ERROR ESTIMATES FOR A CLASS OF FINITE DIFFERENCE-QUADRATURE SCHEMES FOR FULLY NONLINEAR DEGENERATE PARABOLIC INTEGRO-PDES I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
Abstract. Error estimates are derived for a class of monotone finite differencequadrature schemes approximating viscosity solutions of nonlinear degenerate parabolic integro-PDEs with variable diffusion coefficients. The relevant equations can be viewed as Bellman equations associated to a class of controlled jump-diffusion (L´ evy) processes. Our results cover both finite and infinite activity cases.
1. Introduction In this article we consider error estimates for finite difference type numerical schemes for degenerate and fully nonlinear parabolic integro-partial differential equations (integro-PDEs henceforth) of Bellman type. We write the equation in the following abstract form, (1.1)
ut (t, x) + F (t, x, u(t, x), Du(t, x), D2 u(t, x), u(t, ·)) = 0 in QT
where T > 0 is a constant and QT = [0, T )×Rd and we impose a terminal condition, u(T, x) = u0 (x) for all x ∈ Rd .
(1.2)
The nonlocal feature of the equation is indicated by the term u(t, ·). For any t, x, r, p, X ∈ R × Rd × R × Rd × Sd and for any ‘sufficiently well behaved’ ϕ, the nonlinearity F is defined as follows F (t, x, r, p, X, ϕ(·)) n1 o = sup tr aα (t, x)X + bα (t, x) · p + I α ϕ − cα (t, x)r + f α (t, x) , α∈A 2 where the integral operator I α is defined as (1.3)
I α ϕ(t, x) Z = ϕ(t, x + η α (x, z)) − ϕ(t, x) − 1|z| K, δτT u(t, x) =
p∈h2 ZM M
where p ∈ h2 Z and kp ≥ 0 are the nodes and weights respectively. Since kp ≥ 0, this scheme is monotone. This assumption is crucial for the analysis and natural since the measure ν is positive. Note that the sum is finite since kp = 0 for |p| > K, and this is also natural since the measure ν has support in |p| ≤ K. We also require the following consistency estimate (error estimate) Z (3.2) | f ν(dz) − Ih2 (f )| ≤ ν(E)Lf h2 E
for every Lipschitz function f with Lipschitz constant Lf . Remark 3.1. Many classical quadrature rules satisfy these assumptions, the simplest example being the Riemann sum approximation, X Ih2 (f ) = f (p)ν(p + [0, h2 ]M ). p∈⊂h2 Zm
Other examples include the Newton-Cotes quadratures of order less than 9. We refer to [18] for a more detailed discussion. Now we are in a position to introduce the implicit difference-quadrature scheme: h i α α δτT u(t, x) + sup Lα (3.3) (t, x)u + f (t, x) + J u = 0 in QT , h1 h2 α∈A
with the terminal condition (1.2), where α α α Lα h1 u = ak ∆h1 ,lk u + bk δh1 ,lk u − c u,
Jhα2 u = Ih2 (u(t, x + η α (x, z)) − u(t, x)). As a simple consequence of Taylors theorem, we have the following consistency bound (truncation error) α ∗ (3.4) |Lα h21 sup |Dy4 g| + h1 sup |Dy2 g| , h1 g(x) − L g(x)| ≤ N y∈BK (x)
y∈BK (x)
for every four times differentiable function g and h1 ≤ 1, where N ∗ is constant which only depends on K, d1 and BK (x) = {|x| ≤ K}. Remark 3.2. The solution u of the approximation scheme (3.3) is defined on QT , and not merely on a fixed grid. In part this is a technical trick to simplify the analysis, and the numerical solution defined on a grid should simply be the restriction of u to the h1 -grid. Indeed, in the local PDE context the numerical scheme would be well-defined for functions defined only on the h1 -grid. However, due to the choice of numerical quadrature, this is not the case in our nonlocal setting, and the present scheme cannot be implemented on a computer as it stands. Nevertheless, this can be remedied easily by replacing the integrand by a suitable interpolant over the h1 grid. If piecewise linear interpolation is used, monotonicity of the scheme is preserved and all the estimates obtained in this paper would still hold. From a mathematical point of view, the essential difficulties are already
DEGENERATE PARABOLIC INTEGRO-PDES
7
present in the scheme (3.3), so to avoid increasing the length of this paper we will defer the analysis of the scheme with “interpolation” to future work. Remark 3.3. The effect of using the difference operator δτT in (3.3) is “piecewise constant interpolation in time of the solutions”. It is equivalent to using the scheme with the operator δτ and constant-in-time initial data on the strip [−τ, 0] × Rd . We have the following lemma ensuring the existence of unique solution for the finite difference equation (3.3). Lemma 3.1. Assume (A.1), (A.2), (A.3), and (3.1) hold. Then there is a unique bounded function u(t, x) solving (3.3)/ (1.2). Proof. For each time-level t, existence of such a solution can be proven if one know that such a solution exits for t + τT by the contraction mapping argument used in the stationary case in Lemma 3.1 in [11]. Iterations, starting from terminal time T then complete the proof. Remark 3.4. It follows from the proof that the function u(t, x) is continuous in x but in general it will be discontinuous in t. However, u will satisfy a discrete H¨older bound in t (Theorem 3.4), so the size of discontinuities decrease to 0 as τ → 0. For fixed τ > 0, define ¯ T = {nτ ∧ T : n = 0, 1, 2, 3......} × Rd M ¯ T ∩ [0, T ) × Rd . The scheme is well-defined on MT , and often we will and MT = M ¯ T and subsequently translate them to the deduce properties of the scheme on M whole space QT = MT + [0, τ ] × {0}. We have the following lemma whose proof is postponed to the next section. Lemma 3.2. Assume (A.1), (A.2), (A.3), (3.1), (3.2) and h1 < 1. Let C be a ¯ T , continuous in x for each t, and for constant and u1 , u2 functions defined on M some constant µ > 0, sup |ui (t, x)e−µ|x| | < ∞,
i = 1, 2.
MT
If u1 (T, x) ≤ u2 (T, x) and
(3.5)
h i α α δτT u1 + sup Lα u + f (t, x) + J u h1 1 h2 1 + C α∈A h i α α ≥ δτT u2 + sup Lα h1 u2 + f (t, x) + Jh2 u2 , α∈A
then there exists a constant τ ∗ > 0 depending only on K, d1 , µ, ν(E) such that if τ ∈ (0, τ ∗ ), ¯ T. (3.6) u1 ≤ u2 + (T + τ )C+ in M Furthermore, τ ∗ (K, d1 , µ, ν(E)) → ∞ as µ ↓ 0, and if u1 , u2 are bounded, (3.6) holds for all τ > 0. Corollary 3.3. Assume (A.1), (A.2), (A.3), (3.1), (3.2) and h1 < 1. Then the solution vτ,h of (3.3)-(1.2) satisfies |vτ,h |0 ≤ K(T + τ ) + |u0 |0 . Proof. The function ±[K(T − t) + |u0 |0 ] is supersolution/subsolution of (3.3)-(1.2) (remember cα ≥ λ ≥ 0), so the result follows from Lemma 3.4.
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I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
Consider the terminal value problem h 2 Lα δτT u + sup h1 (t + r, x + y)u(t, x) (α,r,y)∈A×Λ×B1
(3.7) + f α (t + 2 r, x + y) +
X
i kp u(t, x + η α (x + y, p)) − u(t, x) = 0 in QT ,
p
with terminal data (1.2). This is the difference-quadrature scheme corresponding to (2.2). By Lemma 3.1 and Corollary 3.3 there exists a unique bounded solution vτ,h of this problem. We have the following theorem, whos proof will be given in the next section. Theorem 3.4. Assume (A.1), (A.2), (A.3), (3.1), (3.2), 0 ≤ h1 ≤ 1, and 0 ≤ τ ≤ τ0 . If τ0 is small enough, there exists a constant N (depending only on τ0 , λ, T, K, d1 , and ν(E)); such that for all ∈ R |vτ,h (t, x) − vτ,h (t, x)| ≤ N ||,
(3.8) (3.9)
|vτ,h (t, x) − vτ,h (t, y)| + |vτ,h (t, x) − vτ,h (t, y)| ≤ N |x − y|, 1
1
|vτ,h (t, x) − vτ,h (s, x)| + |vτ,h (t, x) − vτ,h (s, x)| ≤ N (|t − s| 2 + τ 2 ), ¯T . for all (t, x), (s, y) ∈ Q
(3.10)
Now, with the help of the results stated above, we are in a position to prove the main contribution of this paper, namely Theorem 3.5. Assume (A.1), (A.2), (A.3), (3.1), (3.2), 0 < h1 , h2 , and 0 < τ ≤ τ0 . If τ0 , h1 , h2 are small enough, there exists a constant N1 (depending only on τ0 , λ, d1 , T, K, ν(E)), such that 1 1 |v − vτ,h |0 ≤ N1 τ 4 + h12 + h2 . 1
Proof. Take = (τ + h21 + h42 ) 4 and let τ0 , h1 , h2 be sufficiently small such that ε < 1. If T < 22 then the theorem holds because by (3.10) and (2.3) and the definition of ε, sup |vτ,h − v| ≤ sup |vτ,h − u0 | + sup |u0 − v| ¯T Q
¯T Q
¯T Q
1 2
1 2
1
≤ N (T + τ ) ≤ N (τ + h21 + h42 ) 4 . Next we consider the case T > 22 . First we prove the upper bound 1
1
v − vτ,h ≤ N (τ 4 + h12 + h2 ).
(3.11)
For each α ∈ A, r ∈ (−1, 0) and |y| < 1, equation (3.7) implies (3.12)
2 α δτT vτ,h (t − 2 r, x − y) + Lα h1 (t, x)vτ,h (t − r, x − y) + f (t, x) i X h + kp vτ,h (t − 2 r, x + η α (x, p) − y) − vτ,h (t − 2 r, x − y) ≤ 0 p
¯ T −2 . for (t, x) ∈ Q Now use Krylov’s technique i.e. multiply inequality (3.12) with a mollifier and convolve. Let ζ ∈ C0∞ (Rd+1 ) be our mollifier, a positive function with unit integral and having support in Λ × B1 . Also denote, Z s y () (−d−2) u (t, x) = u(t − s, x − y)ζ 2 , ds dy Rd+1
DEGENERATE PARABOLIC INTEGRO-PDES
9
Then multiplying (3.12) by −d−2 ζ(s/ε2 , y/ε) and integrating with respect to (s, y) ¯ T −22 we obtain, for each α ∈ A, on Q ()
()
α δτT vτ,h + Lα h1 (t, x)vτ,h + f +
X
()
()
kp vτ,h (t, x + η α (x, p)) − vτ,h
≤ 0.
p
From (3.4), (3.2), and Taylor’s formula we have Z ∂ () () () () α α vτ,h + L (t, x)vτ,h + f + vτ,h (t, x + η α (x, z)) − vτ,h ν(dz) ∂t RM \{0} () () 2 () 2 ≤ N τ |Dt vτ,h |0,Q¯ T −22 + h1 |Dx4 vτ,h |0,Q¯ T −22 + h1 |Dx2 vτ,h |0,Q¯ T −22 () ¯ T −22 . + h2 |Dx vτ,h |0,Q¯ T −22 := I in Q ()
Clearly vτ,h + (T − 22 − t)I is a classical supersolution to the equation (2.1) and hence a viscosity supersolution as well in QT −22 . Now using the comparison principle (Theorem 2.1) we have (3.13)
()
v ≤ vτ,h + (T − 22 − t)I +
sup
|v − v () |.
{(T −22 )×Rd }
Using properties of convolutions and regularity of vτ,h (Theorem 3.4), ()
|vτ,h − vτ,h | ≤ N
()
()
and 2n−1 |Dtn vτ,h |0,Q¯ T −22 + n−1 |Dxn vτ,h |0,Q¯ T −22
≤ N.
By the same reasoning as in the beginning of the proof, we also find |v(T − 22 , x) − v () (T − 22 , x)| ≤ N . By the above estimates, Theorem 3.4, and recalling that ε4 = τ +h21 +h42 , we obtain h1 τ + h21 + + h2 v ≤ vτ,h +N + 3 h1 τ + h21 + + h ≤ vτ,h + N + 2 3 1 2 ¯ T −22 . ≤ vτ,h + N (τ + h1 + h42 ) 4 in Q By the regularity of v, vτ,h and the argument given in the case T < 2ε2 , this ¯ T . Moreover, it can be checked that all constants estimate in fact holds in all of Q N only depend on τ0 , λ, ν(E), d1 , d, K and T . This completes the proof of (3.11). The lower bound 1 1 (3.14) vτ,h − v ≤ N τ 4 + h12 + h2 , can be proved in a similar way. Interchange the role of the finite difference scheme and the equation (2.1) in the argument leading to (3.11). Now it can be shown that v () is a classical supersolution of (2.1) in QT −2 . We skip the arguments since they are similar to the arguments for stationary integro-PDEs given in [18], see also [20, 15] for time-dependent pure PDEs. By consistency (3.2) and (3.4), regularity of v () , and properties of mollifiers, it follows that τ + h2 h1 1 () δτT v () + sup Lα + f α + Jhα2 v () ≤ N + + h2 h1 v 3 α∈A
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I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
¯ T −2 −τ , and the comparison result (Lemma 3.2) then gives in Q τ + h2 h1 1 vτ,h ≤ v () + sup (vτ,h − v () )+ + N + + h2 . (3.15) 3 QT \QT −2 −τ Since τ ≤ ε4 ≤ ε2 ≤ 1 and hence QT \QT −2 −τ ⊂ QT \QT −22 , (vτ,h − v () )+ ≤
sup QT \QT −2 −τ
sup
|vτ,h − v () | ≤ N ,
QT \QT −22
where the last inequality was proved at the start of this proof. This estimate, (3.15), and the definition of ε, implies the lower bound (3.14). It can be checked that N depends only on ν(E), K, d1 , d and T . 4. Proofs of the results stated in Section 3 In this section we prove comparison and Lipschitz continuity results for the solution of the difference-quadrature scheme (3.3), (1.2). As an application of the Lipschitz result we derive a continuous dependence estimate for the scheme. Although the basic ideas behind our proofs come from Krylov [21], the nonlocal nature of the problem adds to some extra difficulties and they do not allow us to adopt the “local” approach of Krylov. Our approach is more direct and we employ some new techniques. We begin by stating some auxiliary results. To this end, we need the translation operator Th1 ,l u(x) := u(x + h1 l). We now give some technical lemmas whose proofs can be found in [21]. Lemma 4.1. For any functions u(x), v(x), h1 > 0. and l ∈ Rd we have Th1 ,−l Th1 ,l u = u, Th1 ,l δh1 ,−l = δh1 ,−l Th1 ,l = −Th1 ,−l δh1 ,l = −δh1 ,l Th1 ,−l = −δh1 ,−l , δh1 ,l (uv) = v δh1 ,l (u) + Th1 ,l u δh1 ,l (v), = uδh1 ,l (v) + vδh1 ,l (u) + h1 (δh1 ,l (v))(δh1 ,l (u)) ∆h1 ,l (uv) = u∆h1 ,l (v) + v∆h1 ,l (u) + (δh1 ,l (v))(δh1 ,l (u)) + (δh1 ,−l (v))(δh1 ,−l (u)). In particular, ∆h1 ,l (u2 ) = 2u∆h1 ,l u + (δh1 ,l u)2 + (δh1 ,−l u)2 . Lemma 4.2. Let u,v, w be functions on Rd , l, x0 ∈ Rd , h1 > 0. Assume that v(x0 ) ≤ 0 . Then at x0 it holds (4.1)
−δh1 ,l v ≤ δh1 ,l (v− ), −∆h1 ,l v ≤ ∆h1 ,l (v− ),
(4.2)
|∆h1 ,l u| ≤ |δh1 ,−l ((δh1 ,l u)− )| + |δh1 ,l ((δh1 ,−l u)− )|, |∆h1 ,l u| ≤ |δh1 ,−l ((δh1 ,l u)+ )| + |δh1 ,l ((δh1 ,−l u)+ )|.
Now we prove Lemma 3.2. Proof of Lemma 3.2: Let T 0 be the smallest jτ which exceeds T , where j ∈ N. A ¯ T could be viewed as a solution to the same solution to the equation (3.3) on M ¯ on MT 0 after trivially redefining the function on {T 0 } × Rd . So without loss of generality we assume that T = T 0 .
DEGENERATE PARABOLIC INTEGRO-PDES
11
From (3.5) we then have in MT , for u = u1 − u2 h X i α δτ u + sup Lα u + k + C ≥ 0. u(t, x + η (x, p)) − u(t, x) p h1 α∈A
p∈h2 ZM
Let w = u − C+ (T − t) and note that h X i α δτ w + sup Lα w + k w(t, x + η (x, p)) − w(t, x) p h1 α∈A
p∈h2 ZM
h
α∈A
i kp u(t, x + η α (x, p)) − u(t, x) + C+ + λC+ (T − t)
X
≥ δτ u + sup Lα h1 u +
p∈h2 ZM
≥ 0, and hence for ε > 0, h w + δτ w + sup Lα h1 w + α∈A
i kp w(t, x + η α (x, p)) − w(t, x) ≥ w.
X p∈h2 ZM
For any ψ ≥ w, we can choose small enough so that in MT , h X i ψ + δτ ψ + sup Lα kp ψ(t, x + η α (x, p)) − ψ(t, x) h1 ψ + α∈A
(4.3)
≥ w + δτ w + sup α∈A
p∈h2 ZM
h
Lα h1 w
X
+
√
p∈h2
kp w(t, x + η α (x, p)) − w(t, x)
i
≥ w.
ZM
For a constant γ and hxi := 1 + x2 , we define the functions ξ(t), β(x), and ¯ T in the following way: ζ(t, x) on M ξ(T ) = 1, ξ(t) = γ −1 ξ(t + τT (t)) if t ∈ [0, T ) β(x) = cosh (µhxi), ζ(t, x) = ξ(t)β(x). Note that ξ is recursively defined. By straightforward computations we have, h X i α sup Lα β + k β(x + η (x, p)) − β(x) p h1 α∈A
p∈h2 ZM α
≤ sup L β + N1 (h21 + h1 ) cosh(µhxi + K) + N2 (ν(E), µ) cosh(µhxi + K) α∈A
≤ N2 cosh(µhxi + K). γ−1 τ ξ(t)
and cosh(µhxi + K) ≤ eK cosh(µhxi), it follows that h X i δτ ζ + sup Lα kp ζ(t, x + η α (x, p)) − ζ(t, x) h1 ζ +
Since δτ ξ(t) =
α∈A
≤τ
−1
p∈h2 ZM
(γ − 1)ζ + N3 ζ = κ(γ)ζ,
where N3 = N2 eK and κ(γ) = τ −1 (γ − 1) + N3 . We take τ ∗ = N3−1 and let τ < τ ∗ . Then κ(0) < 0 and κ(1) ≥ 0 and hence we can choose γ so that κ < 0 and 1+κ > 0 for all ε small enough. w+ Now set N = supM ¯ T ζ . Taking ψ = N ζ and ε small enough, (4.3) leads to N ζ(1 + κ) = N ζ + kN ζ h ≥ ψ + δτT ψ + sup Lα h1 ψ + α∈A
X p∈h2 ZM
i kp ψ(t, x + η α (x, p)) − ψ(t, x) ≥ w.
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I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
¯ T. in MT . But w(T, x) is negative by definition, so the inequality holds on entire M By the definition of N , we then have N (1 + κ) ≥ N . Since κ < 0, we conclude that N = 0 and hence w ≤ 0 and (3.6) follows. The remaining part of the lemma becomes obvious if we choose N1 = N2 = N3 = 0 Next we state and prove the key technical result of this paper. Theorem 4.3. Assume (A.1), (A.3), (A.2), (3.1) and (3.2) hold. Let u(t, x) be ¯ T solving (3.3) with |u(T, ·)|1 < ∞. There is a constant N > 0, a function on M depending only on K, d1 , d and the L´evy measure ν, such that, if there is a number c0 ≥ 0 satisfying (4.4)
λ+
1 − e−c0 τ > N, τ
then for every 0 < < Kh1 , l ∈ Rd , |δ,±l u(t, x)| ≤N1 (1 ∨ |l|) 1 + sup |δh1 ,lk u(T, ·)| + sup |δ,±l u(T, ·)|
in
¯ T, M
x
k,x
where N1 only depend on T, λ, c0 , K, d, d1 , ν(E). Proof. We start by introducing a few additional notations. Let r and k be indices running through {±1, ±2....., ±(d1 + 1)} and {±1, ±2, .... ± d1 } respectively, let 0 < ε ≤ Kh1 , and define hk = h1 , k = ±1, ±2, ....., ±d1 , h±(d1 +1) = , l±(d1 +1) = ±l. Choose a constant c0 ≥ 0, let T 0 be the least nτ, n = 1, 2, 3, . . ., such that nτ ≥ T , and define ξ(t) = ec0 t
0
if t < T 0 , ξ(T ) = eco T otherwise;
v = ξu; vr = δhr ,lr v
if
r = ±1, ±2, ...... ± d1 ;
v(t, x ± l) − v(t, x) ; (1 ∨ |l|) sup |v(t, x)|, M1 = sup |vr |.
v±(d1 +1) = M=
¯T (t,x)∈Q
r,x,l,t
Now define W (t, x, l) =
X (vr− )2
and V (t, x, l) = W (t, x, l) − δC(x),
r
where δ > 0 and C(x) ∈ C 2 (Rd ) is positive, convex, and satisfy lim C(x) = ∞ and |DC|0 + |D2 C|0 < ∞.
|x|→∞
To prove the theorem we have to find a bound on M1 which is independent of the discretization constants. We will derive such a bound for W , and towards the end of the proof we will show that this bound implies the sought after bound on M1 . From the properties of C(x), it is clear that V (t, x, l) is bounded above and that ¯ T × Rd such that there exists a point (t0 , x0 , l0 ) ∈ M V (t0 , x0 , l0 ) = sup V (t, x, l). (t,x,l)
DEGENERATE PARABOLIC INTEGRO-PDES
13
If t0 = T , then (4.5)
V (T, ·) ≤ W (T, ·) ≤ N (d1 )e2c0 T
0
2 sup ur (T, x) , r,x
and the theorem is true by Lipschitz continuity of u(T, x). From now on we take t0 < T . By the definition of supremum, there is a sequence of control parameters (αn ) ∈ A (depending on the maximum point (x0 , t0 , l0 )) such that i h αn αn n (t , x ) + J u(t , x ) (t , x )u(t , x ) + f lim Lα 0 0 0 0 0 0 0 0 h h 2 1 n→∞ h i α α α = sup Lh1 (t0 , x0 )u(t0 , x0 ) + f (t0 , x0 ) + Jh2 (u(t0 , x0 )) . α∈A
By assumption (A.3) and the Arzela-Ascoli theorem there is a subsequence {αn } and functions a ¯k , ¯bk , c¯, f¯, η¯, such that αn αn n aα , f αn , η αn → a ¯k , ¯bk , c¯, f¯, η¯ locally uniformly. k , bk , c Obviously, a ¯k , ¯bk , c¯, f¯, η¯ also satisfy (A.1) and (A.3). Moreover since u solve (3.3), (4.6) δτT u + a ¯k ∆h1 ,lk u + ¯bk δh1 ,lk u − c¯u + f¯ X + kp u(t0 , x0 + η¯(x0 , p)) − u(t0 , x0 ) = 0, p∈h2 ZM
at the point (t0 , x0 ), while at the points (t0 , x0 + hr lr ), δτT u + a ¯k ∆h1 ,lk u + ¯bk δh1 ,lk u − c¯u + f¯ X (4.7) + kp u(·, · + η¯(·, p)) − u(·, ·) p∈h2
ZM
≤ 0.
(t0 ,x0 +hr lr )
The last inequality holds at every point in QT . For simplicity we now drop the 0 subscript and rename the maximum point (x, t, l). Replacing u by ξ −1 v in (4.6) and (4.7) we get δτT (ξ −1 v) + ξ −1 a ¯k ∆h1 ,lk v + ¯bk δh1 ,lk v − c¯v + f¯ X (4.8) + kp v(t, x + η¯(x, p)) − v(t, x) = 0 p∈h2 ZM
at the point (t, x) and for each r, and at the points (t, x + hr lr , ) we have h δτT (ξ −1 v) + ξ −1 a ¯k ∆h1 ,lk v + ¯bk δh1 ,lk v − c¯v + f¯ X i (4.9) + kp v(·, · + η¯(·, p)) − v(·, ·) ≤ 0. p∈h2 ZM
(t,x+hr lr )
Subtracting (4.8) from (4.9) and dividing the result by hr , for r = ±1, ±2, ...., ±d1 , and by (|l|∨1) for r = ±(d1 +1), and using the product rule for difference quotients (Lemma 4.1) we get h i (4.10) δτT (ξ −1 vr ) + ξ −1 a ¯k ∆hk ,lk vr + I1r + I2r + I3r + I4r + I5r ≤ 0, where there is no summation with respect to r. Here ( (δhr ,lr a ¯k )∆hk ,lk v, if r 6= ±(d1 + 1) I1r = 1 ¯k )∆hk ,lk v, if r = ±(d1 + 1), (1∨|l|) (δhr ,lr a
14
I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
I2r = hr (δhr ,lr a ¯k )∆hk ,lk vr , ( (Thr ,lr ¯bk )δhk ,lk vr + (δhr ,lr ¯bk )δhk ,lk v, if r 6= ±(d1 + 1) I3r = 1 (Thr ,lr ¯bk )δhk ,lk vr + (1∨|l|) (δhr ,lr ¯bk )δhk ,lk v, if r = ±(d1 + 1), ( −(δhr ,lr c¯)v − (Thr ,lr c¯)vr + ξδhr ,lr f¯, if r 6= ±(d1 + 1) I4r = 1 1 ¯ − (1∨|l|) (δhr ,lr c¯)v − (Thr ,lr c¯)vr + (1∨|l|) ξδhr ,lr f , if r = ±(d1 + 1), P k v(t,x+hr lr +¯η(x+hr lr ,p))−v(t,x+¯η(x,p)) − ν(E)v , if r 6= ±(d + 1) p r 1 hr I5r = Pp v(t,x+hr lr +¯ η (x+hr lr ,p))−v(t,x+¯ η (x,p)) − ν(E)vr , if r = ±(d1 + 1). p kp hr (1∨|l|) The last term is of particular relevance to this paper as it comes from the discretization of the integral term. Now multiply (4.10) by ξvr− and sum up with respect to r. The main part of the proof involves the estimation of each of the above terms as they appear after summation had been done. P − 2 − − P We start with the term r vr I5r . Note that vr vr = (vr ) and moreover that p kp = ν(E) by (3.2). We get X vr− I5r r
X
= ν(E)W +
p, r6=±(d1 +1)
X
+
kp vr−
p, r=±(d1 +1)
kp vr−
v(t, x + hr lr + η¯(x + hr lr , p)) − v(t, x + η¯(x, p)) hr
v(t, x + hr lr + η¯(x + hr lr , p)) − v(t, x + η¯(x, p)) . hr (1 ∨ lr )
For r = ±1, ±2, ........, ±d1 we have v(t, x + h l + η¯(x + h l , p)) − v(t, x + η¯(x, p)) r r r r hr 0 v(t, x + η¯(x, p) + hr l ) − v(t, x + η¯(x, p)) hr = ( )(| l0 | ∨ 1) h r 0 hr (1 ∨ | l |) ≤ K|l0 |M1 , since ε ≤ Kh1 and where l 0 = lr +
η¯(x + hr lr , p) − η¯(x, p) hr
and hence
|l0 | ≤ |lr | + |lr ||∇¯ η (·, p)|L∞ .
For r = ±(d1 + 1) similarly we have, 0 v(t, x + l + η¯(x + l, p)) − v(t, x + η¯(x, p)) ≤ M1 1 ∨ (|l |) (1 ∨ l) 1 ∨ |l| where η¯(x + l, p) − η¯(x, p) and hence |l0 | ≤ |l|(1 + |∇¯ η (·, p)|L∞ ). Putting the above pieces together and using Cauchy-Schwartz inequality we get, X (4.11) vr− I5r ≥ ν(E)W − N (d1 , K)ν(E)M12 . l0 = l +
r
DEGENERATE PARABOLIC INTEGRO-PDES
P Next, we estimate the term r vr− I3r : X X (4.12) vr− I3r = vr− Thr ,lr ¯bk δhk ,lk vr + r
r
vr− (δhr ,lr ¯bk )δhk ,lk v
r6=±(d1 +1)
X
+
X
15
r=±(d1 +1)
1 v − (δhr ,lr ¯bk )δhk ,lk v. 1 ∨ |l| r
At the maximum point (t, x, l) for V , Lemma 4.1 and (4.1) yields for each k X X 0 ≥ δhk ,lk (vr− )2 − δδhk ,lk C(x) ≥ −2 vr− δhk ,lk vr − δδhk ,lk C(x), r
r
which could be rewritten as X vr− δhk ,lk vr + r
δ δhk ,lk C(x) ≥ 0. 4(d1 + 1)
Since bk ≥ 0, this inequality implies that X X δ Thr ,lr ¯bk δhk ,lk C(x). Thr ,lr ¯bk vr− δhk ,lk vr ≥ − 4(d1 + 1) r,k r,k P Combining this inequality with (4.12) we get the desired estimate for r vr− I3r , X (4.13) vr− I3r ≥ −δN (d1 , K) − N (d1 , K)M12 . r
P Now we consider the term r vr− I4r . h i X X vr− I4r = − vr− (δhr ,lr c¯)v − (Thr ,lr c¯)vr + ξδhr ,lr f¯ r
r6=±(d1 +1)
X
−
r=±(d1 +1)
vr−
h
i 1 1 (δhr ,lr c¯)v − (Thr ,lr c¯)vr + ξδhr ,lr f¯ . (1 ∨ |l|) (1 ∨ |l|)
We see that X
Thr ,lr c¯(−vr )vr− =
X
r
Thr ,lr c¯(vr− )2 ≥ λ
r
X
(vr− )2 = λW.
r
Young’s inequality and the definition of M then gives X 0 (4.14) vr− I4r ≥ −N (K, d1 )M1 (ec0 T + M ) + λW. r
P Consider the r vr− δτT (ξ −1 v) term. Once more using that (t, x, l) is a maximum point of V , Lemma 4.1, and (4.1), we get X X X X 0 ≤ −δτT (vr− )2 = −2 vr− δτT (vr− ) − τ (δτT (vr− ))2 ≤ 2 vr− δτT (vr ). r
r
r
r
We conclude that X X − −1 ξvr− δτT ξ −1 vr = ξvr ξ (t + τT (t))δτT vr + vr δτT ξ −1 r
r
= e−c0 τT (t)
X
vr− δτT vr − W ξδτ ξ −1
r −c0 τ
(4.15)
≥W
1−e τ
.
16
I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
Using the relations (4.15), (4.14), (4.13), and (4.11), we obtain from (4.10) 1 − e−c0 τ + λ + ν(E)V − ν(E)N (d1 , K)M12 − δN (d1 , K) − N (d1 , K)M12 τ X 0 − N (d1 , K)M1 (ec0 T + M ) + vr− a ¯k ∆hk ,lk vr + I1r + I2r ≤ 0,
W
r,k
i.e., 1 − e−c0 τ + λ + ν(E)V − ν(E)N (d1 , K)M12 − δN (d1 , K) τ 0 − N (d1 , K)M12 − N (d1 , K)M1 (ec0 T + M ) X X ≤− vr− a ¯k ∆hk ,lk vr − vr− (δhr ,lr a ¯k )∆hk ,lk v
W
k,r6=±(d1 +1)
r,k
X
−
(4.16)
k,r=±(d1 +1)
X 1 vr− (δhr ,lr a ¯k )∆hk ,lk v − hr vr− (δhr ,lr a ¯k )∆hk ,lk vr . 1 ∨ |l| r,k
Once again, using the fact that (t, x, l) is a point of maxima for V , along with the discrete product rule (Lemma 4.1) and (4.1), we have for each k, X 0 ≥ ∆hk ,lk (vr− )2 − δ∆hk ,lk C(x) r
=2
X
vr− ∆hk ,lk vr−
+
r
≥ −2
X (δhk ,lk vr− )2 + (δhk ,l−k vr− )2 − δ∆hk ,lk C(x) r
X
vr− ∆hk ,lk vr
X + (δhk ,lk vr− )2 + (δhk ,l−k vr− )2 − δ∆hk ,lk C(x).
r
r
We rewrite this as X X (4.17) 2 vr− ∆hk ,lk vr + δ∆hk ,lk C(x) ≥ (δhk ,lk vr− )2 + (δhk ,l−k vr− )2 , r
r
and conclude that 2
(4.18)
X
vr− ∆hk ,lk vr + δ∆hk ,lk C(x) ≥ 0.
r
Multiplying (4.17) by a ¯k and summing up with respect to k we get X X Xδ a ¯k ∆hk ,lk C(x) ≥ a ¯k (δhk ,lk (vr− ))2 . vr− a ¯k ∆hk ,lk vr + 2 r,k
r,k
k
Using this inequality, (4.16) becomes 1 − e−c0 τ + λ + ν(E)V − ν(E)N (d1 , K)M12 − δN (d1 , K) τ 0 − N (d1 , K)M12 − N (d1 , K)M1 (ec0 T + M ) δX ≤ J1 + J2 + a ¯k ∆hk ,lk C, 4
W
(4.19)
k
where J1 =
X r,k
vr− |(δhr ,lr a ¯k )∆hk ,lk v| −
1X a ¯k (δhk ,lk vr− )2 4 r,k
DEGENERATE PARABOLIC INTEGRO-PDES
J2 =
X
vr− hr |(δhr ,lr a ¯k )∆hk ,lk vr | −
r,k
17
1X − 1X vr a ¯k ∆hk ,lk vr − a ¯k (δhk ,lk vr− )2 . 2 4 r,k
r,k
Now we estimate J1 . By (4.2), X X (4.20) |∆hk ,lk v| ≤ |δhk ,lk vr− | + |δhk ,l−k vr− |, r
r
and by Lemma 4.1 and Young’s inequality, we get X X vr− |(δhr ,lr (¯ σk )2 )∆hk ,lk v| = vr− |(2¯ σk δhr ,lr σ ¯k + hr (δhr ,lr σ ¯k )2 )∆hk ,lk v| r,k
≤
r,k
X
3
M1 K|¯ σk ∆hk ,lk v| + K 2(d1 + 1)M1
X
r,k
≤ N M1
X
(4.20)
|¯ σk ∆hk ,lk v| + N M12 ≤ N
k
≤N
h1 |∆hk ,lk v|
k
X r,k
X
M1 |¯ σk δhk ,lk vr− | + N M12
r,k
1X 1 |¯ σk δhk ,lk vr− |2 + N M12 ≤ a ¯k (δhk ,lk vr− )2 + N M12 , 8N M12 + 8N 4 r,k
N M12 .
which implies that J1 ≤ The next step is to get a similar estimate on J2 . Note that |a| = 2a− + a,
hr ≤ Kh1 ,
h2r |∆hk ,lk vr | ≤ M1 .
We get X
vr− hr |(δhr ,lr a ¯k )∆hk ,lk vr |
r,k
≤
X
≤
X
≤
X
vr− hr |2(δhr ,lr σ ¯k )¯ σk ∆hk ,lk vr + hr (δhr ,lr σ ¯k )2 ∆hk ,lk vr |
r,k
N1 |vr− hr σ ¯k ∆hk ,lk vr | +
r,k
X
N2 h2r vr− |∆hk ,lk vr |
r,k
2N1 hr vr− |¯ σk |(∆hk ,lk vr )−
+
r,k
X
N1 hr vr− |¯ σk |∆hk ,lk vr + N2 M12 .
r,k
In the above inequality the summation over r may be restricted to the cases where vr− 6= 0 or vr < 0. From (4.1) and hk ∆hk ,lk = δhk ,lk + δhk ,−lk , we then get hk (∆hk ,lk vr )− = hk max(−∆hk ,lk vr , 0) ≤ hk |∆hk ,lk vr− | ≤ |δhk ,lk vr− | + |δhk ,l−k vr− |. The last two estimates give X vr− hr |(δhr ,lr a ¯k )∆hk ,lk vr | r,k
≤ N2 M12 +
X 1 r,k
4
a ¯k (δhk ,lk vr− )2 + N1 vr− h|¯ σk |∆hk ,lk vr ,
and hence √ 1 J2 ≤ N2 M12 − (¯ ak − 2N1 h a ¯k )vr− ∆hk ,lk vr . 2 Let
√ A= k: a ¯k − 2N1 (K)h1 a¯k ≥ 0 .
18
I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
and note that if k ∈ / A, then √ a¯k ≤ 2N1 (K)h1 , a ¯k ≤ 4N12 h21 ,
√ |¯ ak − 2N1 (K)h1 a¯k | ≤ N (K)h21 .
By (4.18) we then get √ 1X − a ¯k − 2N2 (K)h1 a¯k vr− ∆hk ,lk vr 2 r,k X √ − 1 X ¯k vr ∆hk ,lk vr + a ¯k − 2N2 (K)h1 a =− 2 r,k∈A
r,k∈A /
X √ − 1 X ≤− a ¯k − 2N2 (K)h1 a ¯k vr ∆hk ,lk vr + N (K) vr− h21 ∆hk ,lk vr 2 r,k∈A /
r,k∈A
√ 1 X a ¯k − 2N2 (K)h1 a ¯k ∆hk ,lk C(x) + N (d1 , K)M12 ≤ δ 2 r,k∈A
≤ N (K, d1 )(M12 + δ), which gives the estimate J2 ≤ N M12 + δ . The bounds on J1 and J2 along with (4.19) and the definition of V (V = W −δC) give 0 1 − e−c0 τ )V (t, x, l) ≤ N δ + (ec0 T + M1 + M )M1 , τ when t < T . Combining this estimate with the estimate for t = T (4.5) we see that (λ +
V (t, x, l) ≤ (λ +
0 0 1 − e−c0 τ −1 ) N δ + ec0 T + M1 + M M1 + N e2c0 T (sup ur (T, x))2 , τ r,x
¯ T × RN and δ > 0. Using the definition of V and sending for every (t, x, l) ∈ M δ → 0 then give for every t, x, l, W (t, x, l) (4.21)
≤ (λ +
0 0 1 − e−c0 τ −1 ) N ec0 T + M1 + M M1 + N e2c0 T (sup ur (T, x))2 . τ r,x
¯ T and for each Let Wmax = sup(t,x,l)∈M ¯ T ×Rd W (t, x, l). For each (t, x) ∈ M r, either vr√ (t, x) ≤ 0 or −vr (t, x) = v−r (t, x + hr lr ) ≤ 0. In any case we have |vr (t, x)| ≤ Wmax and hence p p 1 (4.22) M1 ≤ Wmax and |δ,±l u| ≤ Wmax . 1 ∨ |l| In view of (4.21) p p 0 1 − e−c0 τ −1 ) N ec0 T + Wmax + M Wmax τ 0 + N e2c0 T (sup ur (T, x))2 .
Wmax ≤ (λ +
r,x 0
By this estimate, Young’s inequality, and M ≤ ec0 T |u|0 , we obtain Wmax ≤ (λ +
1 − e−c0 τ −1 ) N 1 + Wmax + |u|20 + N (sup ur (T, x))2 . τ r,x
DEGENERATE PARABOLIC INTEGRO-PDES
If (λ +
1−e−c0 τ τ
19
) ≥ N + 1, then we conclude that Wmax ≤ N 0 1 + M02 + (sup ur (T, x))2 . r,x
Along with (4.22) this estimate proves the theorem.
Next, following [21], we prove a continuous dependence estimate for the scheme. 1 α 2 Let σ ˆkα , ˆbα ˆα , fˆα , u ˆα ˆα be functions from A × R × Rd to R and set a ˆα σk | . 0,η k,c k = 2 |ˆ Theorem 4.4. Assume (A.1), (A.3), (A.2), (3.1) and (3.2) hold. Let σ ˆkα , ˆbα ˆα , fˆα , ηˆα k,c ¯ satisfy assumptions (A.1) – (A.3). Let u and u ˆ be functions on MT satisfyα α α α ˆα α ˆα α ing (3.3) with coefficients σkα , bα , c , f , η and σ ˆ ˆ , f , ηˆ respectively and k k , bk , c |u(T, ·)|1 + |ˆ u(T, ·)|1 ≤ K. Define n o α α α α α ˆα − f α | + |ˆ := sup |ˆ σkα − σkα | + |ˆbα − b | + |ˆ c − c | + | f η − η | . k k ¯ T ,A,k M
Then if there exists a c0 ≥ 0 satisfying (4.4), there exists a constant N depending only on K, d1 , λ, c0 , T, ν(E) such that ¯ T, (4.23) |u − u ˆ| ≤ N I on M where I := 1 + max |δh1 ,lk u(T, ·)|0 + max |δh1 ,lk u ˆ(T, ·)|0 + −1 |u(T, ·) − u ˆ(T, ·)|0 . k
k
Proof. First we show that it is sufficient to prove the result assuming ≤ h1 . For each θ ∈ [0, 1], let uθ be the (unique) solution of θα θα θα δτT u + sup aθα k ∆h1 ,lk u + bk δhk ,lk u − c u + f α∈A X + kp u(t, x + η θα (x, p)) − u(t, x) = 0 in MT , p θ
with u (T, x) = (1 − θ)u(T, x) + θˆ u(T, x) and where θα θα θα θα θα α α α α α α α α σk , bk , c , f , η = (1 − θ) σkα , bα +θ σ ˆk , ˆbk , cˆ , fˆ , ηˆ . k,c ,f ,η By uniqueness, u0 = u and u1 = u ˆ. Also note that for any θ1 , θ2 ∈ [0, 1], α, k, |σkθ1 α − σkθ2 α |0 + |bθk1 α − bθk2 α |0 + |cθ1 α − cθ2 α |0 + |f θ1 α − f θ2 α |0 + |η θ1 α − η θ2 α |0 ≤ |θ1 − θ2 |. Therefore if we assume the result holds for ≤ h1 , then for any satisfying |θ1 − θ2 | ≤ h1 , we have |uθ1 − uθ2 | ≤ N1 |θ1 − θ2 |I(θ1 , θ2 )
(4.24) where
I(θ1 , θ2 ) = 1 + max |δh1 ,lk uθ1 (T, ·)|0 + max |δh1 ,lk uθ2 (T, ·)|0 k
+
−1
|θ1 − θ2 |
k
−1
θ1
θ2
|u (T, ·) − u (T, ·)|0 .
Clearly I(θ1 , θ2 ) ≤ 4I, so by dividing the interval [0, 1] into sufficient number of sub-intervals θ1 , . . . , θn , we can conclude the theorem (with 4N instead of N ), by writing n X u−u ˆ= (uθi − uθi−1 ), i=1
20
I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
and using (4.24) to estimate each uθi − uθi−1 . Henceforth we assume that ≤ h1 . We will now show that the continuous dependence estimate (4.23) is a consequence of the Lipschitz estimate Theorem (4.3). To this end, we consider Rd as subspace of Rd+1 and write, Rd+1 = (x0 , xd+1 ) : x0 ∈ Rd , xd+1 ∈ R , ¯ T (d + 1) = [0, T ] × Rd+1 , M ¯ T (d + 1) = {jτ ∧ T : j = 0, 1, 2.....} × Rd+1 , Q QT (d + 1) = [0, T ) × Rd+1 ,
¯ T (d + 1). and MT (d + 1) = QT (d + 1) ∩ M
Let ρ ∈ C 1 (R) be a bounded function on R such that ρ(−1) = 1, ρ(0) = 0, ρ0 (p) = ρ0 (q) = 0 if p ≤ −1, q ≥ 0. Now define xd+1 xd+1 + σkα (t, x0 ) 1 − ρ , ˜α ˜(T, ·). Define η˜α : Rd+1 × RM 7→ Rd+1 as and in a similar way, ˜bα ˜α k,c k , fk , u xd+1 xd+1 η˜α (x0 , xd+1 ; z) = ηˆα (x0 , z), 0 ρ + η α (x0 , z), 0 1 − ρ . ˜α ˜α and u We would like to show that, σ ˜kα , ˜bα ˜α ˜(T, ·) all satisfy the assumptions k,c k , fk , η k ¯ (A.1) and (A.3) in QT (d + 1). All other properties apart from Lipschitz continuity in (d + 1)-th direction are straight forward. For σ ˜kα along (d + 1)-direction we have ˆkα (t, x0 )ρ σ ˜kα (t, x0 , xd+1 ) := σ
α d+1 d+1 ∂σ α 0 α 0 0 x ˜k = 1 σ ≤ ρ0 ( x ˆ (t, x ) − σ (t, x ) ρ ( ) ) ≤ K. k k ∂xd+1 A similar conclusion holds for the other functions. ¯ T (d+1), which Therefore by Lemma 3.1 there exists a function u ˜τ,h , defined on M α ˜α α ˜α α solves (3.3) with the new family of coefficients σ ˜k , bk , c˜k , fk , η˜ and terminal data u ˜(T, ·). Furthermore, by uniqueness, we must have
u ˜τ,h (t, x0 , −) = u ˆ(t, x0 ),
u ˜τ,h (t, x0 , 0) = u ˜τ,h (t, x0 , ε) = u(t, x0 ).
Now we choose l = (0, 0, ...., 1), the unit vector along the (d + 1)-th direction. For this l, by Theorem 4.3 (since ε ≤ h1 ) we conclude that there exists a constant N , depending only on d, d1 , K, c0 , λ, T , and ν, such that u h i ˜τ,h (t, x0 , 0) 1 ˜τ,h (t, x0 , −) − u |δ,±l u ˜(T, x)| , ˜(T, x)| + ≤ N sup 1 + |δh1 ,lk u 1 ∨ |l| k,x,l which gives |ˆ u − u| ≤ N I.
Next we establish Lipschitz continuity property of vτ,h in the x-variable. To do ,S this, let S ⊂ B1 = x ∈ Rd : |x| < 1 be nonempty, ∈ R, and vτ,h be the unique solution of the equation h α δτT u + sup Lα h1 (t, x + y)u(t, x) + f (t, x + y) (α,y)∈A×S
+
X
i kp u(t, x + η α (x + y, p)) − u(t, x) = 0
p
with the terminal data (4.25)
u(T, x) = sup u0 (x + y). y∈S
in QT ,
DEGENERATE PARABOLIC INTEGRO-PDES
21
,S Note that if S is a singleton {y}, then by uniqueness vτ,h (t, x) = vτ,h (t, x + y).
Lemma 4.5. Assume (A.1), (A.3), (A.2), and (3.1),(3.2) hold. There is a constant N depending only on K, d1 , ν(E), T , so that if the condition (4.4) is satisfied for a constant c0 ≥ 0, then for every ∈ R, ,S |vτ,h − vτ,h | ≤ N1 ||
(4.26)
on
¯T , Q
where N1 only depend on K, d1 , ν(E), T, λ, c0 . Furthermore, by choosing S := { (y−x) |y−x| }, = |y − x|, we have (4.27)
|vτ,h (t, x) − vτ,h (t, y)| ≤ N1 |y − x|
for all
¯T . (t, y), (t, x) ∈ Q
¯ T . We use of Theorem 4.4, where we Proof. It is sufficient to prove (4.26) on M choose A × S, (σ, b, c, f, η) and (σ, b, c, f, η)(x + y, t/z) in places of A, (σ, b, c, f, η) and σ ˆ , ˆb, cˆ, fˆ, ηˆ, respectively. The contribution from the difference of the terminal data can be bounded by N . A step in the direction of establishing Theorem 3.4 is to prove Lemma 4.6. Assume (A.1), (A.2), (A.3) and (3.1),(3.2) hold. Let h1 , h2 , τ ≤ K. ¯ T and set Let (s0 , x0 ) ∈ M L :=
sup x∈RN ,x6=x0
|vτ,h (s0 , x) − vτ,h (s0 , x0 )| . |x − x0 |
Then for all (t0 , x0 ) ∈ MT satisfying s0 − 1 ≤ t0 ≤ s0 and
1 τT
∈ N, we have 1
|vτ,h (s0 , x0 ) − vτ,h (t0 , x0 )| ≤ N (L + 1)|s0 − t0 | 2 , where N depends only on K, d1 and ν(E). Proof. Without loss of generality we may restrict ourselves to the case 0 < s0 < 1 and t0 = 0. This claim follows by shifting the origin and observing that there holds 1 |vτ,h (s0 , x0 ) − vτ,h (t0 , x0 )| ≤ 2|vτ,h |0 |s0 − t0 | 2 whenever |t0 − s0 | > 1. ¯ s . On M ¯ s we define Fix a constant γ > 0. We are going to work with M 0 0 h i ψ = γL ζ + κ(s0 − t) + K(s0 − t) + γ −1 L + vτ,h (s0 , x0 ), where ξ(t) = es0 −t ,
β(x) = |x − x0 |2 ,
ζ = ξη,
and the constant κ will be chosen later. We will show that if κ is big enough, then ψ is a supersolution of (3.3). On Ms0 we have, δτso ξ = −θξ where, θ := τ −1 (1 − e−τ ) ≥ K −1 (1 − e−K ). Furthermore, using (A.1) – (A.3) we get X Lα kp β(t, x + η α (x, p)) − β(t, x) h1 β(t, x) + p
=
2 2aα k (t, x)|lk |
+
X
α + bα k (t, x)(lk , 2(x − x0 ) + h1 lk ) − c β(t, x)
kp η α (x, p); 2(x − x0 ) + η α (x, p)
p
≤ N1 (d1 , K, ν(E))(1 + |x − x0 |),
22
I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
and hence δτs0 ζ(t, x) + Lα h1 ζ(t, x) +
X
kp ζ(t, x + η α (x, p)) − ζ(t, x)
p
≤ N (d1 , K, ν(E))(1 + |x − x0 |) − θ(τ )|x − x0 |2 . Applying the same operator on ψ and using the above estimates, we have X α δτs0 ψ + Lα kp ψ(t, x + η α (x, p)) − ψ(t, x) h1 ψ + f + p
≤ γL N2 (1 + |x − x0 |) − θ|x − x0 |2 − κ + f α − K. 2
Since |f α | ≤ K and by suitably applying Young’s inequality: 2ab ≤ ra2 + br , it is clear that there exist κ depending only on N2 such that the right hand side of the last inequality is negative. So, for this choice of κ we have X i α α δτs0 ψ + sup Lα ψ + f + k ψ(t, x + η (x, p)) − ψ(t, x) ≤0 p h1 α∈a
p
and ψ(s0 , x) = L(γ|x − x0 |2 + γ −1 ) + vτ,h (s0 , x0 ) ≥ L|x − x0 | + vτ,h (s0 , x0 ) ≥ vτ,h (s0 , x). We now apply Lemma 3.2 on Ms0 and conclude that vτ,h (t, x0 ) ≤ ψ(t, x0 ) = γLκ(s0 − t) + γ −1 L + K(s0 − t) + vτ,h (s0 , x0 ). Minimizing with respect to the γ > 0 and using the fact (s0 − t) ≤ 1, we conclude 1
1
1
1
vτ,h (t, x0 ) − vτ,h (s0 , x0 ) ≤ 2Lκ 2 |s0 − t| 2 + Ks02 |s0 − t| 2 1
≤ N (d1 , T, K, ν(E))(L + 1)|s0 − t| 2 . The estimate for the other side is obtained similarly.
We close this section by giving the proof of Theorem 3.4. Lemma 4.7. Assume (A.1), (A.2), (A.3),(3.1), (3.2), h1 , h2 , τ ≤ K, and L :=
sup ¯ T ,x6=y (t,x),(t,y)∈Q
|vτ,h (t, x) − vτ,h (t, y)| . |x − y|
Then we have 1
1
|vτ,h (s, x) − vτ,h (t, x)| ≤ N (1 + L)(|s − t| 2 + τ 2 ), where N depends only on K, d1 and ν(E). 1
Proof. If |t−s| ≥ 1, then |vτ,h (t, x)−vτ,h (s, x)| ≤ 2|vτ,h |0 |t−s| 2 . We may therefore assume |t − s| ≤ 1. Assume (without loss of generality) that s − t = nτ + γ where γ ∈ [0, τ ) and n is a natural number. If γ = 0, then we apply Lemma 4.6 on (t, 0) + Mnτ and conclude (4.28)
1
|vτ,h (t, x) − vτ,h (t + nτ, x)| ≤ N |nτ | 2 .
Now, for other case when 0 < γ < τ we have, |vτ,h (t, x) − vτ,h (s, x)| ≤ |vτ,h (t, x) − vτ,h (t + nτ, x)| + |vτ,h (s − γ, x) − vτ,h (s, x)| 1
≤ N |nτ | 2 + |vτ,h (s − γ, x) − vτ,h (s, x)|.
DEGENERATE PARABOLIC INTEGRO-PDES
23
Therefore, we have to estimate |vτ,h (s−γ, x)−vτ,h (s, x)| for γ ∈ (0, τ ) and s−γ ≥ 0. ¯ T −s : Define the following functions on (s, 0) + M α α α α ˆα ˆα [ˆ σkα , ˆbα k , c , f ](r, y) = [σk , bk , c , f ](r − γ, y)
and u = vτ,h ; u ˆ(r, y) = vτ,h (r − γ, y), ¯ ¯ T −s , u for all (r, y) ∈ (s, 0) + MT −s . Then, on (s, 0) + M ˆ satisfies (3.3) constructed α ˆα ˆα ˆα from σ ˆk , bk , c , f and unchanged jump amplitudes η α . Noticing the fact that the 1 parameter in Theorem 4.4, after using (A.3), is less than N γ 2 and also using the ¯ T −s , x-Lipschitz continuity of vτ,h we have, after applying Theorem 4.4 on (s, 0)+ M (4.29)
|vτ,h (s, x) − vτ,h (s − γ, y)| = |u(s, x) − u ˆ(s, y)| 1
ˆ(T, y)| ≤ N γ 2 + sup |u(T, y) − u y∈Rd 1 2
= N γ + |vτ,h (T, x) − vτ,h (T − γ, y)|. Lastly, we are left with estimating |vτ,h (T, x) − vτ,h (T − γ, y)|. To this end, ¯τ consider the grid Mτ . With a slight abuse of the notation, we define u ˆ(r, x) on M by u ˆ(0, x) = vτ,h (T − γ, x); u ˆ(τ, x) = vτ,h (T, x). Then u ˆ solves (3.3) on Mτ , with an obvious shift (by a quantity t−γ in the backward direction) in the time variable of the coefficients, and therefore by Lemma 4.6 we must have 1 |ˆ u(τ, x) − u ˆ(0, x)| ≤ N τ 2 , i.e., (4.30)
1
|vτ,h (T, x) − vτ,h (T − γ, x)| ≤ N τ 2 .
Finally, we conclude by combining the estimates (4.28), (4.29), and (4.30).
Proof of Theorem 3.4. Estimates (3.8), (3.9) follow from Theorem 4.4, and estimate (3.10) from Lemma 4.7 if τ is small enough (then L < ∞ by Theorem 4.3). ´vy measures and optimal error bounds in a special case 5. Singular Le In this section we address the case of unbounded (singular) L´evy measures. Specifically, in a special case, we introduce a modified difference-quadrature scheme for which we obtain an optimal (under our assumptions) convergence rate. In general the L´evy measure need not be bounded and/or compactly supported, but it always satisfies the condition Z Z 2 (5.1) |z| ν(dz) + eK|z| ν(dz) < ∞, |z|≤1
|z|>1
for some constant K ≥ 0. Under this condition, the jump amplitude η α must satisfy 1 (5.2) |η α (x, z)| + sup |η α (x + h, z) − η α (x, z)| ≤ N |z|1|z|≤1 + eK|z| 1|z|>1 . h |h|>0 Conditions (5.1) and (5.2) along with (A.1) and (A.3) ensure that the underlying stochastic control problem is well-defined. Moreover, the initial value problem (1.1) and (1.2), with I defined in (1.3), possesses a unique H¨older continuous viscosity
24
I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN
solution. We refer to [16] for the proof of this result and for the precise definition of viscosity solutions in this setting. To solve such problems numerically the first step is often (see, e.g., [12, 18]) to approximate the L´evy measure ν by a finite and compactly supported measure of the form νr,R (z) = 1r 1}, we obtain the error bound |I α w − Ihα w| ≤ N (|Dx3 w|0 + |Dx w|0 )h2 + |Dx2 w|0 h1 . (5.8) Denote by vτ,h the unique solution of (5.6), (1.2) and by v the unique viscosity solution of (1.1), (1.2). In view of (5.8) and the discussion above, if we repeat the proof of Theorem 3.5 we will eventually find that 1 1 1 1 h1 τ + h21 1 + ) ≤ N (τ 4 + h12 + h23 ), |v − vτ,h | ≤ N min( + h2 ( + 2 ) + 3 for sufficiently small τ, h1 , h2 . Summarizing, we have proved
Theorem 5.1. Assume that (A.1), (A.3), (5.1), (5.2), (5.3), (5.4), (5.5) hold. Let v and vτ,h be the solutions of (1.1), (1.2) and (5.6), (1.2), respectively. Then there exists a constant N , depending only on d, d1 , and K, such that 1
1
1
|v − vτ,h | ≤ N (τ 4 + h12 + h23 ). Remark 5.1. Theorem 5.1 appears to be the first result on convergence rates for numerical schemes of Bellman equations with singular L´evy measures. Note that the convergence rate in Theorem 5.1 does not depend on the strength of the singularity at z = 0 of the L´evy measure. Furthermore, the result is most likely optimal under the current conditions. However, if we have further information about ν, e.g., if ν(dz) ≤ N |z|−γ−M dz in a neighborhood of z = 0 for some γ ∈ (0, 2), then the rate 1/3 can be improved. We leave this to the interested reader. References [1] O. Alvarez and A. Tourin. Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 13(3):293–317, 1996. [2] A. L. Amadori. Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach. Differential Integral Equations, 16(7):787–811, 2003. [3] G. Barles, R. Buckdahn, and E. Pardoux. Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep., 60(1-2):57–83, 1997. [4] G. Barles and E. R. Jakobsen. On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations. M2AN Math. Model. Numer. Anal., 36(1):33–54, 2002. [5] G. Barles and E. R. Jakobsen. Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal., 43(2):540–558 (electronic), 2005. [6] G. Barles and E. R. Jakobsen. Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Math. Comp.,76(240):1861-1893, 2007. [7] G. Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal., 4(3):271–283, 1991.
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[8] F. E. Benth, K. H. Karlsen, and K. Reikvam. Optimal portfolio management rules in a nonGaussian market with durability and intertemporal substitution. Finance Stoch., 5(4):447– 467, 2001. [9] F. E. Benth, K. H. Karlsen, and K. Reikvam. Optimal portfolio selection with consumption and nonlinear integrodifferential equations with gradient constraint: a viscosity solution approach. Finance and Stochastic, 5:275–303, 2001. [10] F. E. Benth, K. H. Karlsen, and K. Reikvam. Portfolio optimization in a L´ evy market with intertemporal substitution and transaction costs. Stoch. Stoch. Rep., 74(3-4):517–569, 2002. [11] I. H. Biswas, E. R. Jakobsen, and K. H. Karlsen. Error estimates for finite differencequadrature schemes for a class of nonlocal Bellman equations with variable diffusion. Contemp. Math., volume 429, Amer. Math. Soc., pages 19-31, 2007. [12] R. Cont and P. Tankov. Finacial modeling with jump processes. Chapman & Hall/CRC, Boca Raton, FL, 2004. [13] M. G. Crandall, H. Ishii, and P.-L. Lions. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 27(1):1–67, 1992. [14] W. H. Fleming and H. M. Soner. Controlled Markov processes and viscosity solutions. Springer-Verlag, New York, 1993. [15] E. R. Jakobsen. On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems. Math. Models Methods Appl. Sci., 13(5):613– 644, 2003. [16] E. R. Jakobsen and K. H. Karlsen. Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differential Equations, 212(2):278–318, 2005. [17] E. R. Jakobsen and K. H. Karlsen. A ”maximum principle for semicontinuous functions” applicable to integro-partial differential equations. NoDEA Nonlinear Differential Equations Appl., 13:137-165, 2006. [18] E. R. Jakobsen, K. H. Karlsen, and C. L. Chioma. Error estimates for approximate solutions to Bellman equations associated with controlled jump-diffusions. Submitted to Numer. Math., 2005. [19] N. V. Krylov. On the rate of convergence of finite-difference approximations for Bellman’s equations. Algebra i Analiz, 9(3):245–256, 1997. [20] N. V. Krylov. On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients. Probab. Theory Related Fields, 117(1):1–16, 2000. [21] N. V. Krylov. The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients. Appl. Math. Optim., 52(3):365–399, 2005. [22] R. Mikulyavichyus and G. Pragarauskas. Nonlinear potentials of the Cauchy-Dirichlet problem for the Bellman integro-differential equation. Liet. Mat. Rink., 36(2):178–218, 1996. [23] H. Pham. Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Systems Estim. Control, 8(1):27 pp. (electronic), 1998. (Imran H. Biswas) Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway E-mail address:
[email protected] (Espen R. Jakobsen) Norwegian University of Science and Technology, N-7491, Trondheim, Norway E-mail address:
[email protected] URL: www.math.ntnu.no/~erj/ (Kenneth Hvistendahl Karlsen) Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway E-mail address:
[email protected] URL: folk.uio.no/kennethk/
