ERRATUM: Stochastic evolution equations for large portfolios of stochastic volatility models
aa r X i v : . [ m a t h . P R ] M a y ERRATUM: Stochastic evolution equations for largeportfolios of stochastic volatility models
Ben Hambly ∗ and Nikolaos Kolliopoulos † Mathematical Institute, University of OxfordMay 14, 2019
Abstract
In the article ”Stochastic evolution equations for large portfolios of StochasticVolatility models” ([3], ArXiv ID: 1701.05640) there is a mistake in the proof ofTheorem 3.1. In this erratum we establish a weaker version of this Theorem andthen we redevelop the regularity theory for our problem accordingly. This meansthat most of our regularity results are replaced by slightly weaker ones. We alsoclarify a point in the proof of a correct result.
We will first present the correct results replacing the incorrect ones in a structured way,and then give the proofs. To do this, we require a stronger assumption on the parametersof the CIR volatility process given by (3 . kθξ > .However we now need to impose the stronger condition that kθξ > x ∗ ≈ . x ∗ is the largest root of the equation16 x − x + 24 x − , (E0.1)for our results to hold. We will also clarify an argument in the (correct) proof of Theorem4.1 from the original article in the appendix.Sections and new results/equations in this erratum will be indexed by numbers pre-ceded by the letter “E”. On the other hand, we will refer to everything else as if we werein the original article. E1 The corrected main results
The proof of Theorem 3.1 contains a fatal mistake. We replace the incorrect Theorem 3.1by the the following:
Theorem E1.1.
Suppose that σ is a positive random variable which is bounded awayfrom zero and infinity. Then P - almost surely the conditional probability measure P ( σ t ∈· | B · , G ) possesses a continuous density p t ( · | B · , G ) which is supported in [0 , ∞ ) , for all t > . Moreover, for any T > , any α ≥ and all sufficiently small q > , we have thefollowing integrability condition M αB · , G ( · ) := sup y ≥ (cid:0) y α p · ( y | B · , G ) (cid:1) ∈ L q (Ω × [0 , T ]) ∗ [email protected] † [email protected] (corresponding author) v t,C needs to bereestablished. We state the results in a slightly different way. The two-dimensionaldensity of the above measure-valued process will belong to the following spaces L α = L (cid:16) (Ω , F , P ) × [0 , T ] ; L | y | α (cid:0) R + × R (cid:1)(cid:17) and H α = L (cid:16) (Ω , F , P ) × [0 , T ] ; H ,w ( x ) (cid:0) R + (cid:1) × L | y | α ( R ) (cid:17) for α ≥ w ( x ) = min { , √ x } for x ≥
0, where we write L g ( y ) for the weighted L space with weight function { g ( y ) : y ∈ R } , and H ,g ( x ) ( R + ) for the weighted H ( R + )space with weight function { g ( x ) : x ≥ } in the L norm of the derivative. Apart fromthe integrability conditions, a function u ′ belonging to the second space has to satisfy theboundary condition lim x → + (cid:13)(cid:13) u ′ ( · , x, · ) (cid:13)(cid:13) L | y | α (Ω × [0 , T ] × R ) = 0. Observe that this definition isnot problematic, since k u ′ ( · , x, · ) k L | y | α (Ω × [0 , T ] × R ) has to be continuous in x for x > x = 0), so changing the value of theabove limit gives a different function in an L (Ω × R + × R + × R ) sense. The existenceof a density for v t,C and its regularity are given in the next Theorem, which replacesTheorem 4.3. Theorem E1.2.
Suppose that h is a continuous function taking values in some compactsubset of R + . Suppose also that given G , X has an L -integrable density u ( ·|G ) in R + such that E h k u k L ( R + ) | G i ∈ L q ′ (Ω) for any q ′ > . Suppose finally that k θ ξ > x ∗ and ρ , ∈ ( − , hold for any possible realization of C = ( k , θ , ξ , r , ρ , , ρ , ) ,and that the random variable σ is positive and bounded away from zero and infinity.Then, for any possible realization of C , the measure-valued stochastic process v t,C hasa two-dimensional density u C ( t, · , W · , B · , G ) belonging to the space L α for any α ≥ .Moreover, when ρ := R dW t dB t = 0 and E (cid:20) k u k H ,w x ) ( R + ) | G (cid:21) ∈ L q ′ (Ω) for any q ′ > , the density belongs to H α as well for any α ≥ . Next, we obtain our SPDE exactly as in [3], and we adapt the definition of our initial-boundary value problem to the new regularity results given in the above theorem. Forthis purpose we define the space ˜ L α,w := L y α w ( x ) ( R + × R + ) for any α ≥
0, and then wemodify Definition 5.1 ( α -solution to our problem) as follows Definition E1.3.
For a given real number ρ and a given value of the coefficient vector C , let h : R + −→ R + be a function having polynomial growth, and U be a randomfunction which is extended to be zero outside R + such that U ∈ L (cid:16) Ω; ˜ L α (cid:17) and ( U ) x ∈ L (cid:16) Ω; ˜ L α,w (cid:17) for some α >
0. Given C , ρ , α and the functions h and U , we say that u is an α -solution to our problem when the following are satisfied;1. u is adapted to the filtration { σ (cid:0) G , W t , B t (cid:1) : t ≥ } and belongs to the space H α ∩ L α . 2. u is supported in R + and satisfies the SPDE u ( t, x, y ) = U ( x, y ) − r Z t ( u ( s, x, y )) x ds + 12 Z t h ( y ) ( u ( s, x, y )) x ds − k θ Z t ( u ( s, x, y )) y ds + k Z t ( yu ( s, x, y )) y ds + 12 Z t h ( y ) ( u ( s, x, y )) xx ds + ρ Z t ( h ( y ) √ yu ( s, x, y )) xy ds + ξ Z t ( yu ( s, x, y )) yy ds − ρ , Z t h ( y ) ( u ( s, x, y )) x dW s − ξ ρ , Z t ( √ yu ( s, x, y )) y dB s , (E1.1)for all x, y ∈ R + , where u y , u yy and u xx are considered in the distributional senseover the space of test functions C test = { g ∈ C b ( R + × R ) : g (0 , y ) = 0 , ∀ y ∈ R } . The SPDE of the above definition is satisfied by the density u C for ρ = ξ ρ ρ , ρ , ,where ρ is the correlation between W and B (i.e dW t · dB t = ρ dt ), while the regularityproperties are also satisfied for all α > ρ = 0.Finally we replace Theorem 5.2, which improves the regularity of our two-dimensionaldensity through the initial-boundary value problem, by the following theorem which dif-fers only in the weighted L norm used. Theorem E1.4.
Fix the value of the coefficient vector C , the function h , the real number ρ and the initial data function U . Let u be an α -solution to our problem for all α ≥ .Then, the weak derivative u y of u exists and we have u y ∈ L (cid:16) [0 , T ] × Ω; ˜ L α,w (cid:17) for all α ≥ . E2 The main lemmas needed
To prove Theorem E1.1 which replaces the incorrect Theorem 3.1 from [3], instead ofLemma 3.5 we need the following stronger result, which contains a generalization ofProposition 2.1.1 from page 78 in [7].
Lemma E2.1.
Let B be a Brownian motion defined on [0 , T ] × Ω for some T > andsome probability space (Ω , F , P ) , and let F be a random variable which is adapted to theBrownian motion B . Suppose also that for some q, ˜ q, r, ˜ r, λ, ˜ λ > and α ≥ , with q ≤ and q + q = r + r = q ˜ r ˜ λ + q ˜ rλ = 1 , we have F ∈ L αλ ∨ ˜ q (Ω) ∩ D , λ ∨ q ˜ rq ˜ r − (Ω) ∩ D ,qr (Ω) and also | F | m k D · F k L ,T ]) ∈ L q ˜ r (Ω) for m ∈ { , α } . Then, the domain of the adjoint of the derivative perator D : L ˜ q (Ω) ∩ D , ∨ q ˜ rq ˜ r − (Ω) −→ L ∨ q ˜ rq ˜ r − (cid:0) Ω; L ([0 , T ]) (cid:1) (which is an extensionof the standard Skorokhood integral δ ) contains the process D · F k D · F k L ,T ]) . Moreover, F possesses a bounded and continuous density f F for which we have the estimate sup x ∈ R | x | α f F ( x ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | F | α δ D · F k D · F k L ([0 ,T ]) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) ≤ ( C + 2) E qr (cid:20)(cid:13)(cid:13) D · , · F (cid:13)(cid:13) qrL ( [0 ,T ] ) (cid:21) × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | α k D · F k L ([0 ,T ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C E qr h k D · F k qrL ([0 ,T ]) i × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | α k D · F k L ([0 ,T ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r , (E2.1) for some C > , with the RHS of the above being finite by our assumptions. Next, to prove Theorem E1.2 which replaces Theorem 4.3 from [3], the estimate givenin Theorem 4.1 is not enough. In particular, we need a stronger estimate for the derivativealong with a maximum principle. These are given in the following lemma.
Lemma E2.2.
Let u be the density obtained in Theorem 4.1. For some M > depend-ing only on r and on some compact interval I ⊂ R + containing the minimum and themaximum of σ · , we have the estimate E " sup ≤ t ≤ T k w ( · ) u x ( t, · ) k L ( R + ) ≤ M e MT E h k w ( · ) ( u ) x ( · ) k L ( R + ) i + M e MT E h k u ( · ) k L ( R + ) i (E2.2) where w ( x ) = min { , √ x } for all x ≥ , provided that the RHS is finite. Then, for some M ′ > depending on M and the initial data, we have the maximum principle E " sup ≤ t ≤ T sup x ∈ R + u ( t, x ) ≤ M ′ (E2.3)Finally, the proof of Theorem E1.4 is almost identical to the proof of the correspondingTheorem 5.2 in [3]. The only difference is that the δ -identity contains two extra non-derivative terms, while the weight w ( x ) is introduced to the norms and inner productsinvolved in all the other terms. This is not a problem because the two extra terms are Z t (cid:13)(cid:13) I [0 , × R ( · ) I ǫ, ( s, · ) (cid:13)(cid:13) L ( Ω ; ˜ L δ ) ds (E2.4)and Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , I [0 , × R ( · ) I ǫ, ( s, · ) (cid:29) L ( Ω ; ˜ L δ ) ds (E2.5)4hich do not explode as ǫ → + (by Lemma 5.3 and our regularity assumptions), while forall the other terms we use Lemmas 5.3 and 5.4 for a slightly differently weighted measure µ which gives the weight w ( x ) to the norms and inner products involved. Therefore, weonly need to prove the corrected δ -identity which is stated below. Lemma E2.3 ( the δ -identity ) . The following identity holds for any δ > k I ǫ, ( t, · ) k L ( Ω ; ˜ L δ,w ) = (cid:13)(cid:13)(cid:13)(cid:13)Z D U ( · , z ) φ ǫ ( z, · ) dz (cid:13)(cid:13)(cid:13)(cid:13) L ( Ω ; ˜ L δ,w )+ r Z t (cid:13)(cid:13) I [0 , × R ( · ) I ǫ, ( s, · ) (cid:13)(cid:13) L ( Ω ; ˜ L δ ) ds + Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , I ǫ, ( s, · ) (cid:29) L ( Ω ; ˜ L δ,w ) ds + δ (cid:18) k θ − ξ (cid:19) Z t D I ǫ,z − ( s, · ) , I ǫ, ( s, · ) E L ( Ω ; ˜ L δ − ,w ) ds + (cid:18) k θ − ξ (cid:19) Z t (cid:28) I ǫ,z − ( s, · ) , ∂∂y I ǫ, ( s, · ) (cid:29) L ( Ω ; ˜ L δ,w ) ds − δk Z t D I ǫ,z ( s, · ) , I ǫ, ( s, · ) E L ( Ω ; ˜ L δ − ,w ) ds − k Z t (cid:28) I ǫ,z ( s, · ) , ∂∂y I ǫ, ( s, · ) (cid:29) L ( Ω ; ˜ L δ,w ) ds − Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , ∂∂x I ǫ, ( s, · ) (cid:29) L ( Ω ; ˜ L δ,w ) ds − Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , I [0 , × R ( · ) I ǫ, ( s, · ) (cid:29) L ( Ω ; ˜ L δ ) ds − δρ Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , I ǫ, ( s, · ) (cid:29) L ( Ω ; ˜ L δ − ,w ) ds + ρ , Z t (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x I ǫ,h ( z ) ( s, · ) (cid:13)(cid:13)(cid:13)(cid:13) L ( Ω ; ˜ L δ,w ) ds + δ ( δ − ξ Z t k I ǫ, ( s, · ) k L ( Ω ; ˜ L δ − ,w ) ds − ξ (cid:0) − ρ , (cid:1) Z t (cid:13)(cid:13)(cid:13)(cid:13) ∂∂y I ǫ, ( s, · ) (cid:13)(cid:13)(cid:13)(cid:13) L ( Ω ; ˜ L δ,w ) ds. − ( ρ − ξ ρ ρ , ρ , ) × Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , ∂∂y I ǫ, ( s, · ) (cid:29) L ( Ω ; ˜ L δ,w ) ds. (E2.6) All the terms in the above identity are finite. Proof of Lemma E2.1 . Let { F n : n ∈ N } be a sequence of regular enough (in terms ofMalliavin differentiability) random variables which are adapted to the Brownian motion B · in [0 , T ], with F n −→ F in L αλ ∨ ˜ q (Ω) ∩ D , λ ∨ q ˜ rq ˜ r − (Ω) ∩ D ,qr (Ω). Then, D · F n k D · F n k L ,T ]) + ǫ belongs to the domain of the standard Skorokhod integral δ for any ǫ >
0, and for any m ≤ α , by a well-known property of δ (see property (4) on page 40 in [7]) we have thefollowing relationship, | F | m δ D · F n k D · F n k L ([0 ,T ]) + ǫ ! = | F | m δ ( D · F n ) k D · F n k L ([0 ,T ]) + ǫ + | F | m Z T D s F n D s k D · F n k L ([0 ,T ]) + ǫ ! ds = | F | m δ ( D · F n ) k D · F n k L ([0 ,T ]) + ǫ + | F | m Z T D s F n − R T D s ′ F n · D s ′ ,s F n ds ′ (cid:16) k D · F n k L ([0 ,T ]) + ǫ (cid:17) ds = | F | m δ ( D · F n ) k D · F n k L ([0 ,T ]) + ǫ − | F | m R T R T D s F n · D s ′ F n · D s ′ ,s F n ds ′ ds (cid:16) k D · F n k L ([0 ,T ]) + ǫ (cid:17) . (E3.1)Thus, by the triangle inequality, a boundedness property of the operator δ (see Proposition1.5.4 on page 69 in [7]) and H¨older’s inequality, we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | F | m δ D · F n k D · F n k L ([0 ,T ]) + ǫ !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) ≤ E q "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m δ ( D · F n ) k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q +2 E q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m R T R T D s F n · D s ′ F n · D s ′ ,s F n ds ′ ds (cid:16) k D · F n k L ([0 ,T ]) + ǫ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ≤ E qr [ | δ ( D · F n ) | qr ] × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r +2 E q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m R T R T D s F n · D s ′ F n · D s ′ ,s F n ds ′ ds (cid:16) k D · F n k L ([0 ,T ]) + ǫ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ≤ C qr E qr "(cid:18)Z T Z T (cid:12)(cid:12) D s ′ ,s F n (cid:12)(cid:12) ds ′ ds (cid:19) qr × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C qr E qr "(cid:18)Z T | D s ′ F n | ds ′ (cid:19) qr × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r E q | F | m k D · F n k L ([0 ,T ]) (cid:18)R T R T (cid:16) D s ′ ,s F n (cid:17) ds ′ ds (cid:19) (cid:16) k D · F n k L ([0 ,T ]) + ǫ (cid:17) q ≤ C qr E qr "(cid:18)Z T Z T (cid:12)(cid:12) D s ′ ,s F n (cid:12)(cid:12) ds ′ ds (cid:19) qr × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C qr E qr "(cid:18)Z T | D s ′ F n | ds ′ (cid:19) qr × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r +2 E q | F | m (cid:18)R T R T (cid:16) D s ′ ,s F n (cid:17) ds ′ ds (cid:19) k D · F n k L ([0 ,T ]) + ǫ q ≤ ( C qr + 2) E qr "(cid:18)Z T Z T (cid:12)(cid:12) D s ′ ,s F n (cid:12)(cid:12) ds ′ ds (cid:19) qr × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C qr E qr "(cid:18)Z T | D s ′ F n | ds ′ (cid:19) qr × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r = ( C qr + 2) E qr (cid:20)(cid:13)(cid:13) D · , · F n (cid:13)(cid:13) qrL ( [0 ,T ] ) (cid:21) × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C qr E qr h k D · F n k qrL ([0 ,T ]) i × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r , (E3.2)for r, ˜ r > r + r = 1. Then, for a fixed ǫ >
0, we can use the Lipschitz conti-nuity of ǫ + x , H¨older’s inequality and our assumptions, to show that the last expressionconverges as n −→ + ∞ to the finite quantity( C qr + 2) E qr (cid:20)(cid:13)(cid:13) D · , · F (cid:13)(cid:13) qrL ( [0 ,T ] ) (cid:21) × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C qr E qr h k D · F k qrL ([0 ,T ]) i × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r , (E3.3)which implies that for a sequence { k n : n ∈ N } ⊂ N we have also | F | m δ D · F k n k D · F k n k L ([0 ,T ]) + ǫ ! −→ δ mF,ǫ (E3.4)7eakly in L q (Ω) as n −→ + ∞ , for some δ mF,ǫ . Moreover, when m = 0, for any ˜ F ∈ L ˜ q (Ω) ∩ D , ∨ q ˜ rq ˜ r − (Ω) we have E h ˜ F δ F,ǫ i = lim n → + ∞ E " ˜ F δ D · F k n k D · F k n k L ([0 ,T ]) + ǫ ! = lim n → + ∞ E "Z T D s ˜ F D s F k n k D · F k n k L ([0 ,T ]) + ǫ ds = E "Z T D s ˜ F D s F k D · F k L ([0 ,T ]) + ǫ ds . (E3.5)To see this we observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E "Z T D s ˜ F D s F k n k D · F k n k L ([0 ,T ]) + ǫ − D s F k D · F k L ([0 ,T ]) + ǫ ! ds ≤ E " k D · F k n k L ([0 ,T ]) + ǫ ! Z T (cid:12)(cid:12)(cid:12) D s ˜ F (cid:12)(cid:12)(cid:12) | D s F − D s F k n | ds + E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k D · F k n k L ([0 ,T ]) + ǫ − k D · F k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z T D s ˜ F D s F ds ≤ ǫ E − q ˜ r (cid:20)(cid:13)(cid:13)(cid:13) D · ˜ F (cid:13)(cid:13)(cid:13) q ˜ rq ˜ r − L ([0 ,T ]) (cid:21) E q ˜ r h k D s F − D s F k n k q ˜ rL ([0 ,T ]) i + E (cid:12)(cid:12)(cid:12) k D · F k n k L ([0 ,T ]) − k D · F k L ([0 ,T ]) (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) D · ˜ F (cid:13)(cid:13)(cid:13) L ([0 ,T ]) k D · F k L ([0 ,T ]) (cid:16) k D · F k n k L ([0 ,T ]) + ǫ (cid:17) (cid:16) k D · F k L ([0 ,T ]) + ǫ (cid:17) ≤ ǫ E − q ˜ r (cid:20)(cid:13)(cid:13)(cid:13) D · ˜ F (cid:13)(cid:13)(cid:13) q ˜ rq ˜ r − L ([0 ,T ]) (cid:21) E q ˜ r h k D s F − D s F k n k q ˜ rL ([0 ,T ]) i + C ǫ E (cid:20)(cid:12)(cid:12)(cid:12) k D · F k n k L ([0 ,T ]) − k D · F k L ([0 ,T ]) (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) D · ˜ F (cid:13)(cid:13)(cid:13) L ([0 ,T ]) (cid:21) ≤ (cid:18) ǫ + C ǫ (cid:19) E − q ˜ r (cid:20)(cid:13)(cid:13)(cid:13) D · ˜ F (cid:13)(cid:13)(cid:13) q ˜ rq ˜ r − L ([0 ,T ]) (cid:21) E q ˜ r h k D s F − D s F k n k q ˜ rL ([0 ,T ]) i (E3.6)for some C ǫ >
0, which converges to zero as n → ∞ . Furthermore, by a density argument,we can easily show that there is a unique weak limit δ F,ǫ and, since the subsequence { k n : n ∈ N } can be taken to be the same for both m = 0 and m = α , we can also showthat δ αF,ǫ = | F | α δ F,ǫ . Hence, we can define δ (cid:18) D · F k D · F k L ,T ]) + ǫ (cid:19) := δ F,ǫ and then have δ αF,ǫ = | F | α δ (cid:18) D · F k D · F k L ,T ]) + ǫ (cid:19) as well. Thus, we can also use Fatou’s lemma to estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | F | m δ D · F k D · F k L ([0 ,T ]) + ǫ !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) lim inf n → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | F | m δ D · F k n k D · F k n k L ([0 ,T ]) + ǫ !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) ≤ ( C qr + 2) lim inf n → + ∞ E qr (cid:20)(cid:13)(cid:13) D · , · F k n (cid:13)(cid:13) qrL ( [0 ,T ] ) (cid:21) × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F k n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C qr lim inf n → + ∞ E qr h k D · F k n k qrL ([0 ,T ]) i × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F k n k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r = ( C qr + 2) E qr (cid:20)(cid:13)(cid:13) D · , · F (cid:13)(cid:13) qrL ( [0 ,T ] ) (cid:21) × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C qr E qr h k D · F k qrL ([0 ,T ]) i × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F k L ([0 ,T ]) + ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r (E3.7)for both m = 0 and m = α . This means that we can take ǫ ↓ δ (cid:18) D · F k D · F k L ,T ]) (cid:19) can be defined such that E " ˜ F δ D · F k D · F k L ([0 ,T ]) ! = E "Z T D t ˜ F D t F k D · F k L ([0 ,T ]) dt (E3.8)for any ˜ F ∈ L ˜ q (Ω) ∩ D , ∨ q ˜ rq ˜ r − (Ω) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | F | m δ D · F k D · F k L ([0 ,T ]) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) ≤ ( C qr + 2) E qr (cid:20)(cid:13)(cid:13) D · , · F (cid:13)(cid:13) qrL ( [0 ,T ] ) (cid:21) × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F k L ([0 ,T ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + C qr E qr h k D · F k qrL ([0 ,T ]) i × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F | m k D · F k L ([0 ,T ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r . (E3.9)for both m = 0 and m = α . The finiteness of the RHS in both (E3.8) and (E3.9), whichallows us to obtain these relations by using the Monotone Convergence Theorem, followseasily from the assumed regularity of F and ˜ F . Especially for the finiteness of the RHSin (E3.8), we need to apply first the Cauchy-Schwartz inequality in L ([0 , T ]) for the twoMalliavin derivatives, and then, after an obvious cancellation, apply H¨older’s inequalitywith the appropriate exponents to control the RHS by the product of two finite norms.9aking now ψ ( y ) = I [ a, b ] ( y ) for some a, b ∈ R with a < b and φ ( y ) = R y −∞ ψ ( z ) dz , wecan easily show that P - almost surely we have | φ ( F ) | ≤ b − a , and also D · φ ( F ) = ψ ( F ) D · F by the comment after the proof of Proposition 1.2.3 on page 31 in [7] (since by ourassumptions D · F can never be identically zero, we can use Theorem 2.1.2 from page 86 in[7] to obtain absolute continuity). Thus, by the boundedness of ψ ( F ) and our assumptionswe have φ ( F ) ∈ L ∞ (Ω) ∩ D , λ ∨ q ˜ rq ˜ r − (Ω), which is a subspace of L ˜ q (Ω) ∩ D , ∨ q ˜ rq ˜ r − (Ω).Then, we can work as in the proof of Proposition 2.1.1 on page 78 in [7] to deduce that E [ ψ ( F )] = E " φ ( F ) δ D · F k D · F k L ([0 ,T ]) ! , (E3.10)where now δ is the adjoint of the derivative operator D : L ˜ q (Ω) ∩ D , ∨ q ˜ rq ˜ r − (Ω) −→ L ∨ q ˜ rq ˜ r − (cid:0) Ω; L ([0 , T ]) (cid:1) (E3.11)(so an extension of the standard Skorokhod integral) the domain of which contains theprocess D · F k D · F k L ,T ]) as we have shown above. Since H¨older’s inequality implies that E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I F >x δ D · F k D · F k L ([0 ,T ]) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k I F >x k L ˜ q (Ω) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) δ D · F k D · F k L ([0 ,T ]) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) δ D · F k D · F k L ([0 ,T ]) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q (Ω) (E3.12)which is finite, applying Fubini’s Theorem on (E3.10) we find that P ( a ≤ F ≤ b ) = Z ba E " I F >x δ D · F k D · F k L ([0 ,T ]) ! dx. (E3.13)This implies that F has a density f F given by f F ( x ) = E " I F >x δ D · F k D · F k L ([0 ,T ]) ! for all x ∈ R , and by using the boundedness of the indicator function and the DominatedConvergence Theorem, we can show that this density is continuous.Finally, by recalling that the Skorokhod integral always has zero expectation (thisalso holds for its extension by a density argument), for x ≤ | x | α f F ( x )= E " | x | α I F >x δ D · F k D · F k L ([0 ,T ]) ! = − E " | x | α I F
0, and any λ ≥ q ˜ r ,under the conditional probability measure P ( · | B · , G ), since we can show that q, r can bechosen such that σ αt k D · σ t k L ,T ]) ∈ L q ˜ rB · , G (Ω), P -almost surely. To see the last, we use theCauchy-Schwartz inequality and we recall Theorem 3.1 from [4] to obtain E σ q ˜ rαt k D · σ t k q ˜ rL ([0 ,T ]) ≤ ξ p − ρ t q ˜ r E σ q ˜ r ( α − t Z t e R tt ′ (cid:20)(cid:18) kθ − ξ (cid:19) σs + k (cid:21) ds ! q ˜ r dt ′ ≤ ξ p − ρ t q ˜ r E " σ q ˜ r ( α − t e q ˜ r R t (cid:20)(cid:18) kθ − ξ (cid:19) σs + k (cid:21) ds = ξ p − ρ t q ˜ r E " E " σ q ˜ r ( α − t e q ˜ r R t (cid:20)(cid:18) kθ − ξ (cid:19) σs + k (cid:21) ds (cid:12)(cid:12)(cid:12) σ ≤ ˜ Ct q ˜ r γ v − q ˜ r ( α − t E h σ v H (cid:16) − γ t σ e − kt (cid:17)i (E3.18)for some c, ˜ C >
0, provided that 2 q ˜ r (cid:16) kθ − ξ (cid:17) < ξ (cid:16) kθξ − (cid:17) and q ˜ r ( α − > − kθξ − v ,where v = 12 − (cid:18) kθξ − (cid:19) + s(cid:18) kθξ − (cid:19) − q ˜ r (cid:18) kθξ − (cid:19) , (E3.19)12 t = kξ (cid:0) − e − kt (cid:1) − > kξ for all t ≥
0, and H is a hypergeometric function for which wehave the asymptotic estimate of page 17 in [4]. To have 2 q ˜ r (cid:16) kθ − ξ (cid:17) < ξ (cid:16) kθξ − (cid:17) , q ˜ r ( α − > − kθξ − v and qr < kθ ξ for sufficiently small q >
1, it suffices to obtain allthese strict inequalities for q = 1. Since r < kθ ξ is equivalent to ˜ r > x x − for x = kθξ with x x − >
1, we can have this inequality along with 2˜ r (cid:16) kθ − ξ (cid:17) < ξ (cid:16) kθξ − (cid:17) ⇔ ˜ r < (2 x − x − for some ˜ r > x x − < (2 x − x − . The last inequality is satisfiedsince it is equivalent to 16 x − x + 24 x − > x = kθξ > x ∗ . Then, q ˜ r ( α − > − kθξ − v can be obtained for p = 1, for any α ≥ r sufficiently close toits upper bound (2 x − x − , provided that it holds for q = 1 , α = 0 and ˜ r = (2 x − x − , i.e when − (2 x − x − > − x + 12 (2 x −
1) (E3.20)which is also satisfied when x = kθξ > x ∗ (since x ∗ > σ is bounded awayfrom zero, the argument of H in (E3.18) is bounded from below by some m >
0, andthen the estimate of page 17 in [4] gives H ( − z ) ≤ K | z | − v + q ˜ r ( α − for all z ≥ m , for some K >
0. Therefore, from (E3.18) we obtain E σ q ˜ rαt k D · σ t k q ˜ rL ([0 ,T ]) ≤ ˜ Ct q ˜ r e cT E h σ q ˜ r ( α − i (E3.21)for some c >
0, with the RHS of the above being finite, and this implies also E σ q ˜ rαt k D · σ t k q ˜ rL ([0 ,T ]) | B · , G < ∞ (E3.22) P - almost surely. The last means that the assumptions of Lemma E2.1 are indeed satisfied.From the above we deduce that under P ( · | B · , G ), σ t has a density p t ( y | B · , G ) whichis supported in [0 , + ∞ ) (since this CIR process does not hit zero) and which satisfiessup y ∈ R + y α p t ( y | B · , G ) ≤ ( C + 2) E qr (cid:20)(cid:13)(cid:13) D · , · σ t (cid:13)(cid:13) qrL ( [0 ,T ] ) | B · , G (cid:21) × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ αt k D · σ t k L ([0 ,T ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r | B · , G + C E qr h k D · σ t k qrL ([0 ,T ]) | B · , G i × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ αt k D · σ t k L ([0 ,T ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r | B · , G = ( C + 2) E qr (cid:20)(cid:13)(cid:13) D · , · σ t (cid:13)(cid:13) qrL ( [0 ,t ] ) | B · , G (cid:21) E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ αt k D · σ t k L ([0 ,t ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r | B · , G + C E qr h k D · σ t k qrL ([0 ,t ]) | B · , G i × E q ˜ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ αt k D · σ t k L ([0 ,t ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r | B · , G , so raising to the power q , taking expectations and using Holder’s inequality we obtain E " sup y ∈ R + y α p t ( y | B · , G ) ! q ≤ ˜ C E r (cid:20)(cid:13)(cid:13) D · , · σ t (cid:13)(cid:13) qrL ( [0 ,t ] ) (cid:21) × E r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ αt k D · σ t k L ([0 ,t ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r + ˜ C E r h k D · σ t k qrL ([0 ,t ]) i × E r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ αt k D · σ t k L ([0 ,t ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ˜ r (E3.23)for some ˜ C >
0. Next, by Lemmas 3.2 and 3.3 we have E qr h k D · σ t k qrL ([0 ,t ]) i = E qr Z t ξ (cid:0) − ρ (cid:1) e − R tt ′ (cid:20)(cid:18) kθ − ξ (cid:19) σs + k (cid:21) ds σ t dt ′ ! qr ≤ ξ q − ρ E qr Z t sup s ≤ T σ s dt ′ ! qr = ξ q − ρ √ t E qr sup s ≤ T σ s ! qr = C ′ √ t (E3.24)for some C ′ >
0, while by the estimate of Lemma 3.4 for q ′ = qr we have E qr h(cid:13)(cid:13) D · , · σ t (cid:13)(cid:13) qrL ([0 ,t ]) i ≤ E qr " t qr sup ≤ t ′ ,t ′′ ≤ t ≤ T | D t ′ ,t ′′ σ t | qr = C ′′ t (E3.25)for some C ′′ >
0, since qr < kθ ξ . Moreover, for our choice of q and r , by (E3.21) we have E q ˜ r σ q ˜ rαt k D · σ t k q ˜ rL ([0 ,T ]) ≤ C (3) t (E3.26)for some C (3) >
0. Substituting now (E3.24), (E3.25) and (E3.26) in (E3.23), we obtain E " sup y ∈ R + y α p t ( y | B · , G ) ! q ≤ C (4) + C (5) √ t q (E3.27)for some C (4) , C (5) >
0. Since the RHS of the last is integrable in t for t ∈ [0 , T ] (sincewe can take q < roof of Theorem E1.2 . Let f be a smooth function, compactly supported in R , suchthat f vanishes on the y - axis. Theorem E1.1 applied on the (cid:0) W · , W · (cid:1) - driven CIRprocess (cid:8) σ t : t ≥ (cid:9) implies that the last possesses a density p t (cid:0) ·| B · , G (cid:1) for each t ≥ v t,C ( f ) = E (cid:2) f (cid:0) X t , σ t (cid:1) I { T ≥ t } | W · , B · , C , G (cid:3) = E (cid:2) E (cid:2) f (cid:0) X t , σ t (cid:1) I { T ≥ t } | W · , σ t , B · , C , G (cid:3) | W · , B · , C , G (cid:3) = Z R E (cid:2) f (cid:0) X t , y (cid:1) I { T ≥ t } | W · , σ t = y, B · , C , G (cid:3) p t (cid:0) y | B · , G (cid:1) dy. (E3.28)for any t ≥
0. Next, we compute E (cid:2) f (cid:0) X t , y (cid:1) I { T ≥ t } | W · , σ t = y, B · , C , G (cid:3) = E (cid:2) E (cid:2) f (cid:0) X t , y (cid:1) I { T ≥ t } | W · , σ . , C , G (cid:3) | W · , σ t = y, B · , C , G (cid:3) = E (cid:20)Z R + f ( x, y ) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) dx | W · , σ t = y, B · , C , G (cid:21) = Z R + f ( x, y ) E (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | W · , σ t = y, B · , C , G (cid:3) dx, (E3.29)where u (cid:0) t, x, W · , C , G , h (cid:0) σ . (cid:1)(cid:1) is the L (cid:0) Ω × [0 , T ]; H ( R + ) (cid:1) density given by Theo-rem 4.1 when the coefficient vector C is given and the volatility path is h (cid:0) σ . (cid:1) . By(E3.28) and (E3.29) we have that the desired density exists and is given by u C (cid:0) t, x, y, W · , B · , G (cid:1) = p t (cid:0) y | B · , G (cid:1) E (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | W · , σ t = y, B · , C , G (cid:3) (E3.30)which is supported in R + × R + . Using the Cauchy-Schwartz inequality, the law of totalexpectation, Fubini’s Theorem, and the identity (4.3) we obtain for any α ≥ Z R + Z R + y a (cid:0) u C (cid:0) t, x, y, W · , B · , G (cid:1)(cid:1) dydx = Z R + Z R + y α p t (cid:0) y | B · , G (cid:1) E (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | W · , B · , σ t = y, C , G (cid:3) dydx ≤ M αB · , G ( t ) Z R + Z R + p t (cid:0) y | B · , G (cid:1) × E (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | W · , B · , σ t = y, C , G (cid:3) dydx = M αB · , G ( t ) Z R + E (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | W · , B · , C , G (cid:3) dx = M αB · , G ( t ) E (cid:20)Z R + u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) dx | W · , B · , C , G (cid:21) ≤ M αB · , G ( t ) E h k u ( · ) k L ( R + ) | G i , where we have M αB · , G ( · ) = sup y ≥ (cid:0) y α p · ( y | B · , G ) (cid:1) ∈ L q (Ω × [0 , T ]) for all small enough q > C , by Theorem E1.1). Denoting by E C the expectation given C and taking15 ′ > q + q ′ = 1, by the above and by Holder’s inequality we get E C (cid:20)Z T Z R + Z R + y a (cid:0) u C (cid:0) t, x, y, W · , B · , G (cid:1)(cid:1) dydxdt (cid:21) ≤ E C (cid:20)Z T M αB · , G ( t ) E h k u ( · ) k L ( R + ) | G i dt (cid:21) ≤ E q C "(cid:18)Z T M αB · , G ( t ) dt (cid:19) q E q ′ C h E q ′ h k u ( · ) k L ( R + ) | G ii ≤ T q ′ E q C (cid:20)Z T (cid:16) M αB · , G ( t ) (cid:17) q dt (cid:21) E q ′ h E q ′ h k u ( · ) k L ( R + ) | G ii < ∞ , which shows that the density belongs to the space L α for any α ≥
0. Moreover, repeatingthe above computations but for the derivative multiplied by w ( x ), we find that Z R + Z R + w ( x ) y a (cid:18) ∂u C ∂x (cid:0) t, x, y, W · , B · , G (cid:1)(cid:19) dydx ≤ M αB · , G ( t ) E (cid:20)Z R + w ( x ) u x (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) dx | W · , B · , C , G (cid:21) , so when ρ := R dW t dB t = 0, writing E B · C , G for the expectation given C , G and B · and using Lemma E2.2, we obtain E B · C , G "Z R + Z R + w ( x ) y a (cid:18) ∂u C ∂x (cid:0) t, x, y, W · , B · , G (cid:1)(cid:19) dydx ≤ M αB · , G ( t ) E (cid:20)Z R + w ( x ) u x (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) dx | B · , C , G (cid:21) ≤ M αB · , G ( t ) E " sup ≤ s ≤ T Z R + w ( x ) u x (cid:0) s, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) dx | B · , C , G ≤ M e MT M αB · , G ( t ) (cid:16) E h k w ( · ) ( u ) x ( · ) k L ( R + ) | G i + E h k u ( · ) k L ( R + ) | G i(cid:17) . The last implies that E C "Z T Z R + Z R + w ( x ) y a (cid:18) ∂u C ∂x (cid:0) t, x, y, W · , B · , G (cid:1)(cid:19) dydxdt ≤ M e MT E C (cid:20)Z T M αB · , G ( t ) dt E h k w ( · ) ( u ) x ( · ) k L ( R + ) | G i(cid:21) + M e MT E C (cid:20)Z T M αB · , G ( t ) dt E h k u ( · ) k L ( R + ) | G i(cid:21) ≤ M T q ′ e MT E q C (cid:20)Z T (cid:16) M αB · , G ( t ) (cid:17) q dt (cid:21) × E q ′ C h E q ′ h k w ( · ) ( u ) x ( · ) k L ( R + ) | G ii M T q ′ e MT E q C (cid:20)Z T (cid:16) M αB · , G ( t ) (cid:17) q dt (cid:21) E q ′ C h E q ′ h k u ( · ) k L ( R + ) | G ii < ∞ which gives the weighted integrability of the derivative. To obtain the boundary conditionwhen ρ = 0, we work as follows E C (cid:20)Z R + Z R + y α (cid:0) u C (cid:0) t, x, y, W · , B · , G (cid:1)(cid:1) dydt (cid:21) ≤ E C " Z R + Z R + M αB · , G ( t ) p t (cid:0) y | B · , G (cid:1) × E (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | W · , σ t = y, B · , C , G (cid:3) dydt ≤ E C " Z R + M αB · , G ( t ) Z R + p t (cid:0) y | B · , G (cid:1) × E (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | W · , σ t = y, B · , C , G (cid:3) dydt = E C (cid:20)Z R + M αB · , G ( t ) E (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | W · , B · , C , G (cid:3) dt (cid:21) = E C (cid:20)Z R + M αB · , G ( t ) E C (cid:2) u (cid:0) t, x, W · , G , C , h (cid:0) σ . (cid:1)(cid:1) | B · , C , G (cid:3) dt (cid:21) , where we can use the maximum principle given in Lemma E2.2, the integrability of M αB · , G ( · ) and the Dominated Convergence Theorem, to show that the RHS of the lasttends to zero as x −→ + . This completes the proof of the Theorem. Proof of Lemma E2.3 . The finiteness of all the terms in the identity we are provingis a consequence of Lemma 5.3 and the assumed weighted integrability of u and u x .Multiplying equation (5.6) by w ( x ) (cid:0) y δ (cid:1) + , applying Ito’s formula for the L ( R + ) norm(Theorem 3.1 from [6] for the triple H ⊂ L ⊂ H − , with Λ( u ) = w ( · ) u ), and thenintegrating in y over R + , we obtain the equality k I ǫ, ( t, · ) k L δ,w = (cid:13)(cid:13)(cid:13)(cid:13)Z D U ( · , z ) φ ǫ ( z, · ) dz (cid:13)(cid:13)(cid:13)(cid:13) L δ,w − r Z t (cid:28) ∂∂x I ǫ, ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds + Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds − k θ Z t (cid:28) ∂∂y I ǫ,z − ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds + k Z t (cid:28) ∂∂y I ǫ,z ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds + Z t (cid:28) ∂ ∂x I ǫ,h ( z ) ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds + ξ Z t (cid:28) ∂ ∂y I ǫ, ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds + ρ Z t (cid:28) ∂ ∂x∂y I ǫ,h ( z ) ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds ξ ρ ρ , ρ , Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , ∂∂y I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds + ξ Z t (cid:28) ∂∂y I ǫ,z − ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds + ρ , Z t (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x I ǫ,h ( z ) ( s, · ) (cid:13)(cid:13)(cid:13)(cid:13) L δ,w ds + ξ ρ , Z t (cid:13)(cid:13)(cid:13)(cid:13) ∂∂y I ǫ, ( s, · ) (cid:13)(cid:13)(cid:13)(cid:13) L δ,w ds − ρ , Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w dW s − ξ ρ , Z t (cid:28) ∂∂y I ǫ, ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w dB s . (E3.31)Observe now that by the definition of u xx in our SPDE, we have Z R + Z R u xx ( s, x, z ) φ ǫ ( z, y ) w ( x ) f ( x ) dzdx = Z R + Z R u ( s, x, z ) φ ǫ ( z, y ) (cid:0) w ( x ) f ( x ) (cid:1) xx dzdx = − Z R + Z R u x ( s, x, z ) φ ǫ ( z, y ) (cid:0) w ( x ) f ( x ) (cid:1) x dzdx = − Z R + Z R w ( x ) u x ( s, x, z ) φ ǫ ( z, y ) f x ( x ) dzdx − Z [0 , Z R u x ( s, x, z ) φ ǫ ( z, y ) f ( x ) dzdx (E3.32)which equals − Z R + Z R w ( x ) u x ( s, x, z ) φ ǫ ( z, y ) f x ( x ) dzdx + Z [0 , Z R u ( s, x, z ) φ ǫ ( z, y ) f x ( x ) dzdx − Z R u ( s, , z ) φ ǫ ( z, y ) f (1) dz for any smooth function f defined on [0 , + ∞ ). Since u ∈ H α and since f (1) can becontrolled by the H α norm of f (by using Morrey’s inequality near 1), (E3.32) definesa linear functional on the space of smooth functions f (defined on [0 , + ∞ )) which isbounded under the topology of H α . Then, since those functions form a dense subspace of H α , we have that (E3.32) holds also for any f ∈ H α . In particular, for f = I ǫ, ( s, · , y ),multiplying (E3.32) by y δ and then integrating in ( y, t ) over R + × R + , we obtain Z t (cid:28) ∂ ∂x I ǫ,h ( z ) ( s, · ) , I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds = − Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , ∂∂x I ǫ, ( s, · ) (cid:29) ˜ L δ,w ds. − Z t (cid:28) ∂∂x I ǫ,h ( z ) ( s, · ) , I [0 , × R ( · ) I ǫ, ( s, · ) (cid:29) ˜ L δ ds. (E3.33)Next, identities (5.10)-(5.12) from [3] hold also when ˜ L δ ′ is replaced by ˜ L δ ′ ,w for δ ′ ∈{ δ, δ − , δ − } , and their justification is identical. Substituting these and (E3.33) in(E3.31) we obtain the desired result. 18 emark E3.1 . It is equation (E3.33) which adds two extra terms in the δ -identity. Acknowledgement
The second author’s work was supported financially by the United Kingdom Engineeringand Physical Sciences Research Council [EP/L015811/1], and by the Foundation forEducation and European Culture in Greece (founded by Nicos & Lydia Tricha).
A APPENDIX: A clarification on the proof of Theorem 4.1
The last computation in that proof assumes that A is almost surely a continuity set of X t . To see this, consider the process Y satisfying the same SDE and initial conditionas X but without the stopping condition at 0, and observe that it is a Gaussian processgiven the path W · and given G , which implies that P (cid:0) X t ∈ V | W · , G (cid:1) ≤ P (cid:0) Y t ∈ V | W · , G (cid:1) = 0 (A.1)for any Borel set V ⊂ R + of zero Lebesgue measure. References [1] Denis, L and Matoussi, A. Maximum principle for quasilinear spdes on a boundeddomain without regularity assumptions.
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SIAM Journal on Financial Mathematics (2017),962–1014.[4] Hurd, T. R. and Kuznetsov, A. Explicit formulas for Laplace transforms of stochasticintegrals.