Uniqueness in Cauchy problems for diffusive real-valued strict local martingales
aa r X i v : . [ q -f i n . M F ] J u l Uniqueness in Cauchy problems for diffusive real-valuedstrict local martingales ∗ Umut C¸ etin
London School of Economics
Kasper Larsen
Rutgers UniversityJuly 31, 2020
Abstract : For a real-valued one dimensional diffusive strict local mar-tingale, we provide a set of smooth functions in which the Cauchyproblem has a unique classical solution. We exemplify our resultsusing the inverse 2D Bessel process and quadratic normal volatilitymodels.
Keywords : Strict local martingales, Cauchy problem, Sturm-LiouvilleODEs, boundary layer.
Consider a unique weak solution (cid:0) P x (cid:1) x ∈ R of the time homogenous diffusion dX t = σ ( X t ) dB t , X = x, x ∈ R , (1.1) ∗ The authors have benefited from helpful comments from Erik Ekstr¨om, Ioannis Karatzas, MartinLarsson, Dan Ocone, Li-Cheng Tsai, Johan Tysk, and Kim Weston. In particular, many thanksto Johannes Ruf for his many valuable suggestions. The second author has been supported by theNational Science Foundation under Grant No. DMS 1812679 (2018 - 2021). Any opinions, findings,and conclusions or recommendations expressed in this material are those of the author(s) and donot necessarily reflect the views of the National Science Foundation (NSF). The correspondingauthor is Kasper Larsen. Umut C¸ etin has email: [email protected] and Kasper Larsen has email:[email protected]. σ : R → (0 , ∞ ) is a locally H¨older continuous function and B = ( B t ) t ≥ is astandard one-dimensional Brownian motion. For given continuous data H : R → R ,the Cauchy problem corresponding to h ( t, x ) : = E x [ H ( X t )] , t ≥ , x ∈ R , (1.2)is given by h (0 , x ) = H ( x ) , x ∈ R ,h t = σ ( x ) h xx , t > , x ∈ R , (1.3)plus boundary and growth conditions. As we shall see, strict local martingality of(1.1) requires two non-standard boundary conditions as x → ±∞ .When the volatility function σ in (1.1) is of at most linear growth, the solution of(1.1) becomes a martingale. In that case, when the data H has at most polynomialgrowth, the function h in (1.2) is the unique classical solution of (1.3) of at mostpolynomial growth (see, e.g., Theorem 5.7.6 in Karatzas and Shreve [26] and, for amore general result, Theorem 5.5 in Janson and Tysk [20]).On the other hand, when the solution of (1.1) is a strict local martingale, the mean H ( ξ ) := ξ gives an example where uniqueness of (1.3) fails because both h defined in(1.2) and h ( t, x ) := x solve (1.3). When the solution of (1.1) is nonnegative, Bayraktarand Hao [3] link uniqueness in Cauchy problems to martingality of solutions to (1.1).For a nonnegative strict local martingale and for H ( ξ ) of at most strict sublinearasymptotic growth as ξ ↑ ∞ , Theorem 4.3 in Ekstr¨om and Tysk [11] proves that (1.2)is the unique classical solution of (1.3) in the class of strictly sublinearly growingfunctions. To include the mean H ( ξ ) := ξ , ξ ∈ [0 , ∞ ), Theorem 6.2 in C¸ etin [6]allows H to be of at most linear growth and proves uniqueness in the class of strictlysublinear classical solutions of (1.3) when the solution of (1.1) is nonnegative.Our existence and uniqueness result extends [11] and [6] in that we allow thestrict local martingale solving (1.1) to be real-valued. Our extension allows us toinclude two widely used strict local martingale models: The inverse 2D Bessel processand quadratic normal volatility models (the latter models are often used in financialeconomics to model stock bubbles, see, e.g., Z¨uhlsdorff [41], Andersen [1], and Carr,Fisher, and Ruf [5]). A key step in our analysis uses solutions to Sturm-LiouvilleODEs (i.e., smooth λ -harmonic functions) as Radon-Nikodym derivatives. The class2f data functions H covered by our result is given in terms of these Sturm-Liouvillefunctions and while our class always includes H of at most linear growth, our classtypically also includes faster growing data H . For example, for the inverse 2D Besselexample mentioned above, H ( ξ ) can grow super-linearly as ξ ↑ ∞ .Alternatively, h in (1.2) is uniquely characterized via a smallness property (fordetails, see the references and results in Section 5.1 in Karatzas and Ruf [28]). Incontrast, our uniqueness result explicitly pins down the limiting behavior of h ( t, x )as x → ±∞ . From a numerical perspective, the smallness characterization can bedifficult to implement (see [12]) whereas our boundary restrictions are compatiblewith standard numerical procedures such as finite difference methods.For optimization problems, it is known that strict local martingales can producediscontinuities. For example, El Karoui and Quenez [13] consider a set M of positivemartingales Z = ( Z t ) t ≥ with Z = 1 and they showinf Z ∈M E x [ Z t H ( B t )] = inf ξ ∈ R H ( ξ ) , t > , (1.4)for any bounded continuous function H : R → R . In particular, no martingale Z ∈ M can attain the infimum in (1.4). On the other hand, for t = 0, the left-hand-side of (1.4) is H ( x ) which can produce a discontinuity because H ( x ) = inf ξ ∈ R H ( ξ ) ispossible. Larsen, Soner, and ˇZitkovi´c [34] show that similar discontinuities can appearfor strictly convex objectives and link such discontinuities to strict local martingalityof dual utility optimizers. In the current paper, we give conditions under which thevalue function h in (1.2) is a classical solution (in particular, h continuous) of the PDE(1.3). However, we illustrate that the strict local martingality of (1.1) still producesdiscontinuities in the following sense: For H ∈ C ( R ), we illustrate that is possibleto have lim ξ →±∞ H ′ ( ξ ) = lim t ↓ lim ξ →±∞ h x ( t, ξ ) . (1.5)When the solution X of (1.1) is nonnegative, Corollary 6.1 in [6] illustrates a similardiscontinuity.The many applications of strict local martingales for modeling purposes in finan-cial economics include: (i) Basak and Cuoco [2], Hugonnier [18], and Chabakauri [7]show that strict local martingales can appear endogenously in equilibrium theory.3ii) Cox and Hobson [9], Jarrow, Protter, and Shimbo [21], Heston, Loewenstein, andWillard [15], and Andersen [1] use strict local martingales for derivative pricing. (iii)Stochastic portfolio theory as surveyed in Fernholz and Karatzas [14] uses strict lo-cal martingales to model relative arbitrage opportunities. (iv) Karatzas, Lehoczky,Shreve, and Xu [24], Kramkov and Schachermayer [31], and Lowenstein and Willard[35] exemplify that strict local martingales can appear as dual utility maximizers.More recent references based on nonnegative local martingales that include Kar-daras, Kreher, and Nikeghbali [27], Kramkov and Weston [32], and Kardaras and Ruf[28, 29]. Hulley and Platen [16] and Hulley and Ruf [17] are recent references basedon real-valued local martingales. Let R ∆ be the one-point compactification of R with ∆ being the point-at-infinity . Weconsider the path space Ω of right continuous functions ω : R + → R ∆ which satisfy ω ( t ) = ∆ for all t > s whenever ω ( s ) = ∆ and we note that C ( R + , R ) ⊂ C ( R + , R ∆ ) ⊂ Ω. We let X be the coordinate process on Ω. The filtration ( F t ) t ≥ is the universalcompletion of the natural filtration of X and, therefore, is right continuous because X is a strong Markov process (see, e.g., Theorem 4 in Section 2.3 in [8]). All probabilitymeasures are defined on F := W t ≥ F t .The following assumption ensures the existence of a unique weak solution ( P x ) x ∈ R of (1.1). This existence result is due to Engelbert and Schmidt and can be found in,e.g., Theorem 5.5.7 in [26]. Assumption 2.1.
The volatility function σ : R → (0 , ∞ ) is locally H¨older continuouswith H¨older exponent . ♦ Remark . The assumption of H¨older continuity is not needed for the existenceof a unique weak solution but we use H¨older continuity in Theorem 2.2 below in aPDE existence argument. Furthermore, H¨older continuity can be used to upgrade theunique weak solution of (1.1) to a pathwise unique strong solution by Theorem IX.3.5ii) in [38] and Corollary 5.3.23 in [26]. Also, as discussed after Proposition 5.2 in [25],the H¨older exponent being can be relaxed. Moreover, it is worth emphasizing thatH¨older continuity is not needed for the uniqueness part in Theorem 2.2 below. Infact, we can even go beyond the continuous case and use our uniqueness to show the4nalogous result for the solutions of h t = Lh , where L := ddm ddx is the general formfor the infinitesimal generator of X when σ is not necessarily continuous (see Section4.1 in Itˆo and McKean [19]). Note that the basic solutions (see below after (2.2)) of λu = Lu for a constant λ > m ( dξ ) for the time-homogenous diffusion(1.1) is absolutely continuous with Lebesgue-derivate m ′ ( ξ ) given by m ( dξ ) dξ := m ′ ( ξ ) , m ′ ( ξ ) := 2 σ ( ξ ) , ξ ∈ R . (2.1)We emphasize that there are no absorbing states and therefore X is recurrentunder ( P x ) x ∈ R in the sense that P x ( T y < ∞ ) = 1 for all ( x, y ) ∈ R where T y :=inf { t > X t = y } . Furthermore, the lifetime of X is denoted by ζ ; that is, ζ := inf { t > X t = ∆ } , and ζ satisfies P x ( ζ < ∞ ) = 0 for all x ∈ R .For a constant λ > ϕ : R → R , ϕ ∈ C ( R ), the singular Sturm-Liouville ODEs corresponding to the SDE (1.1) are given by λϕ ( ξ ) = 12 σ ( ξ ) ϕ ′′ ( ξ ) , ξ ∈ R . (2.2)The ODE (2.2) is called singular because the domain R is unbounded. Recall thatthe two basic solutions ϕ λ ↑ ( ξ ) and ϕ λ ↓ ( ξ ) solve (2.2), are positive, convex, monotone,and are unique up to multiplicative constants. We refer to, e.g., Chapter 4 in Itˆo andMcKean [19], Appendix 8 in Revuz and Yor [38], and Chapter 5 in Jeanblanc, Yor,and Chesney [22] for more information. In terms of ϕ λ ↑ ( ξ ) and ϕ λ ↓ ( ξ ), we can definethe nonnegative convex functionΦ λ ( ξ ) := ϕ λ ↓ ( ξ ) + ϕ λ ↑ ( ξ ) , ξ ∈ R , (2.3)which is uniformly bounded away from zero.Our main result is the next theorem and it uses Theorem 1 in Kotani [30] whichensures that when the solution of (1.1) is a strict local martingale, at least one of (i) R ∞ ξm ( dξ ) < ∞ and (ii) R −∞ | ξ | m ( dξ ) < ∞ holds. See also Theorem 1.6 in Delbaenand Shirakawa [10] for the case when the solution of (1.1) is nonnegative . Theorem 2.2.
Suppose Assumption 2.1 holds and let the SDE (1.1) have a unique trong integrable solution. Then, for continuous data H : R → R such that sup ξ, | ξ | > | H ( ξ ) | Φ λ ( ξ ) < ∞ , (2.4) we have:(i) If R ∞ ξm ( dξ ) < ∞ and R −∞ | ξ | m ( dξ ) = ∞ , the function h ( t, x ) in (1.2) is theunique classical solution of (1.3) which is bounded by K Φ λ ( x ) for a constant K ≥ and satisfies the boundary condition lim n ↑∞ h ( s n , n ) n = 0 , whenever ∞ > s n ≥ s n +1 and s ∞ := lim n ↑∞ s n > . (2.5) (ii) If R ∞ ξm ( dξ ) = ∞ and R −∞ | ξ | m ( dξ ) < ∞ , the function h ( t, x ) in (1.2) is theunique classical solution of (1.3) which is bounded by K Φ λ ( x ) for a constant K ≥ and satisfies the boundary condition lim n ↑∞ h ( s n , − n ) n = 0 , whenever ∞ > s n ≥ s n +1 and s ∞ := lim n ↑∞ s n > . (2.6) (iii) If R ∞ ξm ( dξ ) < ∞ and R −∞ | ξ | m ( dξ ) < ∞ , the function h ( t, x ) in (1.2) is theunique classical solution of (1.3) which is bounded by K Φ λ ( x ) for a constant K ≥ and satisfies the boundary conditions (2.5) and (2.6) . We note that (2.4) covers continuous data H : R → R of at most linear growth,i.e., sup ξ, | ξ | > | H ( ξ ) || ξ | < ∞ . (2.7)When H satisfies (2.7), Theorem 3.10(i) in [16] ensures that h in (1.2) is also of atmost linear growth. Furthermore, when R −∞ | ξ | m ′ ( ξ ) dξ = ∞ , we can allow for superlinearly growing data H ( ξ ) as ξ ↓ −∞ and when R ∞ ξm ′ ( ξ ) dξ = ∞ we can allow forsuper linearly growing data H ( ξ ) as ξ ↑ ∞ (the latter is the case for the inverse 2DBessel process in Example 2.3 below).Based on Theorem 2.2, the value function h ( t, x ) in (1.2) can exhibit a boundarylayer at t = 0 in the following sense: Consider the mean H ( ξ ) := ξ , ξ ∈ R , which6atisfies (2.7). Then, whenever R ∞ ξm ( dξ ) < ∞ , the value function h ( t, x ) satisfies(1.5) as x ↑ ∞ because Theorem 2.2(i) giveslim x ↑∞ lim t ↓ h ( t, x ) x = 1 , lim t ↓ lim x ↑∞ h ( t, x ) x = 0 . (2.8)Similarly, whenever R −∞ | ξ | m ( dξ ) < ∞ , the value function h ( t, x ) in (1.2) satisfies as x ↓ −∞ because Theorem 2.2(ii) giveslim x ↓−∞ lim t ↓ h ( t, x ) x = 1 , lim t ↓ lim x ↓−∞ h ( t, x ) x = 0 . (2.9) Example 2.3 (2D Bessel) . For σ ( ξ ) := e − ξ , ξ ∈ R , the dynamics (1.1) become thoseof the inverse 2-dimensional Bessel process that solves the SDE dX t = e − X t dB t , t ≥ , X = x ∈ R . (2.10)Eq. (2.10) has a unique strong solution for a given Brownian motion B = ( B t ) t ≥ .We claim that E x [ X t ] = x + Z ∞ e x r e − r t dr, x ∈ R , t ≥ . (2.11)To see this, we define the function h ( t, x ) := x + Z ∞ e x r e − r t dr, x ∈ R , t ≥ . (2.12)By computing t and x derivatives in (2.12), we see that the PDE in (1.3) holds.Furthermore, for t >
0, L’Hopital’s rule produces the limitlim x ↓−∞ x Z ∞ e x r e − r t dr = − lim x ↓−∞ e − e x t = − . (2.13)7herefore, for t >
0, the function h in (2.12) has the limit in (2.6). Because Z ∞ ξe ξ dξ = ∞ , Z −∞ | ξ | e ξ dξ < ∞ , (2.14)we can use the uniqueness part of Theorem 2.2(ii) to see that (2.11) holds. Conse-quently, the boundary layer limits in (2.9) hold.As an aside, the limit in (2.6) trivially holds because we havelim x ↓−∞ E x [ X t ] = 12 (cid:0) log(2) + log( t ) − γ (cid:1) ∈ R , t > , (2.15)where γ is the Euler-Mascheroni constant ( γ ≈ . ♦ Example 2.4.
Quadratic normal volatility models use dynamics defined by dX t := (cid:0) α + α X t + α X t (cid:1) dB t , X ∈ R , (2.16)and have been widely used in financial economics (see Carr, Fisher, and Ruf [5] for anoverview). Depending on the root configuration ( α + α ξ + α ξ = 0 , ξ ∈ R ) relativeto the initial value X , the solution to the SDE (2.16) is bounded or unboundedfrom above and/or below. For example, in a Radner equilibrium model with limitedstock-market participation, the following SDE is endogenously derived in Eq. (27) inBasak and Cuoco [2]: dX t = − X t (1 + X t ) σdB t , X > , (2.17)for a constant σ ∈ (0 , ∞ ). The dynamics (2.17) produce a nonnegative strict localmartingale. Another specification of (2.16) is the no-real-root specification used foroption pricing in Section 3.6 in Z¨uhlsdorff [41] and Eq. (4.1) in Andersen [1]. Thisprocess is exogenously given by the dynamics dX t = b (cid:16) (cid:0) X t − ab (cid:1) (cid:17) dB t , X ∈ R , (2.18)for constants ( a, b ) with b ∈ (0 , ∞ ). The dynamics (2.18) produce a real-valued strict When X is an inverse 3D Bessel process (which is positive), Example 3.6 in [9] gives a limitsimilar to (2.15) and shows in particular that lim x ↑∞ E x [ X t ] < ∞ for t > Z ∞ ξ (cid:0) b + ( a − ξ ) (cid:1) dξ < ∞ , Z −∞ | ξ | (cid:0) b + ( a − ξ ) (cid:1) dξ < ∞ , (2.19)we see from Theorem 2.2(iii) that h ( t, x ) in (1.2) vanishes as x → ±∞ for t >
0. Inthis case, the mean function H ( ξ ) := ξ , ξ ∈ R , produces a double boundary layer inthe sense that for t > ♦ Finally, we note that Theorem 2.2 is stated under an integrability condition. Whileall nonnegative local martingales are also supermartingales (hence, integrable), thefollowing example shows that real-valued strict local martingales can fail to be inte-grable.
Example 2.5.
Let ( Y t ) t ≥ be the inverse 3D Bessel process with dynamics dY t := − Y t dB t , t ∈ (0 , ∞ ) , Y = y > . (2.20)Eq. (2.20) has a unique strong solution for a given Brownian motion B = ( B t ) t ≥ .This is the classical example due to Johnson and Helms [23] of a strict local martingale.Furthermore, from, e.g., p.74 in Protter [37], the second moment satisfies E y [ Y t ] < ∞ while E y [ h Y i t ] = ∞ for t ∈ (0 , ∞ ). Consequently, the real-valued local martingale X t := Y t − h Y i t , t ≥ , (2.21)is not integrable. In particular, ( X t ) t ≥ is a strict local martingale too. ♦ There are two main ingredients in our proof. In the first subsection, we relate the strictlocal martingale property of the solution to (1.1) to growth properties of solutionsto the corresponding (singular) Sturm-Lioville ODEs (2.2) denoted by ϕ λ ↑ and ϕ λ ↓ .Second, to prove uniqueness, we construct suitable non-equivalent measures ( P ϕ,x ) x ∈ R .The third and last subsection contains the proof of Theorem 2.2.The space Ω is of path type and is projective (see Definitions 23.10 and 62.4 andthe following paragraph in Sharpe [39]). Thus, given a supermartingale multiplicativefunctional M = ( M t ) t ≥ as defined in Eqs. (54.1) and (54.7) in [39], Theorem 62.19 in939] establishes the existence of Markov kernels ( P M,x ) x ∈ R such that for any stoppingtime T , we have E M,x [ F T <ζ ] = E x [ F M T T <ζ ] , F ∈ b F T , x ∈ R . Because X is recurrent, we have P x ( ζ = ∞ ) = 1 for all x ∈ R but — as we shall seebelow — it is entirely possible that P M,x ( ζ = ∞ ) < x ∈ R .For a constant λ >
0, we recall that a continuous function u : R → (0 , ∞ )is called a λ -harmonic function if the process (cid:0) e − λt u ( X t ) (cid:1) t ≥ is a local martingale(strictly positive). In that case, the normalized process M t := e − λt u ( X t ) u ( X ) , t ≥ Lemma 3.1.
Suppose Assumption 2.1 holds and let u : R → (0 , ∞ ) be a λ -harmonicfunction for a constant λ > . Then:(i) There exist Markov kernels ( P u,x ) x ∈ R ∆ such that for any stopping time T E u,x [ F T <ζ ] = 1 u ( x ) E x (cid:2) F e − λT u ( X T )1 T <ζ (cid:3) , F ∈ b F T , x ∈ R . (3.1) (ii) ( P u,x ) x ∈ R ∆ is the law of a regular diffusion with values in R ∆ with a null killingmeasure, and scale function and speed measure given by s u ( z ) = Z z u ( ξ ) dξ, z ∈ R , m u ( dξ ) = u ( ξ ) m ( dξ ) , ξ ∈ R . (3.2) (iii) The mapping ( t, x ) P u,x ( ζ > t ) is jointly continuous on [0 , ∞ ) × R .Proof. (i) is a direct consequence of Theorem 62.19 in [39]. (ii) Let E be an inde-pendent exponentially distributed random variable with parameter λ >
0. Then, u is λ -excessive for X if and only if u is excessive for ˜ X defined by˜ X t := X t , t < E, ∆ , t ≥ E, (3.3)10or t ∈ [0 , ∞ ). Because we can write (3.1) as E u,x [ F T <ζ ] = 1 u ( x ) E x (cid:2) F u ( ˜ X T )1 T < ˜ ζ (cid:3) , where ˜ ζ := ζ ∧ E is the lifetime of ˜ X , the formulas in (3.2) follow from Theorem 8.3 inLanger and Schenk [33] where we note that the killing measure under P u,x is null since u is a λ -harmonic function. (iii) Define the process ˜ X t := s u ( X t ) for t ≥
0. Then,˜ X is a local martingale under P u,x with volatility coefficient s ′ u (cid:0) s − u ( ˜ X t ) (cid:1) σ (cid:0) s − u ( ˜ X t ) (cid:1) .Proposition 4.3 in Karatzas and Ruf [25] gives continuity of ( t, x ) P u,x ( ˜ ζ > t ) where˜ ζ := inf (cid:8) t > X t ∈ { s u ( −∞ ) , s u ( ∞ ) } (cid:9) . Then, (iii) follows because ˜ ζ = inf { t > X t = ∆ } = ζ . ♦ In the setting of the inverse 2D Bessel process, the next example gives the basicsolutions to the Sturm-Liouville ODE (2.2):
Example 3.2 (Continuation of Example 2.3) . Let ( X t ) t ≥ be the inverse 2 dimen-sional Bessel process (2.10). For a constant λ >
0, the corresponding basic solutionsof (2.2) are ϕ λ ↑ ( ξ ) := I ( e ξ √ λ ) , ξ ∈ R , lim ξ ↑∞ ϕ λ ↑ ( ξ ) ξ = ∞ ,ϕ λ ↓ ( ξ ) := K ( e ξ √ λ ) , ξ ∈ R , lim ξ ↓−∞ ϕ λ ↓ ( ξ ) ξ = − , (3.4)see, e.g., Jeanblanc, Yor, and Chesney ([22], p.279). The super linear growth limitin (3.4) (as ξ ↑ ∞ ) and Theorem 2.2 in Urusov and Zervos [40] ensure that the localmartingale (cid:0) e − λt ϕ λ ↑ ( X t ) (cid:1) t ≥ is a martingale. The linear growth limit in (3.4) (as ξ ↓ −∞ ) and Theorem 2.2 in [40] ensure that the local martingale (cid:0) e − λt ϕ λ ↓ ( X t ) (cid:1) t ≥ is a strict local martingale. ♦ The dichotomy between linear growth and strict local martingality exhibited inthe above 2D Bessel example holds in general because strict local martingality iscompletely determined by the growth behavior of the basic solutions ϕ λ ↓ ( ξ ) and ϕ λ ↑ ( ξ )of (2.2). In the next result, we recall that the speed measure m ( dξ ) corresponding to(1.1) is defined in (2.1). 11 heorem 3.3 (Urusov and Zervos [40]) . The following are equivalent:(i) R ∞ ξm ( dξ ) < ∞ (resp. R −∞ | ξ | m ( dξ ) < ∞ ),(ii) ϕ λ ↑ ( ξ ) (resp. ϕ λ ↓ ( ξ ) ) has linear growth at ξ = ∞ (resp. ξ = −∞ ) .(iii) The process (cid:0) e − λt ϕ λ ↑ ( X t ) (cid:1) t ≥ (resp. (cid:0) e − λt ϕ λ ↓ ( X t ) (cid:1) t ≥ ) is a strict local martin-gale under P x for all x ∈ R .Proof. First, the boundary point ∞ is inaccessible because Z ∞ Z y m ( dξ ) dy = ∞ . (3.5)Furthermore, the point ∞ is a natural or entrance boundary depending on whether Z ∞ m (( y, ∞ )) dy = Z ∞ Z ξ dym ( dξ )= Z ∞ ξm ( dξ ) (3.6)is infinite or not. Thus, the claimed equivalences follow from Theorem 2.2 in [40]. ♦ In the below Lemma 3.5, the limits (3.9) and (3.11) extend Corollary 6.1 in C¸ etin[6] to particular time inhomogeneous diffusions which we need in the next section.Furthermore, property (3.15) below has been proven in various settings including [27],[25], and [6]. The following Example 3.4 shows that the limits (3.9) and (3.11) do nothold for arbitrary strict local martingales.
Example 3.4.
Fix a probability measure P under which B = ( B t ) t ≥ is a P -Brownianmotion and denote by Y y = ( Y yt ) t ≥ the unique strong solution of (2.20) for initialvalues y ∈ (0 , ∞ ). Then, the process ˜ Y t := ˜ yY t for t ≥ y > ˜ y ↑∞ E P [ ˜ Y t ]˜ y = E P [ Y t ] ∈ (0 , , t ∈ (0 , ∞ ) . (3.7)Note that the dependence on the initial value ˜ y in the dynamics d ˜ Y t = − y ˜ Y t dB t implies that Corollary 6.1 in C¸ etin (2018) cannot be applied. ♦ emma 3.5. Suppose Assumption 2.1 holds and assume the unique strong solutionof (1.1) is integrable. Then, we have:(i) If R ∞ ξm ( dξ ) < ∞ , the strict local martingale Y t := e − λt ϕ λ ↑ ( X t ) , t ≥ , λ > , (3.8) has the asymptotic expectation lim n ↑∞ E n [ Y s n ] ϕ λ ↑ ( n ) = lim n ↑∞ E n [ Y s n ] n = 0 , ∞ > s n ≥ s n +1 , s ∞ := lim n ↑∞ s n > . (3.9) (ii) If R −∞ | ξ | m ( dξ ) < ∞ , the strict local martingale Y t := e − λt ϕ λ ↓ ( X t ) , t ≥ , λ > , (3.10) has the asymptotic expectation lim n ↑∞ E n [ Y s n ] ϕ λ ↑ ( − n ) = lim n ↑∞ E n [ Y s n ] n = 0 , ∞ > s n ≥ s n +1 , s ∞ := lim n ↑∞ s n > . (3.11) Proof. (i):
Step 1/3:
From Theorem 3.3, we know that (3.8) is a strict local mar-tingale and ϕ λ ↑ ( ξ ) is of linear growth as ξ ↑ ∞ . It follows from Lemma 3.1 that thereexist Markov kernels ( P ϕ,x ) x ∈ R ∆ such that (3.1) holds with u := ϕ λ ↑ . Moreover, s λ ↑ ( z ) := Z z ϕ λ ↑ ( ξ ) dξ, z ∈ R , m λ ↑ ( dξ ) := 2 ϕ λ ↑ ( ξ ) σ ( ξ ) dξ, ξ ∈ R , (3.12)is a scale function and speed measure for X under ( P ϕ,x ) x ∈ R ∆ . Since s λ ↑ ( −∞ ) = −∞ and s λ ↑ ( ∞ ) < ∞ , the limit X ∞ exists and P ϕ,x ( X ∞ = ∞ ) = 1 for all x ∈ R . Step 2/3:
Define the stopping times T n := inf { t > X t ≥ n } , n ∈ N , (3.13)and observe that T n increases to X ’s lifetime ζ under P ϕ,x as n ↑ ∞ . Moreover, P x ( T n < ∞ ) = 1 for all n ∈ N due to X ’s recurrence. Then, for t ∈ [0 , ∞ ), Lemma13.1 yields P ϕ,x ( t < T n ) = Y E x [ Y T n t For n ∈ N , let T n be defined in (3.13) and let t ∈ [0 , ∞ ). Then, we have P ϕ,x ( ζ > t ) = P ϕ,x ( ζ > t, T n ≤ t ) + P ϕ,x ( ζ > t, T n > t )= E ϕ,x h T n ≤ t P ϕ,X Tn ( ζ > t − u ) | u = T n i + P ϕ,x ( T n > t )= E ϕ,x h T n ≤ t w (cid:0) t − T n , n (cid:1)i + P ϕ,x ( T n > t ) , (3.17)where the second line follows from the strong Markov property of X under P ϕ,x inview of Lemma 3.1 (see, e.g., Exercise 6.12 in [39]).Let ( s n ) n ∈ N ⊂ (0 , ∞ ) be a non-increasing sequence with a positive limit s ∞ ∈ (0 , ∞ ). By replacing t with s n in (3.17), we can use dominated convergence whenpassing n ↑ ∞ in (3.17) to see0 = lim n ↑∞ E ϕ,x h T n ≤ s n w (cid:0) s n − T n , n (cid:1)i = E ϕ,x h ζ ≤ s ∞ lim n ↑∞ w (cid:0) s n − T n , n (cid:1)i . (3.18)Therefore, because w ≥ 0, we have1 ζ ≤ s ∞ lim n ↑∞ w (cid:0) s n − T n , n (cid:1) = 0 , P ϕ,x -a.s. (3.19)By using (3.15) and the strict local martingale property of (3.8) we see that14 ( t, x ) < t ∈ (0 , ∞ ) and x ∈ R . Therefore, because s ∞ ∈ (0 , ∞ ), we have P ϕ,x ( ζ ≤ s ∞ ) = 1 − w ( s ∞ , x ) > . (3.20)Consequently, the set ( ζ ≤ s ∞ ) is not P ϕ,x -null. Because s n is non-increasing withlimit s ∞ ∈ (0 , ∞ ) and T n ( ω ) is non-decreasing with limit ζ ( ω ) for ω ∈ ( ζ ≤ s ∞ ), wehave T n ( ω ) ≤ ζ ( ω ) ≤ s ∞ ≤ s n , ω ∈ ( ζ ≤ s ∞ ) . Thus, 1 ζ ≤ s ∞ lim n ↑∞ w (cid:0) s n − T n , n (cid:1) ≥ ζ ≤ s ∞ lim n ↑∞ w (cid:0) s n , n (cid:1) , (3.21)where the inequality uses that t → w ( t, x ) is non-increasing for all x ∈ R . Combining(3.19) and (3.21) yields lim n ↑∞ w (cid:0) s n − T n , n (cid:1) = 0 because the set ( ζ ≤ s ∞ ) is not P ϕ,x -null. Finally, using (3.15), we arrive atlim n ↑∞ w ( s n , n ) = lim n ↑∞ ϕ λ ↑ ( n ) E n [ Y s n ] = 0 . (ii): This proof is similar to the proof of (i) and is omitted. ♦ Proof of Theorem 2.2(i): Step 1/2: In this first step, we consider H positive; that is, H : R → [0 , ∞ ). First,we prove that h defined in (1.2) is a classical solution of (1.3) bounded by K Φ λ andsatisfies (2.5). Second, we prove h is the only such function.Because H satisfies (2.4), we can find two positive constants ( r , r ) such that H ( x ) ≤ r + r Φ λ ( x ) for all x ∈ R . Therefore, we have the upper bound E x [ H ( X t )] ≤ r + r E x [Φ λ ( X t )] ≤ r + re λT Φ λ ( x ) , (3.22)where the last inequality follows from (cid:0) e − λt Φ λ ( X t ) (cid:1) t ≥ being a P x -supermartingale.The second inequality (3.22) and the fact that Φ λ ( x ) > h ( t, x ) := h ( t, x )Φ λ ( x ) , t ≥ , x ∈ R , (3.23)is uniformly bounded.The property R ∞ ξm ( dξ ) < ∞ and Theorem 3.3 ensure that ϕ λ ↑ ( ξ ) satisfies c := lim sup y ↑∞ ϕ λ ↑ ( y ) y ∈ (0 , ∞ ) . (3.24)Then, for a non-increasing sequence ( s n ) n ∈ N ⊂ (0 , ∞ ) with a positive and finite limit s ∞ := lim n ↑∞ s n ∈ (0 , ∞ ), the limit (3.9) in Lemma 3.5(i) gives us the limit in (2.5)because lim n ↑∞ h ( s n , n ) n ≤ lim n ↑∞ r + r E n [Φ λ ( X s n )] ϕ λ ↑ ( n ) ϕ λ ↑ ( n ) n ≤ lim n ↑∞ r + re λs n ϕ λ ↓ ( n ) + r E n [ ϕ λ ↑ ( X s n )] ϕ λ ↑ ( n ) ϕ λ ↑ ( n ) n ≤ c lim n ↑∞ r ′ + r E n [ ϕ λ ↑ ( X s n )] ϕ λ ↑ ( n )= 0 , (3.25)where the constant c is from (3.24) and r ′ is some positive irrelevant constant.To see that the PDE (1.3) holds, we change coordinates. To this end, first assumethat H satisfies H ( ξ ) ≤ r + rϕ λ ↑ ( ξ ) , ξ ∈ R , (3.26)for positive constants ( r , r ). Then, the continous function F ( y ) := H (cid:0) ϕ − λ ↑ ( y ) (cid:1) , y > y := lim ξ ↓−∞ ϕ λ ↑ ( ξ ) , (3.27)is of at most linear growth and satisfies lim y ↓ y F ( y ) < ∞ . Furthermore, for a fixedconstant T ∈ (0 , ∞ ), we define the process Y t := e λ ( T − t ) ϕ λ ↑ ( X t ) , t ∈ [0 , T ] , λ > , (3.28)16ith the local martingale dynamics dY t = e λ ( T − t ) ϕ ′ λ ↑ ( X t ) σ ( X t ) dB t = e λ ( T − t ) ϕ ′ λ ↑ (cid:0) e − λ ( T − t ) ϕ − λ ↑ ( Y t ) (cid:1) σ (cid:0) e − λ ( T − t ) ϕ − λ ↑ ( Y t ) (cid:1) dB t = α ( t, Y t ) dB t , (3.29)where we have defined the volatility function α ( t, y ) := e λ ( T − t ) ϕ ′ λ ↑ (cid:0) e − λ ( T − t ) ϕ − λ ↑ ( y ) (cid:1) σ (cid:0) e − λ ( T − t ) ϕ − λ ↑ ( y ) (cid:1) , y > y, t ≥ . (3.30)Theorem 3.2 in Ekstr¨om and Tysk [11] guarantees that the function f ( t, y ) := ¯ E t,y [ F ( Y T )] , t ∈ [0 , T ] , y > y, (3.31)where ¯ E t,y denotes the expectation with respect to the law of ( Y u ) u ∈ [ t,T ] conditionalon Y t = y , is a classical solution of the Cauchy problem f ( T, y ) = F ( y ) , y > y, f t + α f yy , y > y, t ∈ (0 , T ) . (3.32)Then, the function h ( t, x ) from (1.2) satisfies the PDE (1.3) because ϕ λ ↑ ( ξ ) solves theSturm-Liouville ODE (2.2) and we have the relation f (cid:0) t, y (cid:1) = ¯ E t,y [ F ( Y T )]= E y ◦ ( t ) [ H ( X T − t )]= h (cid:0) T − t, ϕ − λ ↑ ( e − λ ( T − t ) y ) (cid:1) , (3.33)where we have defined y ◦ ( t ) := ϕ − λ ↑ ( e − λ ( T − t ) y ) for t ∈ [0 , T ] and y > y .Second, a similar argument but replacing (3.26) with H ( ξ ) ≤ r + rϕ λ ↓ ( ξ ) , ξ ∈ R , (3.34)and replacing (3.28) with Y t := e λ ( T − t ) ϕ λ ↓ ( X t ) when changing coordinates, shows that h ( t, x ) from (1.2) satisfies the PDE (1.3) again.17hird, by writing H ( ξ ) = H ( ξ ) + H ( ξ ) − H (0) , H ( ξ ) := H ( ξ ∨ , H ( ξ ) := H ( ξ ∧ , (3.35)and noting that when H satisfies (2.4), H satisfies (3.26) and H satisfies (3.34).Then, for i ∈ { , } , the functions h i ( t, x ) : = E x [ H i ( X t )] , t ≥ , x ∈ R , (3.36)satisfy the PDEs h i (0 , x ) = H i ( x ) , x ∈ R ,h it ( t, x ) = σ ( x ) h ixx ( t, x ) , t > , x ∈ R . (3.37)Therefore, h ( t, x ) := h ( t, x ) + h ( t, x ) − H (0) is the function in (1.2) and by usingthe PDEs in (3.37), we see that h satisfies (1.3).Next, we turn to uniqueness. Let h satisfy the PDE (1.3) as well as the conditionsof the theorem. For a fixed constant T ∈ [0 , ∞ ), we establish the uniqueness claimby proving the following representation E Φ ,x (cid:20) H ( X T ) e − λT Φ λ ( X T ) 1 T <ζ (cid:21) = ˜ h ( T, x ) , x ∈ R , (3.38)where ˜ h is defined in (3.23) and the Markov kernels ( P Φ ,x ) x ∈ R ∆ are from Lemma 3.1with u := Φ λ where Φ λ is defined in (2.3). Similarly to (3.12), the scale functionassociated with the diffusion X under P Φ ,x is given by s Φ ( z ) := Z z λ ( ξ ) dξ, z ∈ R . (3.39)Because lim z ↓−∞ s Φ ( z ) > −∞ and lim z ↑∞ s Φ ( z ) < ∞ , the limit of X satisfies X ∞ ∈{−∞ , ∞} , P Φ ,x -a.s., for x ∈ R . We also introduce the stopping times ν n := inf { t > | X t | ≥ n } , n ∈ N , (3.40)and observe that lim n ↑∞ ν n = ζ , P Φ ,x -a.s., for all x ∈ R where we recall that ζ denotes X ’s lifetime. 18o see that (3.38) holds, observe that the boundedness of ˜ h ensures that theprocess (cid:0) e λt ∧ ν n ˜ h ( T − t ∧ ν n , X t ∧ ν n ) (cid:1) t ∈ [0 ,T ] is a P Φ ,x -martingale for all n ∈ N . This givesus ˜ h ( T, x ) = E Φ ,x (cid:2) e λT ∧ ν n ˜ h ( T − T ∧ ν n , X T ∧ ν n ) (cid:3) = E Φ ,x (cid:2) e λT ˜ h (0 , X T )1 T <ν n (cid:3) + E Φ ,x (cid:2) e λν n ˜ h ( T − ν n , X ν n )1 T ≥ ν n (cid:3) . (3.41)Because ˜ h is uniformly bounded, we can use dominated convergence in (3.41) whenpassing n ↑ ∞ to see that˜ h ( T, x ) = E Φ ,x (cid:2) e λT ˜ h (0 , X T )1 T <ζ (cid:3) + lim n ↑∞ E Φ ,x (cid:2) e λν n ˜ h ( T − ν n , X ν n )1 T ≥ ν n (cid:3) = E Φ ,x (cid:20) e λT H ( X T )Φ λ ( X T ) 1 T <ζ (cid:21) + E Φ ,x (cid:2) lim n ↑∞ e λν n ˜ h ( T − ν n , X ν n )1 T ≥ ν n (cid:3) , (3.42)where the second equality uses the initial condition ˜ h (0 , x ) = H ( x )Φ λ ( x ) . Therefore, therepresentation in (3.38) follows as soon as we showlim n ↑∞ ˜ h ( T − ν n , X ν n )1 T ≥ ν n = 0 , P Φ ,x -a.s. (3.43)First, on the set ( T ≥ ν n ), we have X ν n ∈ {− n, n } . Therefore, ϕ λ ↓ ( x ) = lim n ↑∞ E x (cid:2) e − λT ∧ ν n ϕ λ ↓ ( X T ∧ ν n )(1 T ≥ ν n + 1 T <ν n ) (cid:3) ≥ lim n ↑∞ (cid:16) ϕ λ ↓ ( − n ) E x [ e − λT ∧ ν n T ≥ ν n X νn = − n ] + E x (cid:2) e − λT ∧ ν n ϕ λ ↓ ( X T ∧ ν n )1 T <ν n (cid:3)(cid:17) = lim n ↑∞ (cid:16) ϕ λ ↓ ( − n )Φ λ ( − n ) E x (cid:2) e − λT ∧ ν n Φ λ ( X T ∧ ν n )1 T ≥ ν n X νn = − n (cid:3) + E x (cid:2) e − λT ϕ λ ↓ ( X T )1 T <ν n (cid:3)(cid:17) = Φ λ ( x ) lim n ↑∞ ϕ λ ↓ ( − n )Φ λ ( − n ) P Φ ,x ( T ≥ ν n , X ν n = − n ) + E x (cid:2) e − λT ϕ λ ↓ ( X T ) (cid:3) = Φ λ ( x ) lim n ↑∞ P Φ ,x ( T ≥ ν n , X ν n = − n ) + ϕ λ ↓ ( x ) . (3.44)The last equality uses the martingale property in Theorem 3.3 and the propertylim n ↑∞ ϕ λ ↓ ( − n )Φ λ ( − n ) = 1 (this limit holds because lim n ↑∞ ϕ λ ↑ ( − n ) ∈ [0 , ∞ )). Therefore,19ecause ˜ h is bounded, we havelim n ↑∞ ˜ h ( T − ν n , − n )1 T ≥ ν n X νn = − n = 0 , P Φ ,x -a.s. (3.45)Second, because the set ( ζ = T ) is P Φ ,x -null by Lemma 3.1(iii), the sets ( T ≥ ζ )and ( T > ζ ) differ only by a P Φ ,x -null set. Then, we can use the boundary condition(2.5) and the linear growth of ϕ λ ↑ ( ξ ) and Φ λ ( ξ ) as ξ ↑ ∞ to seelim n ↑∞ ˜ h ( T − ν n , n )1 T ≥ ν n X νn = n = 1 T ≥ ζ X ζ = ∞ lim n ↑∞ ˜ h ( T − ν n , n )= 1 T >ζ X ζ = ∞ lim n ↑∞ ˜ h ( T − ν n , n )= 0 , (3.46) P Φ ,x -a.s. The two observations (3.45) and (3.46) establish (3.43). Step 2/2: We consider H : R → R and write H ( ξ ) = H + ( ξ ) − H − ( ξ ) where H + , H − : R → [0 , ∞ ) are defined by H + ( ξ ) := H ( ξ ) ∨ H − ( ξ ) := − (cid:0) H ( ξ ) ∧ (cid:1) for ξ ∈ R . 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