u )d u (cid:19) ; t ≥ , (8.8)it follows that ( ζ, χ x,ε ) solve the SDE dζ t = ( m + 2 ξ ) ( ζ t )d t + σ ( ζ t )d W t ,dχ ε,xt = (cid:18) f ( ζ t ) − χ εt (cid:18) a − θ ′ c ∇ pp − ∇ · ( cθ ) (cid:19) ( ζ t ) (cid:19) d t + χ εt (cid:16) θ ′ σ ( ζ t )d W u + η ε ( ζ t ) ′ (d B t , d ˆ B t ) (cid:17) . (8.9)Since | η ε | ≥ √ ε >
0, Proposition 8.2 shows for f ∈ C ( E ; R + ) the generator L ε,R associated to(8.9) is positive recurrent with invariant density π ε and thus for all ( z, x ) ∈ F and all boundedmeasurable functions h on F (note that conditioned upon χ = x we have χ ε,x = χ x = x = χ ):(8.10) lim T →∞ T Z T h ( ζ t , χ x,εt ) dt = Z F hdπ ε ; Q z,x a.s.. With all the notation in place, Theorem 2.4 is the culmination of a number of lemmas, whichare now presented. The first lemma implies that π ε converges weakly to π as ε ↓ Lemma 8.3.
Let Assumption 1.7 hold. Define X ε as in (8.7) . Then X ε converges to X in P -measure as ε → .Proof of Lemma 8.3. Denote by G the sigma-field generated by Z , W and B . Set δ εt := D εt /D t = E (cid:16) √ ε ˆ B t (cid:17) . By the independence of δ ε and G : E [ | X ǫ − X | | G ] ≤ Z ∞ E [ | δ ǫt − | | G ] D t f ( Z t )d t = Z ∞ E [ | δ ǫt − | ] D t f ( Z t )d t. Now, set h εt := √ e εt −
1. Note that h ε is monotone increasing in ε with lim ε → h ε = 0. Further-more, E [ | δ εt − | ] ≤ E (cid:2) | δ εt − | (cid:3) / = p exp( εt ) − h εt . By assumption, P [ X < ∞ ] = 1. Since for any ε >
0, sup t ≥ δ εt < ∞ P a.s., it thus follows that P [ X ε < ∞ ] = 1. The dominated convergence theorem applied path-wise (recall that there existsa κ > e κt D t → P almost surely) then gives that lim ε → E [ | X ε − X | | G ] = 0, whichshows that the pair ( Z , X ε ) converges in probability to ( Z , X ), finishing the proof. (cid:3) Next, define C as the class of (Borel measurable) functions h which are bounded and Lipschitzin x , uniformly in z ; in other words,(8.11) C := (cid:26) h ∈ B ( E ; R ) | ∃ K ( h ) > s.t. ∀ x , x > , sup z ∈ E | h ( z, x ) − h ( z, x ) | ≤ K ( h ) (1 ∧ | x − x | ) (cid:27) . The next Lemma gives a weak form of the convergence in Theorem 2.4 for regular f . Note thatthe notation Q - lim T →∞ stands for the limit in Q probability as T → ∞ . Lemma 8.4.
Let Assumption 1.7 hold. Assume additionally that f ∈ C ( E ; R + ) . Then for all x > and all h ∈ C : (8.12) Q - lim T →∞ T Z T h ( ζ t , χ xt ) dt = Z F hdπ. Proof of Lemma 8.4.
For ease of presentation we adopt the following notational conventions. First,for any measurable function f and probability measure ν on F set(8.13) h h, ν i := Z F hdν. Next, similarly to ˆ π xT in (2.14), we define ˆ π ε,xT to be the empirical measure of ( ζ, χ ε,x ) on [0 , T ] for χ ε,x as in (8.8). Thus, we write1 T Z T h ( ζ t , χ xt ) dt = h h, ˆ π xT i ; 1 T Z T h ( ζ t , χ ε,xt ) dt = h h, ˆ π ε,xT i . Proposition 8.2 implies for all x > ε > Q - lim T →∞ (cid:10) h, ˆ π ε,xT (cid:11) = (cid:10) h, π ε (cid:11) . Indeed, (8.10) gives for all ( z, x ) ∈ F :(8.14) lim T →∞ h h, ˆ π ε,xT i = (cid:10) h, π ε (cid:11) ; Q z,x a.s.. Thus, the above limit holds Q almost surely, and hence in probability.To prove (8.12) we need to show that for any increasing R + -valued sequence ( T n ) n ∈ N such thatlim n →∞ T n = ∞ , there is a sub-sequence ( T n k ) k ∈ N such that Q - lim k →∞ (cid:10) h, ˆ π xT nk (cid:11) = (cid:10) h, π (cid:11) , as this implies (8.12) by considering double sub-sequences. To this end, let ( ε k ) k ∈ N be any strictlypositive sequence that converges to zero, and assume that ε < κ , where κ > T n k large enough so that k/T n k → Q (cid:20)(cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π ε k ,xT nk (cid:11) − (cid:10) h, π ε k (cid:11)(cid:12)(cid:12)(cid:12) > k (cid:21) ≤ k . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 29
As argued above, this is possible since (cid:10) h, ˆ π ε k ,xT (cid:11) converges to (cid:10) h, π ε k (cid:11) in Q probability. SinceLemma 8.3 implies lim ε → (cid:10) h, π ε k (cid:11) = (cid:10) h, π (cid:11) it follows that Q - lim k →∞ (cid:10) h, ˆ π ε k ,xT nk (cid:11) = (cid:10) h, π (cid:11) . Since (cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π xT nk (cid:11) − (cid:10) h, π (cid:11)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π xT nk (cid:11) − (cid:10) h, ˆ π ε k ,xT nk (cid:11)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π ε k ,xT nk (cid:11) − (cid:10) h, π (cid:11)(cid:12)(cid:12)(cid:12) , it suffices to show Q - lim k →∞ (cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π ε k ,xT nk (cid:11) − (cid:10) h, ˆ π xT nk (cid:11)(cid:12)(cid:12)(cid:12) = 0 . In fact, the claim is that lim k →∞ E Q h(cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π ε k ,xT nk (cid:11) − (cid:10) h, ˆ π xT nk (cid:11)(cid:12)(cid:12)(cid:12)i = 0 , or the even stronger (recall (cid:10) h, ˆ π xT (cid:11) = (1 /T ) R T h ( ζ t , χ xt )d t , (cid:10) h, ˆ π ε k ,xT (cid:11) = (1 /T ) R T h ( ζ t , χ ε k ,xt )d t ):(8.15) lim k →∞ (cid:18) T n k Z T nk E Q [ | h ( ζ t , χ ε k ,xt ) − h ( ζ t , χ xt ) | ] d t (cid:19) = 0 . From (8.11):(8.16) 1 T n k Z T nk E Q [ | h ( ζ t , χ ε k ,xt ) − h ( ζ t , χ xt ) | ] d t ≤ KT n k Z T nk E Q [1 ∧ | χ ε k ,xt − χ xt | ] d t. Furthermore, recall that χ xt = ∆ t (cid:18) x + Z t u f ( ζ u )d u (cid:19) , χ ε k ,xt = ∆ ε k ,xt (cid:18) x + Z t ε k u f ( ζ u )d u (cid:19) , where ∆ ε k is from (8.8). With δ ε k := E (cid:16) √ ε k ˆ B (cid:17) it follows that under Q | χ ε k ,xt − χ xt | ≤ x | ∆ ε k t − ∆ t | + Z t (cid:12)(cid:12)(cid:12)(cid:12) ∆ ε k t ∆ ε k u − ∆ t ∆ u (cid:12)(cid:12)(cid:12)(cid:12) f ( ζ u )d u = x ∆ t | δ ε k t − | + Z t ∆ t ∆ u (cid:12)(cid:12)(cid:12)(cid:12) δ ε k t δ ε k u − (cid:12)(cid:12)(cid:12)(cid:12) f ( ζ u )d u. With G now denoting the σ field generated by ζ , W and B , by the independence of ˆ B and G itfollows that(8.17) E Q [ | χ ε k ,xt − χ xt | | G ] ≤ x ∆ t h ε k t + Z t ∆ t ∆ u h ε k t − u f ( ζ u )d u. where for any ε > h ε is from Lemma 8.3. Since ζ is stationary under Q , it holds for all t > t under Q coincides with the distribution of D t under P and the distributionof R t (∆ t / ∆ u ) h ε k t − u f ( ζ u )d u under Q is the same as the distribution of R t D u h ε k u f ( Z u )d u under P .We next claim there exists a sequence δ k → t ∈ [ k, ∞ ) P (cid:20) ∧ (cid:18) xD t h ε k t + Z t D u h ε k u f ( Z u )d u (cid:19) > δ k (cid:21) ≤ δ k , ∀ k ∈ N . This is shown at the end of the proof. Admitting this fact, and using E Q [1 ∧ | χ ε k ,xt − χ xt | | G ] ≤ ∧ E Q [ | χ x,ε k t − χ xt | | G ], it follows thatlim k →∞ sup t ∈ [ k, ∞ ) E Q [1 ∧ | χ ε k ,xt − χ xt | ] ! = lim k →∞ sup t ∈ [ k, ∞ ) E Q h E Q [1 ∧ | χ ε k ,xt − χ xt | | G ] i! ≤ lim k →∞ sup t ∈ [ k, ∞ ) E (cid:20) ∧ (cid:18) xD t h ε k t + Z t D u h ε k u f ( Z u )d u (cid:19)(cid:21)! ≤ lim k →∞ δ k = 0 . In the above, the first inequality holds because of (8.17) and the second by (8.18) and the factthat for any r.v. Y , E [1 ∧ Y ] ≤ δ + P [1 ∧ Y > δ ]. The last equality follows by construction of δ k .Recall that T n k was chosen so that lim k →∞ ( k/T n k ) = 0 , it follows thatlim sup k →∞ (cid:18) T n k Z T nk E Q [1 ∧ | χ ǫ k ,xt − χ xt | ] d t (cid:19) ≤ lim sup k →∞ kT n k + T n k − kT n k sup t ∈ [ k, ∞ ) E Q [1 ∧ | χ ε k ,xt − χ xt | ] ! = 0 . which in view of (8.16) implies (8.15), finishing the proof. Thus, it remains to show (8.18). Sincefor any a, b >
0, 1 ∧ ( a + b ) ≤ ∧ a + 1 ∧ b the two terms on the right hand side of (8.18) are treatedseparately. Let δ k >
0. First we have P [1 ∧ xD t h ε k t > δ k ] ≤ P [ xD t h ε k t > δ k ]= P (cid:2) xD t e κt > δ k e κt /h ε k t (cid:3) Now, h ε k t ≤ e ε k / t so on t ≥ k , e κt /h ε k t ≥ e ( κ − ε k / t ≥ e ( κ − ε k / k since ε k / < κ . So, for any δ k > e − ( κ − ε k / k/ it follows that P [ xD t h ε k t > δ k ] ≤ P h xD t e κt ≥ e ( κ − ε k / k/ i Set ˜ δ k := sup t ≥ k P (cid:2) xD t e κt ≥ e ( κ − ε k / k/ (cid:3) . Since D t e κt goes to 0 in P probability, it follows that˜ δ k →
0. Thus, taking δ k to be maximum of ˜ δ k and e − ( κ − ε k / k/ it follows that P [1 ∧ χD t h ε k t > δ k ] ≤ δ k . Turning to the second term in (8.18), it is clear that1 ∧ Z t D u h ε k u f ( Z u )d u ≤ ∧ Z ∞ D u h ε k u f ( Z u )d u As shown in the proof of Lemma 8.3, R ∞ D u h ε k u f ( Z u )d u goes to 0 as k → ∞ almost surely. Thusby the bounded convergence theorem, E (cid:2) ∧ R ∞ D u h ε k u f ( Z u )d u (cid:3) → k → ∞ . Since P (cid:20) ∧ Z ∞ D u h ε k u f ( Z u )d u > δ k (cid:21) ≤ δ k E (cid:20) ∧ Z ∞ D u h ε k u f ( Z u )d u (cid:21) , ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 31 upon defining δ k := q E (cid:2) ∧ R ∞ D u h ε k u f ( Z u )d u (cid:3) it follows that P (cid:2) ∧ R ∞ D u h ε k u f ( Z u )d u > δ k (cid:3) ≤ δ k and δ k →
0. This concludes the proof since to combine the two terms one can take δ k to betwice the maximum of the δ k ’s for individual terms. (cid:3) The next lemma proves the convergence in Lemma 8.4 for f ∈ L ( E, p ), not just f ∈ C ( E ; R + ). Lemma 8.5.
Let Assumption 1.7 hold. Then for all x > and all h ∈ C : (8.19) Q - lim T →∞ T Z T h ( ζ t , χ xt ) dt = Z F hdπ. Proof of Lemma 8.5.
By mollifying f , since p is tight in E there exists a sequence of functions f n ∈ C ( E ) ∩ L ( E, p ) with f n ≥ Z E | f n ( z ) − f ( z ) | p ( z ) dz ≤ n − − n . Note that E (cid:20)Z ∞ ne − t/n | f n ( Z t ) − f ( Z t ) | d t (cid:21) = Z ∞ ne − t/n E [ | f n ( Z t ) − f ( Z t ) | ] d t = Z ∞ ne − t/n (cid:18)Z E | f n ( z ) − f ( z ) | p ( z ) dz (cid:19) d t ≤ Z ∞ n − e − t/n − n d t = 2 − n . Thus, by the Borel-Cantelli lemma it follows that P almost surelylim n →∞ Z ∞ ne − t/n | f n ( Z t ) − f ( Z t ) | d t = 0 . For n > κ from Assumption 1.7, let A n = n − sup t ∈ R + ( e t/n D t ). Note that lim n →∞ A n = 0 almostsurely since for each δ > P almost surely finite random variable T = T ( δ ) so that D t ≤ δe − κt for t ≥ T , and hence A n = 1 n sup t ∈ t ∈ R + (cid:16) e t/n D t (cid:17) ≤ n e T/n sup t ≤ T D t + δn . Since Z ∞ D t | f n ( Z t ) − f ( Z t ) | d t ≤ A n Z ∞ ne − t/n | f n ( Z t ) − f ( Z t ) | d t we see that(8.21) lim n →∞ Z ∞ D t | f n ( Z t ) − f ( Z t ) | d t = 0; P − a.s. Thus, with X n := R ∞ D t f n ( Z t )d t that lim n →∞ X n = X almost surely and hence if π n is the jointdistribution of ( Z , X n ) then π n converges to π weakly, as n → ∞ . Now, on the same probabilityspace as in Lemma 8.1 define χ x,nt := ∆ t (cid:18) x + Z t ∆ − t f n ( ζ t )d t (cid:19) ; t ≥ . Note that | χ n,xt − χ xt | ≤ ∆ t Z t ∆ − u | f n ( ζ u ) − f ( ζ u ) | d u, ∀ t ≥ , and by construction the law of the process on the right hand side above under Q is the same asthe law of R · D u | f n ( Z u ) − f ( Z u ) | d u under P . It thus follows that for δ > t ∈ R + Q [ | χ n,xt − χ xt | > δ ] ≤ P (cid:20)Z ∞ D u | f n ( Z u ) − f ( Z u ) | d u > δ (cid:21) := φ n ( δ ) . By (8.21) we can find a non-negative sequence ( δ n ) such that δ n → δ → φ n ( δ n ) = 0. Now,for h ∈ C we have almost surely for t ≥ | h ( ζ t , χ n,xt ) − h ( ζ t , χ xt ) | ≤ K (1 ∧ | χ n,xt − χ xt | ) . Therefore, with ˆ π x,nT denoting the empirical law of ( ζ, χ n,x ) we have E Q (cid:2)(cid:12)(cid:12)(cid:10) h, ˆ π x,nT (cid:11) − (cid:10) h, ˆ π xT (cid:11)(cid:12)(cid:12)(cid:3) ≤ KT Z T E Q [1 ∧ | χ n,xt − χ xt | ] d t. Since for any 0 < δ < Y we have E [1 ∧ | Y | ] ≤ δ + P [ | Y | > δ ] it follows thatfor any n sup T ∈ R + E Q (cid:2)(cid:12)(cid:12)(cid:10) h, ˆ π x,nT (cid:11) − (cid:10) h, ˆ π xT (cid:11)(cid:12)(cid:12)(cid:3) ≤ K ( φ n ( δ ) + δ ) , and hence for the given sequence ( δ n ):(8.22) lim sup n →∞ sup T ∈ R + E Q (cid:2)(cid:12)(cid:12)(cid:10) h, ˆ π x,nT (cid:11) − (cid:10) h, ˆ π xT (cid:11)(cid:12)(cid:12)(cid:3) ≤ lim sup n →∞ K ( φ n ( δ n ) + δ n ) = 0 . Now, fix an sequence ( T k ) such that lim k →∞ T k = ∞ . Since Lemma 8.4 implies for each n , Q − lim T →∞ | (cid:10) h, ˆ π x,nT (cid:11) − (cid:10) h, π n (cid:11) | = 0 for each n we can find a T k n so that Q (cid:20)(cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π x,nT kn (cid:11) − (cid:10) h, π n (cid:11)(cid:12)(cid:12)(cid:12) > n (cid:21) < n It thus follows that Q − lim n →∞ (cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π n,xT kn (cid:11) − (cid:10) h, π n (cid:11)(cid:12)(cid:12)(cid:12) = 0 . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 33
Since lim n →∞ (cid:12)(cid:12)(cid:10) h, π n (cid:11) − (cid:10) h, π (cid:11)(cid:12)(cid:12) = 0 it follows by (8.22) that for each γ > Q h(cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π xT kn (cid:11) − (cid:10) h, π (cid:11)(cid:12)(cid:12)(cid:12) > γ i ≤ Q h(cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π xT kn (cid:11) − (cid:10) h, ˆ π x,nT kn (cid:11)(cid:12)(cid:12)(cid:12) > γ i + Q h(cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π x,nT kn (cid:11) − (cid:10) h, π n (cid:11)(cid:12)(cid:12)(cid:12) > γ i + 1 (cid:12)(cid:12)(cid:12) (cid:10) h,π n (cid:11) − (cid:10) h,π (cid:11) (cid:12)(cid:12)(cid:12) > γ ≤ γ sup T ∈ R + E Q (cid:2)(cid:12)(cid:12)(cid:10) h, ˆ π xT (cid:11) − (cid:10) h, ˆ π x,nT (cid:11)(cid:12)(cid:12)(cid:3) + Q h(cid:12)(cid:12)(cid:12)(cid:10) h, ˆ π x,nT kn (cid:11) − (cid:10) h, π n (cid:11)(cid:12)(cid:12)(cid:12) > γ i + 1 (cid:12)(cid:12)(cid:12) (cid:10) h,π n (cid:11) − (cid:10) h,π (cid:11) (cid:12)(cid:12)(cid:12) > γ → n → ∞ . We have just showed that for any sequence ( (cid:10) h, ˆ π xT k (cid:11) ) there is a subsequence ( (cid:10) h, ˆ π xT kn (cid:11) ) whichconverges in Q probability to (cid:10) h, π (cid:11) which in fact proves that ( (cid:10) h, ˆ π xT (cid:11) ) converges in Q probabilityto (cid:10) h, π (cid:11) , proving (8.19). (cid:3) The next lemma strengthens the convergence in Lemma 8.5 to almost sure convergence under Q , but for π almost every x >
0, for h ∈ C from (8.11). Lemma 8.6.
Let Assumption 1.7 hold. Then for all h ∈ C and π almost every x > : (8.23) lim T →∞ T Z T h ( ζ t , χ xt ) dt = Z F hdπ ; Q a.s. . Proof of Lemma 8.6.
We again use the notation in (8.13). Recall χ from Lemma 8.1 and defineˆ π T as the empirical law of ( ζ, χ ) on [0 , T ]. Given that ( ζ, χ ) is stationary under Q , the ergodictheorem implies that for all bounded measurable functions h on F that there is a random variable Y such that(8.24) lim T →∞ (cid:10) h, ˆ π T (cid:11) = Y ; Q a.s. . By Lemma 8.5 it holds that for h ∈ C , Y = (cid:10) h, π (cid:11) with Q probability one. Indeed, let δ > Q (cid:2)(cid:12)(cid:12) Y − (cid:10) h, π (cid:11)(cid:12)(cid:12) ≥ δ (cid:3) ≤ Q (cid:2)(cid:12)(cid:12) Y − (cid:10) h, ˆ π T (cid:11)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:10) h, ˆ π T (cid:11) − (cid:10) h, π (cid:11)(cid:12)(cid:12) ≥ δ (cid:3) ≤ Q (cid:20)(cid:12)(cid:12) Y − (cid:10) h, ˆ π T (cid:11)(cid:12)(cid:12) ≥ δ (cid:21) + Q (cid:20)(cid:12)(cid:12)(cid:10) h, ˆ π T (cid:11) − (cid:10) h, π (cid:11)(cid:12)(cid:12) ≥ δ (cid:21) The first of these two terms goes to 0 by (8.24). As for the second, denote by π | x the marginal of π with respect to χ . Then Q (cid:20)(cid:12)(cid:12)(cid:10) h, ˆ π T (cid:11) − (cid:10) h, π (cid:11)(cid:12)(cid:12) ≥ δ (cid:21) = Z ∞ π | x ( dx ) Q (cid:20)(cid:12)(cid:12)(cid:10) h, ˆ π xT (cid:11) − (cid:10) h, π (cid:11)(cid:12)(cid:12) ≥ δ (cid:21) By Lemma 8.4 the integrand goes to 0 as T → ∞ for all x > Q (cid:20) lim T →∞ (cid:10) h, ˆ π T (cid:11) = (cid:10) h, π (cid:11)(cid:21) = Z ∞ π (cid:12)(cid:12) x ( dx ) Q (cid:20) lim T →∞ (cid:10) h, ˆ π xT (cid:11) = (cid:10) h, π (cid:11)(cid:21) , and thus (8.23) holds for π a.e. x >
0, finishing the proof. (cid:3)
The last preparatory lemma strengthens Lemma 8.6 to show almost sure convergence for allstarting points x >
0, not just π almost every x > Lemma 8.7.
Let Assumption 1.7 hold. Then for all h ∈ C and all x > T →∞ T Z T h ( ζ t , χ xt ) dt = Z F hdπ ; Q a.s. . Proof of Lemma 8.7.
Recall from Remark 2.5 that χ x takes the form(8.26) χ xt = ∆ t (cid:18) x + Z t t f ( ζ t )d t (cid:19) ; t ≥ . Let h ∈ C . By Lemma 8.6, there is some x > x > (cid:12)(cid:12)(cid:10) h, ˆ π xT (cid:11) − (cid:10) h, ˆ π x T (cid:11)(cid:12)(cid:12) ≤ T Z T | h ( ζ t , χ xt ) − h ( ζ t , χ x t ) | d t ≤ KT Z T (1 ∧ | χ xt − χ x t | ) d t = KT Z T (1 ∧ ∆ t | x − x | ) d t ≤ K | x − x | T Z ∞ ∆ t d t We will show below that Q (cid:2)R ∞ ∆ t d t < ∞ (cid:3) = 1. Admitting this it holds that Q almost surely,lim T →∞ | (cid:10) h, ˆ π xT (cid:11) − (cid:10) h, ˆ π x T (cid:11) | = 0 and hence the result follows since (8.25) holds for x .It remains to prove that Q (cid:2)R ∞ ∆ t d t < ∞ (cid:3) = 1. By way of contradiction assume there is some0 < δ ≤ Q (cid:2)R ∞ ∆ t d t = ∞ (cid:3) = δ . Then, for each N it holds that Q (cid:2)R ∞ ∆ t d t > N (cid:3) ≥ δ , which in turn implies lim T →∞ Q hR T ∆ t d t > N i ≥ δ . By construction, for any fixed T > , T ] under Q coincides with the law of D under P on [0 , T ]. It this holdsthat lim T →∞ P hR T D t d t > N i ≥ δ . But, this gives P (cid:2)R ∞ D t d t > N (cid:3) ≥ δ for all N and hence P (cid:2)R ∞ D t d t = ∞ (cid:3) >
0. But this violates Assumptions 1.7 since lim t →∞ e κt D t = 0 P almost surelyfor some κ >
0. Thus, Q (cid:2)R ∞ ∆ t d t < ∞ (cid:3) = 1 finishing the proof. (cid:3) With all the above lemmas, the proof of Theorem 2.4 is now given.
Proof of Theorem 2.4.
We again adopt the notation in (8.13). In view of Lemma 8.1 the remainingstatement Theorem 2.4 which must be proved is that there is a set Ω ∈ F ∞ with Q [Ω ] = 1 suchthat (2.15) holds: i.e. ω ∈ Ω = ⇒ lim T →∞ (cid:10) h, ˆ π xT (cid:11) ( ω ) = (cid:10) h, π (cid:11) for all x > , h ∈ C b ( F ; R ) . Recall the definition of C from (8.11) and let h ∈ C b ( F ; R ) ∩ C . In view of Lemma 8.7 there is aset Ω h ∈ F ∞ such that Q [Ω h ] = 1 and ω ∈ Ω h = ⇒ lim T →∞ (cid:10) h, ˆ π xT (cid:11) ( ω ) = (cid:10) h, π (cid:11) for all x > . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 35
Let the (countable subset) ˜
C ⊂ C be as in the technical Lemma A.2 below and set Ω = ∩ h ∈ ˜ C Ω h .Clearly, Q [Ω ] = 1. Let ω ∈ Ω and h ∈ C b ( F ; R ) with C = sup y ∈ F | h ( y ) | . Let ε > n ≥ ↑ φ nm,k , ↓ φ nm,k and θ n as in Lemma A.2 such that (A.11) therein holds. In what follows the ω will be suppressed, but all evaluations are understood to hold for this ω .Let x >
0. With ν from (A.11) equal to ˆ π xT it follows that (cid:10) ↑ φ nm,k , ˆ π xT (cid:11) − C (cid:10) − θ n − , ˆ π xT (cid:11) − ε ≤ (cid:10) h, ˆ π xT (cid:11) ≤ (cid:10) ↓ φ nm,k , ˆ π xT (cid:11) + 2 C (cid:10) − θ n − , ˆ π xT (cid:11) + 2 ε. With ν from (A.11) equal to π one obtains (cid:10) ↑ φ nm,k , π (cid:11) − C (cid:10) − θ n − , π (cid:11) − ε ≤ (cid:10) h, π (cid:11) ≤ (cid:10) ↓ φ nm,k , π (cid:11) + 2 C (cid:10) − θ n − , π (cid:11) + 2 ε. Putting these two together yields (cid:10) h, ˆ π xT (cid:11) − (cid:10) h, π (cid:11) ≥ (cid:10) ↑ φ nm,k , ˆ π xT (cid:11) − C (cid:10) − θ n − , ˆ π xT (cid:11) − ε − (cid:16)(cid:10) ↓ φ nm,k , π (cid:11) + 2 C (cid:10) − θ n − , π (cid:11) + 2 ε (cid:17) = (cid:10) ↑ φ nm,k , ˆ π xT (cid:11) − (cid:10) ↓ φ nm,k , π (cid:11) − C (cid:0)(cid:10) − θ n − , ˆ π xT (cid:11) + (cid:10) − θ n − , π (cid:11)(cid:1) − ε. Since θ n − , ↑ φ nm,k , ↓ φ nm,k ∈ ˜ C ⊂ C taking T → ∞ giveslim inf T →∞ (cid:10) h, ˆ π xT (cid:11) − (cid:10) h, π (cid:11) ≥ (cid:10) ↑ φ nm,k , π (cid:11) − (cid:10) ↓ φ nm,k , π (cid:11) − C (cid:10) − θ n − , π (cid:11) − ε. Now, from Lemma A.2 we know for fixed m, n that the functions ↑ φ nm,k and ↓ φ nm,k are increasingand decreasing respectively in k and such that a) lim ↓ k →∞ φ nm,k ( y ) − ↑ φ nm,k ( y ) = 0 for y ∈ ¯ F n − andb) | ↑ φ nm,k ( y ) − ↑ φ nm,k ( y ) | ≤ C + 2 ε for all y ∈ F and n, m, k . Therefore, taking k → ∞ in theabove and using the monotone convergence theorem we obtainlim inf T →∞ (cid:10) h, ˆ π xT (cid:11) − (cid:10) h, π (cid:11) ≥ − C + ε ) π (cid:2) ¯ F cn − (cid:3) − C (cid:10) − θ n − , π (cid:11) − ε. From Lemma A.2 we know that 0 ≤ θ n ( y ) ≤
1, lim n →∞ θ n ( y ) = 1 for all y ∈ F . Thus, by thebounded convergence theorem and the fact that π is tight in F it follows that by taking n ↑ ∞ :lim inf T →∞ (cid:10) h, ˆ π xT (cid:11) − (cid:10) h, π (cid:11) ≥ − ε. Taking ε ↓ T →∞ (cid:10) h, ˆ π xT (cid:11) − (cid:10) h, π (cid:11) ≥
0. Thus, we have just shown for ω ∈ Ω , x > h ∈ C b ( F ; R ) that lim inf T →∞ (cid:10) h, ˆ π xT (cid:11) ( ω ) − (cid:10) h, π (cid:11) ≥ . By applying the above to ˆ h = − h ∈ C b ( F ; R ) we see thatlim sup T →∞ (cid:10) h, ˆ π xT (cid:11) ( ω ) − (cid:10) h, π (cid:11) ≤ , which finishes the proof. (cid:3) Appendix A. Some Technical Results
Lemma A.1.
Let Assumptions 1.7 hold, and additionally assume that | η | > , f ∈ C ( E ; R + ) .Recall F from (2.1) and the invariant density p for Z . Let h ∈ C ( F ) be given and set (A.1) φ ( z, x ) := p ( z ) h ( z, x ); ψ ( z, x ) := Z x h ( z, y ) dy. Let the operator L be as in (2.5) and the operator L R = A ij ∂ ij + ( b R ) i ∂ i be as in the proof ofProposition 8.2, where A is from (2.4) and b R is from (8.6) . Let ˜ L R be the formal adjoint of L R .Then ˜ L R φ = p ∂ x ( Lψ ) . In particular, if Lψ = 0 then ˜ L R φ = 0 .Proof. For notational ease, the arguments will be suppressed when writing functions except forthe x appearing in the drifts and volatilities of the operators. Now, recall the dynamics for thereversed process ( ζ, χ ) in (8.2): dζ t = (cid:18) c ∇ pp + div ( c ) − m (cid:19) ( ζ t ) dt + σ ( ζ t ) dW t dχ t = (cid:18) f ( ζ t ) − χ t (cid:18) a − θ ′ c ∇ pp − ∇ · ( cθ ) (cid:19) ( ζ t ) (cid:19) dt + χ t (cid:0) θ ′ c ( ζ t ) dW t + η ( ζ t ) ′ dB t (cid:1) , and note, as is mentioned in the proof of Proposition 8.2, that L R is the generator for ( ζ, χ ). Tofurther simplify the calculations, set(A.2) ξ := 12 (cid:18) c ∇ pp + div ( c ) (cid:19) − m, and(A.3) H ( c, θ ) := ∇ · ( cθ ) − θ ′ div ( c ) . Note that by (7.2) it follows that 0 = ∇ · ( pξ ). With this notation we have that dζ t = ( m + 2 ξ ) ( ζ t ) dt + σ ( ζ t ) dW t dχ t = (cid:0) f ( ζ t ) − χ t (cid:0) a − θ ′ ( m + ξ ) − H ( c, θ ) (cid:1) ( ζ t ) (cid:1) dt + χ t (cid:0) θ ′ c ( ζ t ) dW t + η ( ζ t ) ′ dB t (cid:1) , which in turns yields that(A.4) A = c xcθxθ ′ c x ( θ ′ cθ + η ′ η ) ! ; b R = m + 2 ξf − x ( a − θ ′ ( m + ξ ) − H ( c, θ )) ! . along with(A.5) b = m − f + x ( a + θ ′ cθ + η ′ η ) ! . Lastly, multivariate notation will be used for derivatives with respect to z and single variate notationused for derivatives with respect to x . Thus, for the given φ : ∇ ( z,x ) φ = ( ∇ φ, ˙ φ ); D z,x ) φ = D φ ∇ ( ˙ φ ) ∇ ( ˙ φ ) ′ ¨ φ ! . ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 37
Since φ = p h and p is not a function of x : ∇ ( z,x ) φ = p ∇ h + h ∇ pp ˙ h ! . By definition, ˜ L R φ = ∇ ( z,x ) · (cid:0) (1 / A ∇ ( z,x ) φ + φ div ( z,x ) ( A )) − b R φ (cid:1) . Using (A.4): A ∇ ( z,x ) φ = pc ∇ h + hc ∇ p + px ˙ hcθpxθ ′ c ∇ h + hxθ ′ c ∇ p + px ˙ h ( θ ′ cθ + η ′ η ) ! . Calculation shows div ( z,x ) ( A ) = div ( c ) + cθx ∇ · ( cθ ) + 2 x ( θ ′ cθ + η ′ η ) ! , so that12 ( A ∇ ( z,x ) φ + φ div ( z,x ) ( A ))= 12 pc ∇ h + hc ∇ p + px ˙ hcθ + ph div ( c ) + phcθpxθ ′ c ∇ h + hxθ ′ c ∇ p + px ˙ h ( θ ′ cθ + η ′ η ) + pxh ∇ · ( cθ ) + 2 pxh ( θ ′ cθ + η ′ η ) ! . This gives (1 / A ∇ ( z,x ) φ + φ div ( z,x ) ( A )) − b R φ = ( A , B ) ′ where A = 12 (cid:16) pc ∇ h + hc ∇ p + px ˙ hcθ + ph div ( c ) + phcθ (cid:17) − phm − phξ, B = 12 (cid:16) pxθ ′ c ∇ h + hxθ ′ c ∇ p + px ˙ h ( θ ′ cθ + η ′ η ) + pxh ∇ · ( cθ ) + 2 pxh ( θ ′ cθ + η ′ η ) (cid:17) − phf + pxha − pxhθ ′ ( m + ξ ) − pxhH ( c, θ ) . (A.6)Now, ˜ L R φ = ∇ · A + ˙ B . A is treated first. From (A.2) it follows that p div ( c ) + c ∇ p = 2 p ( m + ξ )and hence 2 A = pc ∇ h + px ˙ hcθ + phcθ − phξ. For a scalar function f and R d valued function g , ∇ · ( f g ) = f ∇ · g + ∇ f ′ g . Using this2 ∇ · A = p ∇ · ( c ∇ h ) + ∇ h ′ c ∇ p + px ˙ h ∇ · ( cθ ) + x ∇ ( p ˙ h ) ′ cθ + ph ∇ · ( cθ )+ ∇ ( ph ) ′ cθ − h ∇ · ( pξ ) − p ∇ h ′ ξ, = p ∇ · ( c ∇ h ) + ∇ h ′ c ∇ p + px ˙ h ∇ · ( cθ ) + px ∇ ( ˙ h ) ′ cθ + x ˙ h ∇ p ′ cθ + ph ∇ · ( cθ )+ p ∇ h ′ cθ + h ∇ p ′ cθ − h ∇ · ( pξ ) − p ∇ h ′ ξ. Using that ∇ · ( c ∇ h ) = tr (cid:0) cD h (cid:1) + ∇ h ′ div ( c ) and collecting terms by derivatives of h gives2 ∇ · A = p tr (cid:0) cD h (cid:1) + px ∇ ( ˙ h ) ′ cθ + ∇ h ′ ( p div ( c ) + c ∇ p + pcθ − pξ ) , + ˙ h (cid:0) px ∇ · ( cθ ) + x ∇ p ′ cθ (cid:1) + h (cid:0) p ∇ · ( cθ ) + ∇ p ′ cθ − ∇ · ( pξ ) (cid:1) . Since p div ( c ) + c ∇ p = 2 p ( m + ξ ), ∇ · ( pξ ) = 0 and ∇ · ( cθ ) = H ( c, θ ) + θ ′ div ( c ), p div ( c ) + c ∇ p + pcθ − pξ = 2 pm + pcθ,px ∇ · ( cθ ) + x ∇ p ′ cθ = 2 pxθ ′ ( m + ξ ) + pxH ( c, θ ) ,p ∇ · ( cθ ) + ∇ p ′ cθ − ∇ · ( pξ ) = 2 pθ ′ ( m + ξ ) + pH ( c, θ ) . Plugging this in and factoring out the p yields2 p ∇ · A = tr (cid:0) cD h (cid:1) + x ∇ ( ˙ h ) ′ cθ + ∇ h ′ (2 m + cθ ) + ˙ h (cid:0) xθ ′ ( m + ξ ) + xH ( c, θ ) (cid:1) + h (cid:0) θ ′ ( m + ξ ) + H ( c, θ ) (cid:1) . (A.7)Turning to B in (A.6). Using p div ( c ) + c ∇ p = 2 p ( m + ξ ) and ∇ · ( cθ ) = H ( c, θ ) + θ ′ div ( c ) yields2 B = pxθ ′ c ∇ h − pxhθ ′ ( m + ξ ) + px ˙ h ( θ ′ cθ + η ′ η ) + 2 pxh ( θ ′ cθ + η ′ η ) − phf + 2 pxha − pxhH ( c, θ ) . Since only h depends upon x ,2 ˙ B = pθ ′ c ∇ h + px ∇ ( ˙ h ) ′ cθ − phθ ′ ( m + ξ ) − px ˙ hθ ′ ( m + ξ ) + 2 px ˙ h ( θ ′ cθ + η ′ η ) + px ¨ h ( θ ′ cθ + η ′ η )+ 2 ph ( θ ′ cθ + η ′ η ) + 2 px ˙ h ( θ ′ cθ + η ′ η ) − p ˙ hf + 2 pha + 2 px ˙ ha − phH ( c, θ ) − px ˙ hH ( c, θ ) . Grouping terms by derivatives of h and factoring out the p yields2 p ˙ B = x ¨ h ( θ ′ cθ + η ′ η ) + x ∇ ( ˙ h ) ′ cθ + h (cid:0) − θ ′ ( m + ξ ) + 2( θ ′ cθ + η ′ η ) + 2 a − hH ( c, θ ) (cid:1) + ∇ h ′ cθ (A.8) + ˙ h (cid:0) − xθ ′ ( m + ξ ) + 4 x ( θ ′ cθ + η ′ η ) − f + 2 xa − xH ( c, θ ) (cid:1) . Putting together (A.7) and (A.8) and using that ˜ L R φ = ∇ · A + ˙ B :1 p ˜ L R φ = 12 tr (cid:0) cD h (cid:1) + x ∇ ( ˙ h ) ′ cθ + 12 x ¨ h ( θ ′ cθ + η ′ η ) + ∇ h ′ ( m + cθ )+ ˙ h (cid:0) x ( θ ′ cθ + η ′ η ) − f + xa (cid:1) + h (cid:0) θ ′ cθ + η ′ η + a (cid:1) . (A.9)Turning now to ψ , since Lψ = 12 tr (cid:0) cD ψ (cid:1) + x ∇ ( ˙ ψ ) ′ cθ + 12 x ¨ ψ ( θ ′ cθ + η ′ η ) + ∇ ψ ′ m + ˙ ψ (cid:0) − f + xa + x ( θ ′ cθ + η ′ η ) (cid:1) , ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 39 it follows that (note : only ψ depends upon x and ˙ ψ = h )˙ Lψ = 12 tr (cid:16) cD ˙ ψ (cid:17) + x ∇ ( ¨ ψ ) ′ cθ + ∇ ( ˙ ψ ) ′ cθ + x ¨ ψ ( θ ′ cθ + η ′ η ) + 12 x ... ψ ( θ ′ cθ + η ′ η )+ ∇ ( ˙ ψ ) ′ m + ¨ ψ (cid:0) − f + xa + x ( θ ′ cθ + η ′ η ) (cid:1) + ˙ ψ (cid:0) a + θ ′ cθ + η ′ η (cid:1) , = 12 tr (cid:16) cD ˙ ψ (cid:17) + x ∇ ( ¨ ψ ) ′ cθ + 12 x ... ψ ( θ ′ cθ + η ′ η )+ ∇ ( ˙ ψ ) ′ ( m + cθ ) + ¨ ψ (cid:0) x ( θ ′ cθ + η ′ η ) − f + xa (cid:1) + ˙ ψ (cid:0) a + θ ′ cθ + η ′ η (cid:1) , = 12 tr (cid:0) cD h (cid:1) + x ∇ ( ˙ h ) ′ cθ + 12 x ¨ h ( θ ′ cθ + η ′ η )+ ∇ h ′ ( m + cθ ) + ˙ h (cid:0) x ( θ ′ cθ + η ′ η ) − f + xa (cid:1) + h (cid:0) a + θ ′ cθ + η ′ η (cid:1) . But, from (A.9) this last term is precisely (1 /p ) ˜ L R φ . (cid:3) Lemma A.2.
Let Assumption 1.7 hold. Let C be as in (8.11) . Recall that F = E × (0 , ∞ ) andlet { F n } n ∈ N be a family of open, bounded, increasing subsets of F with smooth boundary such that F = ∪ n F n . There exists a countable family of functions (A.10) ˜ C := n ↑ φ nm,k , ↓ φ nm,k , θ n | n, m, k ∈ N , n ≥ o ⊂ C such that1) For each n ≥ , ≤ θ n ≤ with θ n = 1 on ¯ F n and θ n = 0 on F cn +1 .2) For each n ≥ and m , the functions ↑ φ nm,k are increasing in k and the functions ↓ φ nm,k aredecreasing in k . Furthermore, for any n ≥ and m , lim k →∞ | ↑ φ nm,k ( y ) − ↓ φ nm,k ( y ) | = 0 for y ∈ ¯ F n − .Additionally, for any h ∈ C b ( F ; R ) set C = C ( h ) := sup y ∈ F | h ( y ) | . Then, for any ε > and anyinteger n ≥ there exits an integer m = m ( ε, n ) such that for all k ∈ N , sup y ∈ F | ↑ φ nm,k ( y ) | ≤ C + ε , sup y ∈ F | ↓ φ nm,k ( y ) | ≤ C + ε . Furthermore, for any Borel measure ν on F : Z F ↑ φ nm,k dν − C Z F (1 − θ n − ) dν − ε ≤ Z F hdν ≤ Z F ↓ φ nm,k dν + 2 C Z F (1 − θ n − ) dν + 2 ε. (A.11) Proof of Lemma A.2.
Fix n ∈ N and let ( φ nm ) m ∈ M be a countable dense (with respect to thesupremum norm) subset of C b ( ¯ F n ; R ). Now, let k ∈ N and define:(A.12) ↑ ˜ φ nm,k ( y ) := inf y ∈ ¯ F n ( φ nm ( y ) + k | y − y | ) ; ↓ ˜ φ nm,k ( y ) := sup y ∈ ¯ F n ( φ nm ( y ) − k | y − y | ) ; y ∈ ¯ F n . As shown in [2, Ch. 3.4], ↑ ˜ φ nm,k and ↓ ˜ φ nm,k are a) increasing and decreasing respectively in k , andb) Lipschitz continuous in ¯ F n with Lipschitz constant k . Furthermore, as k ↑ ∞ , ↑ ˜ φ nm,k ր φ nm and ↓ φ nm,k ց φ nm on ¯ F n . Next, let θ n ∈ C ∞ ( F ; R ) be such that 0 ≤ θ n ≤ θ n ( y ) = 1 on ¯ F n and θ n ( y ) = 0 on F cn +1 .Clearly, θ n ∈ C for each n . Now, assume n ≥ ↑ ˜ φ nm,k and ↓ ˜ φ nm,k from functions on ¯ F n to all of F via ↑ φ nm,k ( y ) = ↑ ˜ φ nm,k ( y ) θ n − ( y ) y ∈ ¯ F n else ; ↓ φ nm,k ( y ) = ↓ ˜ φ nm,k ( y ) θ n − ( y ) y ∈ ¯ F n else Clearly, ↑ φ nm,k and ↓ φ nm,k are Lipschitz on F and, since F n is bounded, it also holds that ↑ φ nm,k and ↓ φ nm,k are in C . Note also that ↑ φ nm,k and ↓ φ nm,k increase and decrease respectively as k ↑ ∞ to a function which is equal to φ nm on ¯ F n − and that ↑ φ nm,k , ↓ φ nm,k are bounded on all of F bysup y ∈ ¯ F n | ↑ ˜ φ nm,k ( y ) | and sup y ∈ ¯ F n | ↓ ˜ φ nm.k ( y ) | respectively. This proves 1) ,
2) above.Now, let h ∈ C b ( F ; R ) with C = sup y ∈ F | h ( y ) | . Let ε > n ≥ m = m ( ε, n ) sothat sup y ∈ ¯ F n | h ( y ) − φ nm ( y ) | ≤ ε . By construction of ↑ ˜ φ nm,k in (A.12) it follows for each k that − ( C + ε ) ≤ inf y ∈ ¯ F n ( φ nm ( y )) ≤ ↑ ˜ φ nm,k ( y ) ≤ φ nm ( y ) ≤ h ( y ) + ε ≤ C + ε ; y ∈ ¯ F n . By definition of ↑ φ nm,k this gives sup y ∈ F | ↑ φ nm,k ( y ) | ≤ C + ε . Furthermore, since θ n − ( y ) = 1 on¯ F n − , we have h ( y ) ≥ ↑ φ nm,k ( y ) − ε on ¯ F n − . Therefore, for any Borel measure ν , using the notationin (8.13): (cid:10) h, ν (cid:11) ≥ (cid:10) (cid:16) ↑ φ nm,k − ε (cid:17) ¯ F n − , ν (cid:11) − Cν (cid:2) ¯ F cn − (cid:3) ≥ (cid:10) ↑ φ nm,k , ν (cid:11) − (cid:10) ↑ φ nm,k ¯ F cn − , ν (cid:11) − ε − Cν (cid:2) ¯ F cn − (cid:3) ≥ (cid:10) ↑ φ nm,k , ν (cid:11) − ( C + ε ) ν (cid:2) ¯ F cn − (cid:3) − ε − Cν (cid:2) ¯ F cn − (cid:3) ≥ (cid:10) ↑ φ nm,k , ν (cid:11) − Cν (cid:2) ¯ F cn − (cid:3) − ε ≥ (cid:10) ↑ φ nm,k , ν (cid:11) − C Z F (1 − θ n − ) dν − ε, where the last inequality follows since 1 ¯ F cn − ( y ) ≤ − θ n − ( y ). This gives the lower bound in (A.11).A similar calculation shows for all k that − ( C + ε ) ≤ h ( y ) − ε ≤ φ nm ( y ) ≤ ↓ ˜ φ nm,k ( y ) ≤ sup y ∈ ¯ F n ( φ nm ( y )) ≤ C + ε ; y ∈ ¯ F n . This gives sup y ∈ F | ↓ φ nm,k ( y ) | ≤ C + ε and h ( y ) ≤ ↓ φ nm,k ( y ) + ε on ¯ F n − . Thus (cid:10) h, ν (cid:11) ≤ (cid:10) (cid:16) ↓ φ nm,k + ε (cid:17) ¯ F n − , ν (cid:11) + Cν (cid:2) ¯ F cn − (cid:3) ≤ (cid:10) ↓ φ nm,k , ν (cid:11) − (cid:10) ↓ φ nm,k ¯ F cn − , ν (cid:11) + ε + Cν (cid:2) ¯ F cn − (cid:3) ≤ (cid:10) ↓ φ nm,k , ν (cid:11) + ( C + ε ) ν (cid:2) ¯ F cn − (cid:3) + ε + Cν (cid:2) ¯ F cn − (cid:3) ≤ (cid:10) ↓ φ nm,k , ν (cid:11) + 2 Cν (cid:2) ¯ F cn − (cid:3) + 2 ε ≤ (cid:10) ↓ φ nm,k , ν (cid:11) + 2 C Z F (1 − θ n − ) dν + 2 ε. ONTINUOUS-TIME PERPETUITIES AND TIME REVERSAL OF DIFFUSIONS 41
Therefore, the upper bound in (A.11) is established. (cid:3)
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Constantinos Kardaras, Department of Statistics, London School of Economics, 10 HoughtonStreet, London, WC2A 2AE, England.
E-mail address : [email protected] Scott Robertson, Department of Mathematical Sciences, Carnegie Mellon University, Wean Hall6113, Pittsburgh, PA 15213, USA.
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