Parametrix approximations for non constant coefficient parabolic PDEs

Transcription

1 MPA Munich Personal epec Archive Parametri approimations for non constant coefficient parabolic PDEs Paolo Foschi and Luca Pieressa and Sergio Polidoro Dept. Matemates, Univ. of Bologna, Italy, Dpt. of Mathematics, Univ. of Bologna, Italy 0. March 008 Online at MPA Paper No. 7943, posted 7. March 008 1: UTC

2 Parametri approimations for non constant coefficient parabolic PDEs L. Pieressa P. Foschi S. Polidoro March 5, Introduction Closed form epressions for the fundamental solution of parabolic partial differential equations (PDE are needed in several application domains ranging from Chemistry to Physics, from Statistics to Finance [, 3, 5, 6]. In this paper a closed form approimation for the fundamental solution of the following parabolic PDE operator is derived LU(, t := a( U(, t t U(, t = 0, (1 on Ω = [0, t]. The operator is uniformly parabolic and its coefficient function a is bounded and Hölder regular. Precisely, it is assumed that H.1 there eists m and M such that 0 < m a( M <, ; H. there eist α (0, 1 and C such that a( a(y C y α,. Under H.1 and H. the fundamental solution Γ(z; ζ = Γ(, t; ξ, τ of (1 eists and is unique [4]. When a is constant, a( v, the fundamental solution is given by Γ v ( ξ, t τ, where Γ v (, t := 1 ep (. 4πvt 4vt In general however, an eplicit epression is not always available and approimations need to be computed. A method for approimating Γ(z; ζ consists on using Levi s parametri series epansion, recently reconsidered as a computational method by Corielli and Pascucci for Dip. di Matematica, Univ. di Bologna P.le Porta S. Donato, Bologna, Italy Dept. Matemates, Univ. of Bologna Viale Filopanti, Bologna, Italy Dip. di Matematica G. Vitali, Univ. di Modena e eggio Emilia Via Campi, 13/b Modena, Italy 1

3 financial applications [3]. The parametri epansion epresses Γ(z; ζ in terms of the fundamental solution Z(z; ζ of the corresponding frozen PDE a(ξu (, t U t (, t = 0, given by Z(, t; ξ, τ = Γ a(ξ ( ξ, t τ. Specifically, Γ is given by the series epansions Γ(z; ζ = Z(z; ζ + n Z k (z; ζ + E n+1 (z; ζ, ( k=1 where Z k (z; ζ := t τ Z(z; z 0 (LZ k (z 0 ; ζdz 0, (3 with (LZ k defined by (LZ 1 (z; ζ = LZ(z; ζ, (LZ k (z; ζ = t τ LZ(z; w(lz k 1 (w; ζdw. The error term E n+1 is of the order O((t τ n+ Γ M+ɛ (z, ζ, with ɛ > 0. In the following Γ(z; ζ is approimated by using the first three terms of the parametri series in (: Γ(z; ζ = Z(z; ζ + Z 1 (z; ζ + Z (z; ζ + O(t Γ M+ɛ (z; ζ. (4 where, for notational convenience and without loss of generality, it has been considered only the case ζ = (ξ, 0, that is τ = 0. The main practical it of the parametric series consists in the fact that Z k is defined in terms of a k dimensional integral which, in general, cannot be computed eplicitly. Here a closed form approimation to Z 1 and Z is derived. That approimation has errors not eceeding the remainder in (4. With this purpose let rewrite where Z 1 (z; ζ = t 0 I 1 (t 0 ; z, ζdt 0 and Z (z; ζ = I 1 (t 0 ; z, ζ = I (t 0, t 1 ; z, ζ = t t0 0 0 I (t 0, t 1 ; z, ζdt 1 dt 0, (5 Z(z; 0, t 0 LZ( 0, t 0 ; ζd 0, (6 Z(z; z 0 LZ(z 0 ; z 1 LZ(z 1 ; ζd 0 d 1, (7 and z i = ( i, t i, i = 0, 1. Now, notice that some of the factors in the integrands in (6 and (7 tend to Dirac s deltas at the corners of the domains of the integrals in (5. Thus, at that points the computation of I 1 (t 0 ; z, ζ and I (t 0, t 1 ; z, ζ reduces to simple

4 function evaluations. Using these values, Z 1 and Z are approimated by a trapezoidal rule as follows and Ẑ 1 (z; ζ = t ( I1 (0 + ; z, ζ + I 1 (t ; z, ζ (8 Ẑ (z; ζ = t 6 ( I (0 +, 0 + ; z, ζ + I (t, 0 + ; z, ζ + I (t, t ; z, ζ, (9 where f( + and f( denote, respectively, the it to from right and left. The errors of approimations in (6 and (7 are, respectively, of order O(t 3 I 1,(0,t and O(t 3 ( t0 t 0 + t0 t 1 + t1 t 1 I,Dt, where D t = {(t 0, t 1 0 < t 0 < t and 0 < t 1 < t 0 } is the domain of the second integral in (5 1. The paper is structured as follows, in net section some results concerning its of Gaussian and parametri functions Γ v and Z 1 are reported. These results will be used in section 3 to derive epressions for the terms appearing in (8 and (9. Preinary results The first lemma gives point-wise bounds on the non-constant parameter Gaussian function in terms of constant parameters Gaussians. Γ(, t := Γ g( (, t, Lemma 1. Let g( bounded, that is 0 < m g( M <, then m M M Γ m(, t Γ g( (, t m Γ M(, t, and t > 0. Proof. See [3]. The following results characterize the its of Γ v (, t and Γ(, t when t 0. Lemma. Let f( have it for 0, v > 0 and g( satisfying H.1 and H., f(γ v (, td = f( Γ(, td = f( (10a t 0 t 0 0 { n t 0 v n/ t n/ Γ v(, td = n (k!/k!, n = k, γ((n + 1/ = π k+1 k!/ (10b π, n = k + 1. n 0, if k < n, t 0 g( k/ t Γ(, k/ td = finite and positive, if k = n and (10c, if k > n. where γ is the gamma function [1]. 1 Actually, it can be proved that the errors are of order O(t 3 I 1 Z(z; ζ and O(t 3 I 1 Z(z; ζ. 3

5 Proof. By Lemma 1, equation (10c is a direct consequence of (10b and of the boundedness of g(. emark 1. Notice that (10b is valid not only at the it t 0 but also for any t > 0. In the previous Lemma, (10a characterizes the it of Γ v (, t and Γ(, t as a Dirac s delta. Equations (10b and (10c state that n and t n/ are of the same order when integrated with a Gaussian function. The following Lemma derives a result analogous to point 3 of Lemma for Γ(, t. Lemma 3. Under the hypothesis of Lemma and assuming g continuous and derivable, with bounded first derivative near 0, then H n = t 0 n g( n/ t Γ(, n! n/ td = (n/!, n even, (11 0, n odd, and H n = t 0 1 n+1 t 1/ (g(t Γ(, n+1/ td = (n + 1! n! g (0. (1 g(0 Proof. By (10c in Lemma the it (11 is finite, but in a 0/0 indeterminate form. Thus, let rewrite the it as t H n = g( n/ n Γ(, td t 0 (n/t n/ 1. In the above it, the derivative of Γ(, t is given by ( t Γ(, t = 4g(t 1 Γ(, t, t so that H n becomes ( 1 H n = t 0 n+ n g( n/+1 t n/+1 n Γ(, ng( n/ t n/ td = 1 n H n+ 1 n H n, and thus H n+ = (n + 1H n. Now, since H 0 = 1 it follows that for n even H n = n/ (n 1!! = (n!/(n/!. For n odd, H n = 0 is a direct consequence of (1. Let thus consider (1. Proceeding as above it can be proved that H n = 1 H 4(n+1 n+1 1 H (n+1 n, so that H n = (n+1! n! H 0. It remains to show that H 0 = t 0 t g( Γ(, td = g (0 g(0. 4

6 Since g is bounded, the it can be rewritten as ɛ H 0 = t 0 ɛ t Γ(, td, g( for any ɛ > 0. Furthermore, again by assuming boundness and continuity of g, ɛ > 0 can be choosen such that / g( is monotone in the domain of integration. This allows to consider the change of variables y = / g(, ( dy d = 1 1 g ( g( g( and to rewrite the integral as ɛ ɛ t Γ(, td = g( = = ɛ ɛ ɛ ɛ ɛ ɛ t 1 e g( 4πg(t y t e y 4t 4πt dy + ɛ ɛ t g( g ( g( tg( Γ(, td 4g(t d ( g ( g( 1 e 4πg(t 4g(t d which, by boundedness and continuity assumptions on g, tends to g (0/ g(0 for t 0. Corollary 1. Under the same assumptions of Lemma 3, if g C with g, g and g bounded near the origin, then n+1 t 0 (g(t Γ(, n+1 td = 0. Proof. By hypothesis g and g are bounded in [ ɛ, ɛ], so that, in that interval, g 1/ and g 1/ can be approimated as g( 1/ = g(0 1/ 1 g (0g(0 3/ + O( and g( 1/ = g(0 1/ + O(. Using these approimations it holds g(t = g( 1 g( 1 t = g(0 1 g( 1 t 1 g (0 g( 1 g(0 3 = g(0 1 g( 1 t 1 g (0 g(0 1 g(0 3 g(t + O(3 /t, g(t + O(3 /t thus, ( ɛ t 0 g(t Γ(, td = g(0 1 t 0 ɛ g( 1 t 1 g (0 g(0 1 g(0 3 g(t + O(3 /t Γ(, td The first derivative of / p g( is null in = 0 if and only if 0 g( g ( = 0, a case ruled out because, by hypothesis, g > 0 and g (0 <. 5

7 by Lemma and 3 = 1 g (0 1 g (0 g(0 g(0 g(0 = 0. Finally, by proceeding in a manner analogous to the proof of Lemma 3, the thesis follows for all n. Now, let resume rules for computing its of integrals involving the parametri Z. These results are used in the net section together with those provided in the corollary. Lemma 4. If a( satisfy H.1 and H. then f(, tz(, t; ξ, τd = f(, t, t τ + τ t t τ + ξ f(ξ, τz(, t; ξ, τdξ = τ t ξ f(ξ, τ, (13a (13b provided that the its in the HSs eist and are unique. Furthermore, for n even ( ξ n ( ξ n n! Z(, t; ξ, τd = Z(, t; ξ, τdξ = t τ + (a(ξ(t τ n/ τ t (a(ξ(t τ n/ (n/!. (14a and, for n odd t τ + ( ξ n Z(, t; ξ, τd = (a(ξ(t τ n/ τ t ξ n n Z(, t; ξ, τd = τ t (a(ξ(t τ n/ ( n 1 π ( ξ n Z(, t; ξ, τdξ = 0, (a(ξ(t τ n/ (14b! (14c and τ t ξ n Z(, t; ξ, τdξ < + (a(ξ(t τ n/ (14d Proof. Direct consequence of Lemma and Lemma 3. Corollary. With the same hypothesis of Lemma 4, if f C,1 and 0 and assuming a( twice diffrentiable near = ξ, then ( f(, tlz(, t; ξ, τd = (a( a(ξf(, t t τ + t τ + ξ f(, tγ(, t = ± (15a 6

8 and f(ξ, τlz(, t; ξ, τdξ = (a ( + a ( f(, t. τ t (15b Proof. In order to prove (15a, let notice that LZ = (a( t Z = a( Z a(ξ Z so that (15a follows by integrating by parts: f(, tlz(, t; ξ, τd = f(, t(a( a(ξ Z(, t; ξ, τd ( = (a( a(ξf(, t Z(, t; ξ, τd. The last results, equation (15b, is a bit more involving. Let rewrite LZ(z; ζ = A(z; ζz(z; ζ, where ( a( a(ξ ( ξ A(, t, ξ, τ := a(ξ(t τ 4a(ξ(t τ 1 (16 By a Taylor epansion of a(ξ centered in, A can be rewritten as ( A(z; ζ = a ( ξ ( a(ξ(t τ a ( ( ξ ( ( ξ a(ξ(t τ 4a(ξ(t τ 1 + O ( ( ξ 3 t τ so that, from the Taylor epansion f(ξ, τ = f(, τ ( ξf (1,0 (, τ + O( ξ and defining y = ( ξ/ a(ξ(t τ it gives f(ζa(z; ζ = 1 8 a ((y 4 y f(, τ + 1 (y 3 y 4 a ( f(, τ a(ξ(t τ, Now, since by (14a τ t 1 4 a ((y 4 y f (1,0 (, τ + O(( ξ 3 /(t τ (y 4 y Z(, t; ξ, τdξ = 8, and, by Corollary 1, y n+1 Z(, t; ξ, τdξ = 0, n = 0, 1, τ t a(ξ(t τ it follows that τ t f(ζa(z; ζz(z; ζdξ = a ( f(, t a (f(, t, assuming continuity of f and f (1,0 near (, t. The latter requirement can be weakened by replacing f(, t and f (1,0 (, t with the corresponding its and assuming eistence and unicity of these. 7

9 3 Approimating Z 1 and Z In the following it is assumed that the coefficient a( satisfies assumptions H.1 and H. and has enough regularity near or ξ. 3.1 Computing I 1 (t and I 1 (0 + By rule (13b of Lemma 4 it follows that I 1 (t ;, t; ξ, 0 = t 0 t Z(, t; 0, t 0 LZ( 0, t 0 ; ξ, 0d 0 = LZ(, t; ξ, 0, (17 for t > 0. The second it I 1 (0 + ;, t; ξ, 0 = t Z(, t; 0, t 0 LZ( 0, t 0 ; ξ, 0d 0, t > 0, is computed by using rule (15a in Corollary, which gives I 1 (0 + ;, t; ξ, 0 = a (ξz(, t; ξ, 0 + a (ξ ξ Z(, t; ξ, 0 = (a (ξ + (a (ξ ( ( ξ 1 a(ξ 4a(ξt 3. Computation of I (t, t and I (t, a (ξ ξ a(ξt Let consider the it I (t 0, t 1, by Lemma it follows that t 0 t I (t, t 1 = Z(, t; 0, t 0 LZ(z 0 ; z 1 LZ(z 1 ; ζd 0 d 1 t 0 t = LZ(z; z 1 LZ(z 1 ; ζd 1. Z(, t; ξ, 0. (18 Now, the its (t, t 1 and (t, t 1 are tackled by means of Corollary, t t 1 t equations (15a and (15b, respectively. That is, I (t, 0 + = t LZ(, t; 1, t 1 LZ( 1, t 1 ; ξ, 0d 1 = ( a (ξ + a (ξ ξ LZ(, t; ξ, 0 = (a (ξ + (a (ξ ( ( ξ 1 a(ξ 4a(ξt + a (ξ ξa(, t; ξ, 0 A(, t, ξ, 0 + a (ξ ξ a(ξt LZ(, t; ξ, 0 (19 and I (t, t = LZ(, t; 1, t 1 LZ( 1, t 1 ; ξ, 0d 1 t 1 t = ( a ( + a ( LZ(, t; ξ, 0 (0 8

10 Now, since LZ(, t; ξ, τ = L Z(, t; ξ, τ + a ( Z(, t; ξ, τ = L ξ a Z(, t; ξ, τ + a(ξt = ξ a(ξt LZ(, t; ξ, τ 1 t ( LZ(, t; ξ, τ a( a(ξ ξ a Z(, t; ξ, τ + a(ξt ( LZ(, t; ξ, τ, a( a(ξ the it in (0 can be rewritten as ( I (t, t = a ( a ( ξ a(ξt + a ( LZ(, t; ξ, 0 a( a(ξ 3.3 Computing I (0 +, a ( t ξ Z(, t; ξ, τ, a(ξt Before deriving an epression for I (0 +, 0 + it is convenient to prove the following Lemma. Lemma 5. Let assume f( C on B = [ξ ɛ, ξ + ɛ], then f( ξ Z(, t; ξ, τd = f (ξ. (1 Proof. Firstly, ξ Z(, t; ξ, τ = t τ ( a ( (ξ ( ξ a(ξ 4a(ξ(t τ 1 ξ + Z(, t; ξ, τ. a(ξ(t τ Then, from f( = f(ξ + ( ξf (ξ + O(( ξ f,b it follows that (1 can be rewritten as a f( ξ Z(, t; ξ, τd = t τ B t τ B ( (ξ ( ξ a(ξ 4a(ξ(t τ 1 f(ξz(, t; ξ, τd ( ξ + t τ B a(ξ(t τ f (ξz(, t; ξ, τd. where it has been used the fact that integral of odd terms is null and that the remaining omitted terms vanish for t τ by virtue of Lemma 4, equation (14a. The result in (1 follows by noting that by (14a the first it converges to 0 and the second one to f (ξ. Let consider now the computation of I (0 +, 0 +. From (15a it follows that I (t 0, 0 + = Z(, t; 0, t 0 LZ(z 0 ; z 1 LZ(z 1 ; ζd 0 d 1 t1 0 = Z(z; z 0 (a (ξ + a (ξ ξ LZ(z 0 ; ζd 0. 9

11 so that, I (0 +, 0 + = a (ξz(z; z 0 LZ(z 0 ; ζd 0. + a (ξz(z; z 0 ξ LZ(z 0 ; ζd 0 t0 0 t0 0 [ ] = a (ξ 0 0 (a( 0 a(ξz(z; 0, 0 + a (ξc 0 =ξ where C = t0 0 Z(z; z 0 ξ LZ(z 0 ; ζd 0. Now, since LZ(z 0 ; ζ = (a( 0 a(ξ 0 0 Z(z 0, ζ, C can be rewritten as C = a (ξz(z; z Z(z 0 ; ζd 0 t0 0 + Z(z; z 0 (a( 0 a(ξ 0 0 ξ Z(z 0 ; ζd 0 t0 0 ( = a (ξ ξξ Z(z; ζ (a( 0 a(ξz(z; z 0 ξ Z(z 0 ; ζd 0 t0 0 [ ] = a (ξ ξξ Z(z; ζ (a( 0 a(ξz(z; 0, 0 Thus, I (0 +, 0 + = eferences. 0 =ξ [ ] (a (ξ + a (ξ (a( 0 a(ξz(z; 0, 0 a (ξ ξξ Z(z; ζ. 0 =ξ ( [1] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, ninth dover printing, tenth gpo printing ed., National Bureau of Standards Applied Mathematics Series, vol. 55, Dover, M M (9 #4914 [] Yacine Aït-Sahalia, Maimum likelihood estimation of discretely sampled diffusions: a closed-form approimation approach, Econometrica 70 (00, no. 1, 3 6. M M19660 (004c:617 [3] F. Corielli and A. Pascucci, Parametri approimations for option prices, preprint AMS Acta, Università di Bologna (available on-line at pascucci/web/icerca/pdf/parametri.pdf (007. [4] Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall Inc., Englewood Cliffs, N.J., M 31 #606 10

12 [5] Nancy Makri and William H. Miller, Eponential power series epansion for the quantum time evolution operator, The Journal of Chemical Physics 90 (1989, no., [6] L. C. G. ogers and O. Zane, Saddlepoint approimations to option prices, Ann. Appl. Probab. 9 (1999, no., M M (000b: