Morfismos, Vol 24, Num 1, 2020 by Morfismos, Department of Mathematics, Cinvestav

VOLUMEN 24 NÚMERO 1 ENERO A JUNIO DE 2020 ISSN: 1870-6525

Morfismos Departamento de Matemáticas Cinvestav

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Morfismos

Contents - Contenido Alternative to Euler’s formula for Bernoulli numbers

∞

1 n=1 n2k

with k ∈ Z + and for even indexed

E. Salinas-Hernández, Abelardo Santaella-Quintas, Martha P. Ramı́rez-Torres and Gonzalo Ares de Parga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Computations of hitting time densities for the generalized Cox-Ingersoll-Ross diffusion Jonathan Gutierrez-Pavón . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Morfismos, Vol. 24, No. 1, 2020, pp. 1–14 Morfismos, Vol. 24, No. 1, 2020, pp. 1–14

∞

Alternative to Euler’s formula for n=1 n12k with + k∈ Z and for indexed Bernoulli 1 Alternative to even Euler’s formula for ∞numbers with n=1 n2k + 1 k E.∈Salinas-Hernández Z and for even indexed Bernoulli numbers Abelardo Santaella-Quintas 2 Gonzalo Ares de Parga Martha P. Ramı́rez-Torres 1 E. Salinas-Hernández Abelardo Santaella-Quintas 2 Gonzalo Ares de Parga Martha P. Ramı́rez-Torres Abstract This paper proposes an alternative to get an orig ∞ 1 mechanism + inal result for the expression Abstract , k ∈ Z , the first related 2k n=1 n result was obtained by Leonhard Euler in 1732; later, we will be This paper proposes an alternative mechanism to get an orig ∞ able to reproduce even indexed Bernoulli numbers from both reinal result for the expression n=1 n12k , k ∈ Z + , the first related sults. result was obtained by Leonhard Euler in 1732; later, we will be able to reproduce indexed Bernoulli numbers both re2010 Mathematics Subject even Classification: 11M06, 11B68,from 41A58, 42A16. sults.

Keywords and phrases: Bernoulli numbers, alternative to Euler’s formula. 2010 Mathematics Subject Classification: 11M06, 11B68, 41A58, 42A16. Keywords and phrases: Bernoulli numbers, alternative to Euler’s for1 mula. Introduction Euler’s for the first time in 1736, gives an answer for obtained 1 formula, Introduction 1 , nowdays known as the Riemann Zeta Function the sum value +∞ n=1 n2k evaluated even positive integers It isininteresting toan note that for Euler’sat formula, obtained for theζ(2k). first time 1736, gives answer +∞ 1 Euler’s formula has its origin in the Basilea problem, which consists the sum value n=1 n2k , nowdays known as the Riemann Zeta Function in finding sum of the reciprocals squares oftopositive evaluated atinfinite even positive integers ζ(2k). of It the is interesting note that the +∞ 1 integers . The problem, raised to Leibniz by Oldenburg who 2 Euler’s n=1 formula has its origin in the Basilea problem, which consists n wasinsecretary of the Royal Society in 1673, had been addressed years finding the infinite sum of the reciprocals of the squares of positive +∞ 1 and byWallis. This problem was also addressed agointegers byPietro Mengoli n=1 n2 . The problem, raised to Leibniz by Oldenburg who by Jacob and Johan Bernoulli, who tried to attack using triangular was secretary of the Royal Society in 1673, had itbeen addressed years numbers, but eventually realizing that such a path could not leadaddressed them ago byPietro Mengoli and byWallis. This problem was also 1by Jacob and Johan Bernoulli, who tried to attack it using triangular SNI I CONACYT, EDI and COFAA in the IPN. 2 numbers, but eventually realizing that suchRamı́rez a pathTorres couldtonot leadthe them This work is part of the thesis of Martha Paola obtain Bachelor’s degree of Physics and Mathematics. The thesis advisors are E. Salinas 1 SNI(ESCOM-IPN) I CONACYT, and EDIAbelardo and COFAA in theQuintas IPN. (ESFM -IPN). Hernández Santaella 2 This work is part of the thesis of Martha Paola Ramı́rez Torres to obtain the Bachelor’s degree of Physics and Mathematics. The thesis advisors are E. Salinas Hernández (ESCOM-IPN) and Abelardo 1 Santaella Quintas (ESFM -IPN).

Salinas-Hernández et al

to the correct answer. The problem obsessed Jacob Bernoulli to such an extent that he ended up launching it as an open challenge. It should be noted that he proved the non convergence of the harmonic series (1)

1 1 1 + + + · · · = ∞. 2 3 4

There are works ([1]) where the proof of Euler’s formula is done by 1 π2 applying probabilistic techniques. In [5] it is shown that ∞ n=1 n2 = 6 while in [3] a proof is given directly. Moreover, there are results as in [4] where it is found through formulas which are proved by induction. On the other hand, in [6] and [8], in addition to giving a short proof, they also offer a general expression to calculate ζ(2k + 1). This work is developed as follows: In the second section we start by giving a synthesized explanation of how Euler came up with the expression for ζ(2). The section closes by presenting the first general expression for ζ(2k), which was obtained by Euler in 1748. In the third section we develop our innovative proposal which involves Fourier series and, with it, we obtain a new general formula to calculate ζ(2k). In the fourth section we give an expression to find B2k .

Mechanism used by Euler to generate ζ(2)

By the year 1731, the prodigious Leonhard Euler, had already been able to calculate the first 20 digits of the problem (2) 1 +

1 1 1 1 1 + + + + ··· + = 1.6439345666815598031 22 32 42 52 10002

and of course, it goes without saying that at that time there were no mechanical devices so sophisticated to make such enormous sums and certainly much less calculators. In order to achieve this, he started from the well known geometric series, already well established for that time (3)

1 = 1 + x + x2 + x 3 + x 4 + · · · + x k + · · · 1−x

that converges as long as |x| < 1, which by integrating, can lead to (4)

− ln (1 − x) = x +

xk x2 x3 x4 + + + ··· + + ··· 2 3 4 k

∞

1 n=1 n2k , k

An alternative for

∈ Z+

1 dx, and after careful integrathen integrating the expression 02 ln (1−x) x tion by parts, Euler manages to come up with the famous expression ∞ ∞ 1 1 = + [ln 2]2 2k k−1 n 2 k2

(5)

n=1

k=1

to find the expression for ζ(2) (see [2]). Euler starts from the Taylor series expansion of the function sin x, it is: (6)

sin x = x −

x2k−1 x 3 x5 x7 + − + · · · + (−1)k−1 + ··· 3! 5! 7! (2k − 1)

then dividing by x(x = 0), we obtain (7)

x2k−2 x2 x4 x6 sin x =1− + − + · · · + (−1)k−1 + ··· . x 3! 5! 7! (2k − 1)

Now, considering the change z = x2 , we get (8)

1−

z k−1 z z2 z3 + − + · · · + (−1)k−1 + ··· . 3! 5! 7! (2k − 1)

Since the roots of the sin x function are: ±π, ±2π, ±3π..., ±nπ, then the roots of the previous expression will be ±π 2 , ±4π 2 , ±9π 2 ..., ±n2 π 2 , it is,

(9)

1−

z 3!

= 1−

z2 5!

x2 π2

z3 7!

−

k−1

z + · · · + (−1)k−1 (2k−1) + ···

1−

x2 4π 2

1−

x2 9π 2

··· 1 −

x2 n2 π 2

···

and therefore, we will have that x2 x2 x2 x2 sin x = 1− 2 1− 2 1 − 2 ··· 1 − 2 2 ··· . (10) x π 4π 9π n π Equating the second coefficient of (9) with the corresponding quadratic term in the previous expression, and after careful calculation, we obtain (11)

1 1 1 1 1 1 = 2+ 2+ 2+ + ··· + 2 2 + ··· . 2 3! π 4π 9π 16π n π

Equivalently, (12)

1 1 1 1 1 1 1 = = 2 1+ + + + ··· + 2 + ··· , 3! 6 π 4 9 16 n

Salinas-Hernández et al

from where it is directly obtained ∞ 1 π2 . = ζ(2) = n2 6

(13)

n=1

Due to some critiques he received (for example from Johan Bernoulli in April 1737 where he points out some deficiencies about the proof such as that the only roots of sinx x = 0, were x = nπ, with n = ±1, ±2, ±3, . . .), Euler found other more convincing solutions about this result to finally publish the generalization in his work Introductio in analysin infinitorum in 1748. This was done through the expression (14)

∞ 1 (−1)k−1 (2π)2k B2k , = ζ(2k) = 2(2k!) n2k

n=1

where B2k corresponds to even indexed Bernoulli numbers.

Formula development

It is well known that Jean-Baptiste Joseph Fourier (1768 − 1830) who publishes in 1822 his Theorie analytique de la chaleur (Analytical theory of heat), a treatise in which he established the partial differential equation that governs the diffusion of heat giving it a solution through the use of infinite series of trigonometric functions, introduced the concept that later will be called the Fourier expansion of analytic functions; in fact, any analytic function f (x) defined on the interval [−L, L] can be expressed as an infinite series expansion of functions of sines and cosines (15)

nπx nπx a0 f (x) = an cos + + bn sin 2 L L ∞

n=1

with

(16)

a0 =

1 L

an =

1 L

bn =

1 L

−L f (x)dx

−L f (x) cos

−L f (x)sin

nπx L

dx.

An alternative for

∞

1 n=1 n2k , k

∈ Z+

It is clear that Fourier never imagined that the implications of his results would have a great impact on the development of engineering: Communications, signal processing, and so on. The functions that are built in the next section fulfill the conditions that guarantee uniform convergence in the compact. They fulfill the conditions indicated in the following theorems. Theorem. Let f be a continuous function on the interval −L ≤ x ≤ L, such that f (−L) = f (L) and whose derivative f is quasi-continuous in that interval. Then the series ∞ a2n + b2n

n=1

converges, with an and bn being the Fourier coefficients.

Proof. Can be found in [9].

Theorem. Under the conditions stated in the previous theorem, the convergence of the Fourier series nπx nπx a0 + + bn sin an cos 2 L L ∞

n=1

to f (x) in the interval −L ≤ x ≤ L is absolute and uniform with respect to x in this interval.

Proof. Can be found in [9].

3.1

Obtaining the Formula for ζ(2k)

Let us consider the Fourier series expansion of the function f (x) = x2 defined on the interval [−2, 2]; after some direct calculations, we obtain the expression: +

16 π2

= C2 +

16 π2

f2 (x) = x2 = (17)

4 3

∞

n=1

∞

n=1

(−1)n cos( nπx 2 ) n2 (−1)n cos( nπx 2 ) n2

Salinas-Hernández et al

where

(18)

C2 =

a0 2

an =

(−1)n 16 , π 2 n2

4 3

∀ n ∈ Z +;

bn = 0, ∀ n ∈ Z + Substituting in the previous expression x = 2 or x = −2, we get 22 =

∞ ∞ 4 16 (−1)n cos(nπ) 4 16 (−1)2n + 2 + = 3 π n2 3 π2 n2 n=1

n=1

∞ 4 16 1 = + 2 3 π n2

(19)

n=1

Given that cos(nπ) = cos(−nπ) = (−1)n , it can be written as ∞ 4 16 1 2 = + 2 3 π n2 2

(20)

n=1

when isolating (21)

∞

1 n=1 n2 ,

we obtain ∞ 1 π2 ζ(2) = . = n2 6 n=1

This corresponds to Leonhard Euler’s famous formula that he obtained in 1734 and whose solution was analyzed in the previous section. Substituting x = 0 in (17) we get another important series, namely, (22)

∞ (−1)n

n=1

=−

π2 . 12

From the expression (17), we have (23)

∞ 16 (−1)n cos 4 f2 (x) − C2 = x − = 2 3 π n2 2

n=1

nπx 2

An alternative for

∞

1 n=1 n2k , k

∈ Z+

Clearly the function is integrable. Let’s define x 3 f3 (x) = 0 (f2 (x) − C2 )dx = x3 − x

16 0 π2

= (24) =

32 π3

∞

n=1

∞

n=1

4x 3

(−1)n cos( nπx 2 ) n2

n=1

∞

(−1)n (sin( nπx −sin( nπ0 2 ) 2 )) 3 n (−1)n sin( nπx 2 ) n3

+ C3 .

The above expression coincides with the Fourier series expansion of 3 the polynomial f3 (x) = x3 − 4x 3 on the interval [−2, 2], in this case we have that C3 is integration constant and it is equal to zero (the function is odd), thus an = 0 ∀ n ∈ Z + (25)

+ bn = (−1)n π32 3 n3 n ∈ Z

C3 =

a0 2

As a consequence (26)

f3 (x) =

1 4

2 −2 f3 (x)dx

32 x3 4x − = 3 3 3 π

1 2

∞

2 −2

x3 3

4x 3

−

(−1)n sin

dx = 0.

nπx

n=1

If in the previous equation we evaluate at x =1, we get the series ∞

(27)

n=1

π3 (−1)n = − (2n − 1)3 32

Now, let’s consider the function x 3 f4 (x) = 0 x3 − 4x − C 3 dx = 3 (28)

= − π644 = − π644 = − π644

∞

x4 12

4x2 6

−

(−1)n (cos( nπx −cos( nπ0 2 ) 2 )) 4 n=1 n

∞

(−1)n cos( nπx 2 ) n=1 n4

∞

(−1)n cos( nπx 2 ) n=1 n4

64 π4

∞

+ C4 .

n=1

(−1)n n4

Salinas-Hernández et al

We have that C3 = 0; but we include the constant to generate the algorithm. We have that the previous expression is the Fourier 2 4 expansion of the polynomial f4 (x) = x12 − 4x6 on the interval [−2, 2], where the Fourier coefficients are n

an = − 64(−1) , ∀ n ∈ Z+ π 4 n4 bn = 0, ∀ n ∈ Z +

(29)

C4 =

a0 2

1 4

−2 f4 (x)dx =

1 4

2 x4 −2

−

4x2 6

From equations (28) and (29), we get

dx = − 28 45 .

∞ 64 (−1)n 28 =− . 4 4 π n 45

(30)

n=1

From the previous expression we conclude that ∞ (−1)n

(31)

n=1

=−

7 4 π . 720

Therefore (32)

∞ 64 (−1)n cos x4 4x2 f4 (x) = − =− 4 12 6 π n4 n=1

nπx 2

−

28 . 45

Equivalently (33)

∞ 64 (−1)n cos x4 4x2 28 − + =− 4 f4 (x) − C4 = 12 6 45 π n4 n=1

nπx 2

Substituting in (32) x = 2 o x = −2, we get (34)

∞ ∞ 24 28 4(22 ) 64 (−1)n cos(nπ) 28 64 1 − − . − =− 4 = − 4 4 4 12 6 π n 45 π n 45 n=1

n=1

Equivalently (35)

∞ 24 64 1 28 24 − =− 4 − . 4 12 6 π n 45 n=1

∞

1 n=1 n2k , k

An alternative for

Isolating from the above expression

∈ Z+

∞

1 n=1 n4 ,

we have

∞ 1 π4 ζ(4) = = . n4 90

(36)

n=1

From equation (33), let us obtain f5 (x): f5 (x) =

x 0

= − 128 π5

(37)

x5 60

(f4 (x) − C4 )dx =

= − 128 π5

∞

n=1

∞

n=1

−

4x3 18

28x 45

(−1)n (sin( nπx −sin( nπ0 2 ) 2 )) n5 (−1)n sin( nπx 2 ) n5

+ C5 .

Calculating the Fourier expansion of the previous expression we get C5 = 0. Evaluating the last equation at x = 1, we have the series ∞

(38)

n=1

(−1)n 5π 5 . = − (2n − 1)5 1536

Let’s get the function f6 (x), based on the foregoing x

f6 (x) =

(f5 (x) − C5 )dx =

256 π6

(39)

∞

n=1

∞

n=1

∞

n=1

x6 360

−

x4 18

14x2 45

(−1)n (cos( nπx −cos( nπ0 2 ) 2 )) n6 (−1)n cos( nπx 2 ) n6 (−1)n cos( nπx 2 ) n6

−

256 π6

+ C6 .

∞

n=1

(−1)n n6

Since the previous expression corresponds to the Fourier expansion of the function, we have that C6 is defined by (40)

1 C6 = 4

f6 (x)dx = −2

2 −2

x4 14x2 x6 − + 360 18 45

From the above we obtain that (41)

∞ (−1)n

n=1

=−

31π 6 . 30240

dx =

248 . 945

Salinas-Hernández et al

So we have that f6 (x) is expressed as (42)

∞

x6 x4 14x2 256 (−1)n cos f6 (x) = − + = 6 360 18 45 π n6 n=1

nπx 2

248 . 945

Considering that cos(nπ) = (−1)n and substituting in the previous 1 equation x = 2 or x = −2. Hence, isolating ∞ , we get n=1 n6 (43)

∞ 1 π6 π6 π6 248 [f f = , = (2) − C ] = (2) − 6 6 6 n6 256 256 945 945

n=1

which can be expressed as (44)

ζ(6) =

∞ 1 π6 . = n6 945

n=1

Using the same notation, we get ζ(2) = ζ(2(1)) = ∞ n=1 (45) ζ(4) = ζ(2(2)) = ∞ n=1 ζ(6) = ζ(2(3)) = ∞ n=1

1 n2 1 n4 1 n6

= = =

π2 π2 6 = 16 [f2 (2) − C2 ], 4 π π4 90 = − 64 [f4 (2) − C4 ], π6 π6 945 = 256 [f6 (2) − C6 ].

Based on the above, we find the general expression (46)

ζ(2k) =

∞ 2k 1 k+1 π = (−1) [f2k (2) − C2k ], n2k 4k+1

n=1

where the following expressions are defined recursively: f2 (x) = x 2 x fn (x) = 0 (fn−1 (x) − Cn−1 )dx, ∀ n ≥ 3, with (47)

Cn−1 =

C2k =

1 4

  

1 4

si n − 1 = 2k + 1

−2 fn−1 (x)dx si n − 1 = 2k k+1

k4 −2 f2k (x)dx = (−1) π 2k

Let’s apply the above procedure to verify (48)

ζ(8) = ζ(2(4)) =

∞

n=1

(−1)n n2k

∞ 1 π8 . = n8 9450

n=1

∀n≥3

∀ k ∈ Z+

An alternative for

∞

1 n=1 n2k , k

∈ Z+

We must obtain f8 (2) for which we first need to find f7 (x) in terms of f6 (x). From the equation of fn (x) defined in (47), we obtain x x x6 4 2 248 f7 (x) = 0 (f6 (x) − C6 )dx = 0 360 − x18 + 14x − 45 945 dx (49) x7 2520

−

x5 90

14x3 135

−

248x 945 .

Now, let’s obtain f8 (x) in terms of f7 (x). We have C7 = 0 because the polynomial is an odd function, thus x x x7 5 3 248x f8 (x) = 0 (f7 (x) − C7 )dx = 0 2520 − x90 + 14x − 135 945 dx (50) x8 20160

−

x6 540

14x4 540

−

124x2 945 .

Then f8 (2) = (51) C8 =

1 4

28 20160

2 −2

−

26 540

x8 20160

−

14(24 ) 540

x6 540

124(22 ) 945

−

14x4 540

−

68 = − 315 ,

124x2 945

From the previous expressions, we get

(52)

Therefore (53)

508 . dx = − 4725

68 8 π8 (−1)5 π45 [f8 (2) − C8 ] = − 1024 − 315 +

512 π8 = − 1024 − 4725 = ζ(8) =

π8 9450

508 4725

π8 9450 .

= (−1)5 π45 [f8 (2) − C8 ],

which is what we wanted to prove. We also get (54)

∞ (−1)n

n=1

=−

127 π 8 . 1209600

In order to verify this, let us apply the procedure indicated in the equations (46) and (47) (55)

ζ(10) = ζ(2(5)) =

∞ 1 π 10 . = n10 93555

n=1

Salinas-Hernández et al

We must obtain f10 (2) for which we first have to find f9 (x) in terms of f8 (x). From the equation of fn (x) defined in (47), we obtain x f9 (x) = 0 (f8 (x) − C8 )dx x

(56)

x8 20160

x9 181440

−

x6 540

x7 3780

14x4 540

14x5 2700

−

124x2 945

−

124x3 2835

508 4725

508x 4725 .

Now, let us obtain f10 (x) in terms of f9 (x). We have C9 = 0 because the polynomial is an odd function, thus x f10 (x) = 0 (f9 (x) − C9 )dx =

(57)

x 0

x9 181440

x10 1814400

−

x7 3780

x8 30240

14x5 2700

14x6 16200

−

124x3 2835

124x4 11340

508x 4725

508x2 9450 .

Then f10 (2) = =

210 1814400

28 30240

14(26 ) 16200

−

124(24 ) 11340

508(22 ) 9450

248 2835 ,

(58) C10 =

−

1 4

2 −2

x10 1814400

−

x8 30240

14x6 16200

−

124x4 11340

584 13365 .

From the above expressions, we get 10

(59)

(−1)6 π46 [f10 (2) − C10 ] = =

Hence (60)

π 10 4096

4096 93555

ζ(10) =

π 10 93555

π 10 4096

π 10 93555 .

248 2835

∞ (−1)n

n=1

584 13365

= (−1)6 π46 [f10 (2) − C10 ],

which is what we wanted to prove. We also get (61)

−

n10

=−

73 π 10 . 6842880

508x2 9450

An alternative for

∞

1 n=1 n2k , k

∈ Z+

Bernoulli numbers

In Euler’s formula (62)

ζ(2k) = (−1)k+1

(2π)2k B2k , k ∈ Z +, 2(2k)!

even indexed Bernoulli numbers B2k appear; they define a sequence of rational numbers. Initially they arise from the search of a formula to know the sum of the k th powers of the first n positive integers and were named in honor of Jacob Bernoulli who introduced them for the first time in 1713. In a different way, Euler also established a formula to define Bernoulli numbers.

4.1

Formula for Bernoulli numbers B2k

From equation (62) and equation (46), we obtain (63)

(−1)k+1

(2π)2k B2k π 2k = (−1)k+1 k+1 [f2k (2) − C2k ]. 2(2k)! 4

From the previous expression we deduce (64)

B2k =

(2k)! 22k−1 4k+1

[f2k (2) − C2k ],

k ∈ Z +.

This expression allows us to obtain even indexed Bernoulli numbers. In order to illustrate this, we next display calculations for the first four even indexed Bernoulli numbers: 1 4 − 43 = 16 B2 = 212!42 [f2 (2) − C2 ] = 16 3 1 B4 = 234!43 [f4 (2) − C4 ] = 64 − 43 + 28 45 = − 30 (65) 45 8 248 1 B6 = 256!44 [f6 (2) − C6 ] = 512 − 1568− 945508= 42 1 8! 315 B8 = 27 45 [f8 (2) − C8 ] = 1024 − 315 + 4725 = − 30 Acknowledgments SNI CONACYT, EDI and COFAA of Instituto Politécnico Nacional. E. Salinas-Hernández Departamento de Ciencias Básicas, ESCOM del Instituto Politécnico Nacional, Av. Juan de Dios Bátiz S/N, Gustavo A. Madero, 07738 Ciudad de México, esalinas@ipn.mx

Abelardo Santaella-Quintas Departamento de Matemáticas, ESFM del Instituto Politécnico Nacional, Avenida IPN s/n Edificio 9, U.P. Adolfo López Mateos, Gustavo A. Madero, 07738 Ciudad de México, abelardo.santaella@ipn.mx

Salinas-Hernández et al

Martha Paola Ramı́rez Torres Departamento de Matemáticas, ESFM del Instituto Politécnico Nacional, Avenida IPN s/n Edificio 9, U.P. Adolfo López Mateos, Gustavo A. Madero, 07738 Ciudad de México, mathpaola@gmail.com

Gonzalo Ares de Parga Departamento de Fı́sica, ESFM del Instituto Politécnico Nacional, Avenida IPN s/n Edificio 9, U.P. Adolfo López Mateos Gustavo A. Madero, 07738 Ciudad de México gonzalo@esfm.ipn.mx

References [1] Bao Quoc ta and Chung Pham Van, A probabilistic proof of Euler’s Formula for ζ(2n). MATH. REPORTS 21 (71), (2019), 61–65. [2] E. de Amo, M. Dı́az Carrillo and J. Fernández-Sánchez, Another proof of Euler’s formula for ζ(2k). Proc. Amer. Math. Soc. 139 (2011), 4, 1441–1444. [3] H. Tsumura, An elementary proof of Euler’s formula for ζ(2m). Amer. Math. Monthly 111 (2004), 5, 430–431. [4] Chungang, J. and Yonggao, C., Euler’s formula for ζ(2k), proved by induction on k, Math. Mag. 73 (2000) 154–155. MR1573450. 1 π2 [5] R. C. Boo, An Elementary proof of ∞ n=1 n2 = 6 , Amer. Math. Monthly 94 (1987) 662–663. [6] B. C. Berndt, Elementary evaluation of ζ(2n), Math. Magazine 48 (1975), 148–154. [7] R. Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974) 1067–1086. [8] T.M. Apostol, Another elementary proof of Euler’s formula for ζ(2n). Amer. Math. Monthly 80 (1973), 425–431. [9] R.V. Churchill, Series de Fourier y problemas de contorno. McGraw-Hill (1979).

Morfismos, Vol. 24, No. 1, 2020, pp. 15–34 Morfismos, Vol. 24, No. 1, 2020, pp. 15–34

Computations of hitting time densities for the generalized Cox-Ingersoll-Ross diffusion Computations of hitting time densities for the Jonathan Gutierrez-Pavón 1 diffusion generalized Cox-Ingersoll-Ross Jonathan Gutierrez-Pavón

Abstract We give explicit formulæ for the density function of first hitting time of the so-called generalized Cox-Ingersoll-Ross Abstractprocess. In fact, we treat the several cases of the diffusion depending on the values of the parameters. function and of first hitting time of To We findgive the explicit density formulæ function for we the use density the eigenvalues eigenfunctions the so-called generalized Cox-Ingersoll-Ross process. In fact, we treat associated with the infinitesimal operator. It turns out that a very im- the several cases of the diffusion depending the values of thewhere parameters. portant tool in this analysis is the so-called on Kummer equation, we the solutions; density function we use eigenvalues and eigenfunctions use To thefind known this allows us tothe compute the eigenfunctions in associated with thehypergeometric infinitesimal operator. It turns out that a very imterms of the confluent functions. portant tool in this analysis is the so-called Kummer equation, where we 2010 Mathematics Subject Classification: 60H10,the 34A05. use the known solutions; this allows 60J60, us to compute eigenfunctions in the confluent hypergeometric hitting functions. Keywordsterms and of phrases: Cox-Ingersoll-Ross, times, spectral decomposi-

tion, Kummer differential equation. 2010 Mathematics Subject Classification: 60J60, 60H10, 34A05. Keywords and phrases: Cox-Ingersoll-Ross, hitting times, spectral decomposiKummer differential equation. 1 tion, Introduction As described in [8], the changes of the so-called membrane potential between Introduction two1neurons in the human brain can be modeled using an Itô’s diffusion As described in [8], dX the changes of the so-called membrane potential between t = µ(Xt , t)dt + σ(Xt , t)dBt , two neurons in the human brain can be modeled using an Itô’s diffusion see also [4]. It turns out that this model can be a good approximation of a µ(X + σ(X dXto t = t , t)dt t , t)dBt ,in these applications real phenomenon, according [8]. One of main interests is what people in neuroscience call the interspike intervals, which can be seen see also [4]. It turns out that this model can be a good approximation of a as random variable of the form real phenomenon, according to [8]. One of main interests in these applications is what people in neuroscience call: the interspike f (t)}, intervals, which can be seen τ = inf{t Xt ≥ as random variable of the form where f is a deterministic time function. The importance of this random (t)}, τ =that inf{tthe : Xflow t ≥ fof variable relies on the hypothesis information in the nervous 1where f is is part a deterministic time function. The importance of Department this random This work of the Ph.D. dissertation of the author at the Mathematics of Cinvestav 2017.on Thethe research was partially by CONACYT. variable inrelies hypothesis that supported the flow of information in the nervous 1 This work is part of the Ph.D. dissertation of the author at the Mathematics Department 15 of Cinvestav in 2017. The research was partially supported by CONACYT.

Hitting time for generalized CIR

system is encoded in the timing of spikes. For this kind of phenomenon, a model considered in the literature is the diffusion dXt = (µ − aXt )dt + σ Xt − SdBt ,

where µ, a, σ, S are constants.

Coincidentally, this model is also used in financial mathematics to model interest rates. In fact, according to the theory of pricing, the price of a so-called zero-coupon bond is give by the expectation t E e− 0 Xs ds ,

see e.g. [2]. Again, knowing information of the first time when Xt surpasses a function is of importance in this context. When S = 0, the process X is called the Cox-Ingersoll-Ross process (CIR), see also [6] for more details. In the context of financial mathematics, specifically in the so-called risk theory, an important paradigm is modelling the time of default of a bond based on the so-called intensity models, where it is used a function λ(x) to study the random time t τ = inf{t : λ(Xs )ds ≥ E}, 0

for some constant E, see e.g. [7]. These are some reasons that motivate us to work with a diffusion that is a generalization of the CIR process. Precisely, the solution of dXt = (µ − aXt )dt + σ Xt − SdBt .

In particular, we study the first hitting time of the process X, defined as τy := inf{t > 0 : Xt = y}, where y is a given constant.

Such task of finding the density has been done before using theory of spectral decompositions, in particular we use results in [10]. The case S = 0 is well known and study, see e.g. [11] . Notice that when S is not zero one might try to apply a space transformation and use Itô’s lemma to go back to the case S = 0. However, in this paper we do not do that. Instead do that, we work directly with the differential equations and the spectral decomposition, this helps to see the connection with the so-called Kummer differential equation, which is an important tool for the whole analysis. This way of work has the benefit to see how the spectral theory works when dealing hitting times, and can be applied to other diffusions. Let us mention how the paper is organized. In the coming Section 2 we present the basic theory that we use along the paper. Basic concepts such as

Hitting time for generalized CIR

the speed measure, the scale function and the killing measure are introduced. We also recall the classification of the end-points in the state space. In Section 3 we will state the result of V. Linetsky [10] and we present some tables which summarizes our results. In the section 4 we study the classification of the end-point S. Sections 5, 6 and 7 presents the solution ψ of the equation Lψ + λψ = 0, and we provide the proofs of the results. Finally, in Section 8 we study the first hitting time density of the process Y that is solution of the stochastic differential equation dYt = (β − bYt )dt + σ s − Yt dBt .

Such process is the reflected analogous of X. To this end, we will apply the Itô’s formula to recycle the formulas obtained for X. To finish, in the Section 9 we present a numerical illustration.

Preliminaries

Let {Xt : t ≥ 0} be a one-dimensional diffusion whose state space is some interval I ⊆ R with end-points e1 and e2 . Every diffusion has three basic characteristics that determine the process: speed measure, scale function and killing measure, see [3] for more details. We consider the special case when the three basic characteristics are absolutely continuous with respect to Lebesgue measure in the interior of I, i.e. x s (y)dy, x ∈ (e1 , e2 ). (1) m(dx) = m(x)dx, k(dx) = k(x)dx, s(x) = Then the infinitesimal generator associated is the following

1 2 a (x)f (x) + b(x)f (x) − c(x)f (x), x ∈ (e1 , e2 ). 2 The functions a, b, c are called the infinitesimal parameters of X, and are related to m, k, s through the following formulas x 2b(y) dy − 2c(x) 2 a2 (y) , m(x) = , k(x) = 2 . s (x) = e a2 (x)s (x) a (x)s (x)

(2)

Lf (x) =

The speed measure, the scale function and the killing measure determine the behavior of the diffusion in the interior of the state space I. However the behavior of the diffusion at the boundary points is characterized by boundary conditions. In [3] it is presented a classification of the end-points of I according to the behavior of the diffusion in the neighborhood of these end-points. To explain this, let z be fixed such that e1 < z < e2 . According with (1) we have z x (3) m((x, z)) := m(y)dy, and s(x) := s (y)dy. x

Hitting time for generalized CIR

Now define A:= B:=

e 1z e1

[m((x, z)) + k((x, z))] s (x)dx, [s(z) − s(x)] (m(x) + k(x))dx.

Then the end-point e1 it is classified in the following manner (for e2 is similar, see [12]) i. The end-point e1 is called exit if (4)

A < ∞ and B = ∞.

ii. The end-point e1 is called entrance if (5)

A = ∞ and B < ∞.

iii. The end-point e1 is called regular if (6)

A < ∞ and B < ∞.

In this case, one additionally has the following subclassification: • if m({e1 }) = k({e1 }) = 0 then e1 is called regular reflecting, • if m({e1 }) < ∞, and k({e1 }) = ∞ then e1 is called regular killing, • if 0 < m({e1 }) < ∞, and k({e1 }) = 0 then e1 is called regular sticky, • if m({e1 }) = 0, and k({e1 }) > 0 then e1 is called regular elastic, • if m({e1 }) = ∞, and k({e1 }) ≥ 0 then e1 is called regular absorbing. iv. The end-point e1 is called natural if (7)

A = ∞ and B = ∞.

Remark 2.1. Since in this paper m, k are absolutely continuous with respect to Lebesgue measure, then the boundary condition at a regular boundary can only be reflecting or killing, for more details see [12]. Also, in this paper we do not consider the case when e1 is natural, because in our examples this case is not presented.

Hitting time for generalized CIR

Note that if k((e1 , e2 )) = 0, if we use the previous classification and the sacle function s, it is known that for e1 < x < y   1, if e1 is entrance or regular reflecting;   x (8) Px (τy < ∞) = s (z)dz e1    y s (z)dz , if e1 is exit or regular killing, e1

where τy := inf {t > 0 : Xt = y}; see for instance [10].

Spectral expansion for first hitting time density

Let us present the spectral decomposition theorem of V. Linetsky found in [10]; see also [9]. Consider a diffusion X which is solution of the stochastic differential equation (9) dXt = (µ − aXt )dt + σ Xt − S dBt , where µ, a, S ∈ R and σ > 0. It is known that the infinitesimal operator is (10)

Lf (x) =

σ 2 (x − S) f (x) + (µ − ax)f (x), 2

x ∈ (S, ∞),

acting on twice differentiable functions f : (S, ∞) → R. To specify completely the process X, we also need to specify the behavior of X at S; this fact is important for the following theorem. Theorem 3.0.1. (Linetsky [10]) Let X be a diffusion that is solution of dXt = µ(Xt )dt + σ(Xt )dBt and whose state space I has the end-points e1 and e2 . Define I y := [e1 , y] if e1 is regular reflecting, and I y := (e1 , y] in any other case. Fix X0 = x and y ∈ I such that e1 < x < y < e2 , and suppose that e1 is either regular, entrance or exit. For λ ∈ C and x ∈ I y , let ψ(x, λ) be the unique non trivial solution (up to a multiple independent of x) of the Sturm-Liouville equation Lψ + λψ = 0,with boundary condition at e1 given by (11)

lim ψ(x, λ) = 0 or +

x→e1

lim+

x→e1

ψx (x, λ) = 0. s (x)

Then the spectral expansion of Px (t < τy < ∞), with e1 < x < y takes the form (12)

Px (t < τy < ∞) = −

∞

n=1

e−λn t

ψ(x, λn ) , λn ψλ (y, λn )

where 0 < λ1 < λ2 < λ3 · · · are the simple positive zeros of ψ(y, λ), i.e., ψ(y, λn ) = 0. Note that each λn depends on y.

Hitting time for generalized CIR

Remark 3.0.2. The function ψ(x, λ) appearing in the previous theorem is square-integrable with respect to m in a neighborhood of e1 ; and ψ(x, λ) and ψx (x, λ) are continuous in x and λ in I y × C and entire in λ ∈ C for each x ∈ I y fixed. For more details see [9]. Remark 3.0.3. In our case the state space is (S, ∞), this implies that using the notation of the Theorem 3.0.1 we have that e1 := S and e2 := ∞. If we apply the identity (13)

Px (τy < ∞) = Px (τy < t) + Px (t < τy < ∞),

by Theorem 3.0.1 and (8) we obtain that for e1 < x < y (14)

Px (τy ≤ t) = Px (τy < ∞) +

∞

e−λn t

n=1

ψ(x, λn ) . λn ψλ (y, λn )

An important ingredient for the methodology is solving the equation Lψ + λψ = 0. It turns out that the boundary condition for ψ at e1 = S depends on the classification of S (see [3] for details): i. If e1 is exit or regular killing then the boundary condition is lim ψ(x, λ) = 0.

x→e+ 1

ii. If e1 is entrance or regular reflecting then the boundary condition is lim

x→e+ 1

where ψx (x, λ) :=

ψx (x, λ) = 0, s (x)

∂ ∂x ψ(x, λ).

Remark 3.0.4. When e1 < y < x < e2 , the first hitting time problem is treated similarly. In this case we have I y := [y, e2 ] if e2 is regular reflecting, and I y := [y, e2 ) in any other case. It is also known that  1, if e2 is entrance or regular reflecting;   e2 (15) Px (τy < ∞) = s (z)dz x  , if e2 is exit or regular killing.  e2 s (z)dz y Then the spectral expansion of Px (t < τy < ∞), with y < x < e2 is

(16)

Px (t < τy < ∞) = −

∞

n=1

e−λn t

φ(x, λn ) , λn φλ (y, λn )

where the solution φ(x, λ) of Lu + λu = 0 is square-integrable with respect to m in a neighborhood of e2 and satisfying the appropriate boundary condition at e2 , i.e.

Hitting time for generalized CIR

i. lim φ(x, λ) = 0 if e2 is exit or regular killing, or x→e− 2

ii. lim− x→e2

φx (x, λ) = 0 if e2 is entrance or regular reflecting. s (x)

Also, φ(x, λ) and φx (x, λ) are continuous in x and λ in I y × C and entire in λ ∈ C for each x ∈ I y fixed. The λn are all the simple positive zeros of φ(y, λ). Table 1. Nature of the end-point S Parameters 2aS − 2µ I. −1 < <0 σ2 with a = 0 2aS − 2µ II. ≥0 σ2 with a = 0 2aS − 2µ III. ≤ −1 σ2 with a = 0 IV. a = 0 and −2µ −1 < 2 < 0 σ

S Regular reflecting

Exit

Boundary Condition ψx (x, λ) =0 lim s (x) x→S + lim ψ(x, λ) = 0

x→S +

Entrance

lim+

x→S

Regular killing

ψx (x, λ) =0 s (x)

lim ψ(x, λ) = 0

x→S +

Table 2. Solution ψ Parameters 2aS − 2µ <0 σ2 with a = 0 2aS − 2µ II. ≥0 σ2 with a = 0 2aS − 2µ III. ≤ −1 σ2 with a = 0 I. −1 <

IV. a = 0 and −1 <

−2µ <0 σ2

Solution of Lψ + λψ = 0 −λ 2µ−2aS 2a(x−S) , a , σ2 σ2

ψ(x, λ) = F

ψ(x, λ) =

2a(x−S) 1− σ2

2µ−2aS σ2

ψ(x, λ) = F ψ(x, λ) =

2(x−S) σ2

−λ 2µ−2aS 2a(x−S) , a , σ2 σ2

( 1 − 2

2a(x−S) 2µ−2aS −λ +1,2− 2µ−2aS , σ2 a − σ2 σ2

µ ) σ2

· J(1− 2µ ) σ2

2λ(x−S) 2 σ2

We will apply Theorem 3.0.1 to find the first hitting time density of the process X that is solution of (9). We summarize our results in the following tables. Table 1 shows the nature of the end-point S (second column) and the type of boundary condition (third column), both depending on certain regions for the parameters µ, a, S and σ (first column). Table 2 shows the solution of the equation (11) and Table 3 shows precisely the formula (14). Note that in the end there are four types: I, II, III, IV. The

Hitting time for generalized CIR

proofs of the first two columns of Table 1 are give in Section 4. Section 5 deals with cases I and III of these tables. Section 6 deals with the case II, and the Section 7 presents the case IV. Let us give the notation used in the tables: • F (a, b, x) :=

∞ (a)n xn , where (a)n := a · (a + 1) · · · (a + n − 1). (b)n n! n=0

• Fλ represents the derivative of F with respect to λ. v+2n ∞ (−1)n x2 the Bessel function. • Jv (x) := n!Γ(v + n + 1) n=0 • Jv,n represents a positive zero of the Bessel function Jv . Here Γ is the Gamma function. Table 3. Spectral decomposition Parameters I. −1 <

2aS − 2µ <0 σ2

with a = 0 2aS − 2µ II. ≥0 σ2 with a = 0 2aS − 2µ III. ≤ −1 σ2

Spectral decomposition for the first hitting time density n 2µ−2aS , 2a(x−S) ∞ F −λ a , σ2 σ2 −λn t Px (τy ≤ t) = 1 + e n 2µ−2aS , 2a(y−S) λn Fλ −λ n=1 a , σ2 σ2 x

Px (τy ≤ t) = Sy S

Px (τy ≤ t) = 1 +

with a = 0 IV. a = 0 and −2µ −1 < <0 σ2

v x−S v x−S 2 y−S −2 y−S

∞

−λn t

n=1

−λn t

n=1

Px (τy ≤t)=

∞

s (z)dz

λn Fλ ∞

−λn a

n=0

2µ−2aS 2a(x−S) , σ2 σ2

−λn a

−

ψ(x, λn ) λn ψλ (y, λn )

2µ−2aS 2a(y−S) , σ2 σ2

σ 2 Jv,n t Jv Jv,n x−S y−S 8(y−S) Jv,n ·J1+v (Jv,n )

Classification of the end-point S

We consider the process X that is solution of (9) with state space give by the interval I = [S, ∞) if S is regular reflecting or I = (S, ∞) in any other case. We suppose that X0 = x > S and we consider y ∈ I fixed such that x < y. To study this process we first classified the end-point S. Proposition 4.0.1. The end-point S has the following classification: i. If −1 < ii. If

2aS − 2µ < 0 then S is regular. σ2

2aS − 2µ ≥ 0 then S is exit. σ2

Hitting time for generalized CIR

2aS − 2µ ≤ −1 then S is entrance. σ2 Proof. We prove the first case, the other two cases are similar. Suppose that iii. If

−1 <

2aS − 2µ < 0, σ2

and let z be a fixed value such that S < z < ∞. To verify that S is regular, we calculate the following integrals to see that they are finite: z m((x, z))s(x)dx S z z 2µ−2aS 2aS−2µ 2ax −1 −2 −2ay 2 σ2 σ = 2σ e (y − S) dy e σ2 (x − S) σ2 dx S x z z 2µ−2aS 2aS−2µ −1 2 σ (y − S) dy (x − S) σ2 dx ≤M S Sz z 2µ−2aS 2aS−2µ =M (y − S) σ2 −1 dy · (x − S) σ2 dx S

< ∞,

where M is a constant. In similar way we obtain z s(x)m(x)dx S z z 2ay 2aS−2µ 2µ−2aS −2ax = e σ2 (y − S) σ2 dy 2σ −2 e σ2 (x − S) σ2 −1 dx S

< ∞.

We used, in both cases, the fact that −1 < integrals are finite. Corollary 4.0.2. If a = 0 and −1 <

−2µ σ2

2aS − 2µ < 0, to obtain that the σ2

< 0 then the end-point S is regular.

The condition a = 0 presented in tables shows up in the following sections.

First hitting time density for the cases I and III

According to Theorem 3.0.1, for the cases I and III (in Tables 1, 2 and 3) we have to find a function ψ(x, λ) such that it satisfies (17)

σ 2 (x − S) ψ (x) + (µ − ax)ψ (x) + λψ(x) = 0, and 2

In turn we have the following proposition.

lim+

x→S

ψx (x, λ) = 0. s (x)

Hitting time for generalized CIR

Proposition 5.0.1. If a = 0, then the function −λ 2µ − 2aS 2a(x − S) (18) ψ(x, λ) = F , , , a σ2 σ2 is solution of (17), with F (a, b, x) :=

∞ (a)n xn , (b)n n! n=0

where (a)n := a · (a + 1) · · · (a + n − 1). Proof. Consider z = obtain the equation

σ 2 (x−S) , 2

zg (z) + Now consider w =

2µ 4az 2aS − 4 − 2 σ2 σ σ

4az σ4

at: wh (w) + (

and define g(z) := f

σ2

+ S . From (17) we

4λ g(z) = 0. σ4 4 and define h(w) := g σ4aw . Since a = 0, we arrive g (z) +

2µ − 2aS − w)h (w) − σ2

−λ a

h(w) = 0.

The previous equation is called the Kummer equation (see [15]), and according with [15, p.2] the solution is −λ 2µ − 2aS h(w) = F , ,w . a σ2 Returning to the variable x we obtain the result. Applying the Proposition 5.0.1 and the formula (14), we obtain the following theorem. Theorem 5.0.2. Let X be the process that is solution of (9) with S < x < y, a = 0, such that S is regular reflecting or entrance. Then the first hitting time distribution of the process X is n 2µ−2aS 2a(x−S) ∞ F −λ , , a σ2 σ2 , (19) Px (τy ≤ t) = 1 + e−λn t −λn 2µ−2aS 2a(y−S) λ n Fλ , σ2 n=1 a , σ2 −λn 2µ − 2aS 2a(y − S) , , is where the derivative Fλ a σ2 σ2 k −λn 2a(y−s) ∞ 2 σ −λ −1 n a k φ (20) +k 2µ−2aS a a k! k=0 σ2 k 1 −λn −λn 2µ − 2aS 2a(y − S) + φ F , , , a a a σ2 σ2

Hitting time for generalized CIR

and φ(z) :=

Γ (z) , where Γ is the Gamma function. See [10] for more details. Γ(z)

From the Theorem 5.0.2 we obtain a formula for the first hitting time density for the cases I and III

Px (τy ∈ dt) = −

(21)

∞

e−λn t

Fλ

n=1

∞

e−λn t λn cn ,

−λn 2µ−2aS 2a(x−S) , σ2 a , σ2

−λn 2µ−2aS 2a(y−S) , σ2 a , σ2

n=1

where (22)

cn :=

−F

λ n Fλ

−λn 2µ−2aS 2a(x−S) , σ2 a , σ2

−λn 2µ−2aS 2a(y−S) , σ2 a , σ2

Remark 5.0.3. To be able to carry out a numerical procedure we propose a naive but effective approximation of λn and cn . First consider the following formula found in [15], (23)

F (a, b, x) = π

14 − 2b x b −a Γ(b)e 2 x 2 −1 bπ π b . −a − + 1+O |a| 2 cos 2 x 2 2 4

−1 2

Since the λn are the zeros of ψ(y, λ), with y fixed, we will use the formula (23) to find an approximation of λn for n large given by

σ2 λn ≈ 2a(y − S)

(24)

π(µ − aS) nπ 3π − + 2σ 2 2 8

µ − aS × a. − σ2

Then applying again the formula (23) we obtain an approximation for cn in (22) with n large given by

≈ ×

2a(x−y) 3 1 − µ−aS σ2 (−1)n+1 2π n + µ−aS σ2 − 4 · e x − S 4 σ2 · 2 y−S 4µ−4aS 2a(y−S) 3 π 2 n + µ−aS − − · 2 2 2 σ 4 σ σ π(µ − aS) π µ − aS 3 x−S − . cos π n + − + σ2 4 y−S σ2 4

Hitting time for generalized CIR

First hitting time density for the case II

For the case II, in the tables of section 3, we have to find a function ψ(x, λ) such that (25)

Lψ + λψ = 0 and

lim ψ(x, λ) = 0.

x→S +

We have the following proposition: Proposition 6.0.1. If a = 0, then the function ψ(x, λ) defined by the formula

2a(x − S) σ2

1− 2µ−2aS σ2 −λ 2µ−2aS 2µ−2aS 2a(x−S) − F + 1, 2 − , , a σ2 σ2 σ2

is solution of (25). Proof. The proof is similar to the Proposition 5.0.1. With the same changes of variable we obtain the Kummer equation. Applying the formula (14) and (8) we arrive at x ∞ s (z)dz −λn t ψ(x, λn ) S + , e (26) Px (τy ≤ t) = y λn ψλ (y, λn ) s (z)dz n=1 S 2aS−2µ σ2

2az

where s (z) = e σ2 (z − S) Px (τy ∈ dt) for the case II is −

∞

n=1

e−λn t

(x − S)1− (y − S)1−

2µ−2aS σ2

Fλ

. Therefore the first hitting time density

−λn a

−

2µ−2aS σ2

+ 1, 2 −

2µ−2aS 2a(x−S) , σ2 σ2

−λn a

−

2µ−2aS σ2

+ 1, 2 −

2µ−2aS 2a(y−S) , σ2 σ2

To find an approximation of the λn (with n large), we use again the formula (23), thus 2 µ − aS 2µ − 2aS π π µ − aS λn ≈ −a × y 1 − − −n + −1 . − σ2 8 2 σ2 σ2 We apply again the formula (23) if we wish to approximate cn .

First hitting time density for the case IV

Note that in previous two sections we consider a = 0. Now we present an example of a particular situation when a = 0. In this case the process X is the solution of dXt = µdt + σ Xt − SdBt .

Hitting time for generalized CIR

We consider only the case when −1 < −2µ σ 2 < 0 (The other cases in Proposition 4.0.1 with a = 0 are similar). From Corollary 4.0.2 we have that the end-point S is regular, and we now assume that S is killing. Therefore we have to find a function ψ(x, λ) such that it satisfies

(27)

σ 2 (x − S) ψ (x) + µψ (x) + λψ(x) = 0 and lim ψ(x, λ) = 0. 2 x→S +

Proposition 7.0.1. The function ( 1 − µ ) 2(x − S) 2 σ2 2λ(x − S) · J(1− 2µ2 ) 2 , (28) ψ(x, λ) = σ σ2 σ2 v+2n ∞ (−1)n x2 . is solution of (27), where Jv (x) := n!Γ(v + n + 1) n=0

Proof. It follows using the formula in Table 15 of [13].

Remark 7.0.2. For y fixed such that S < x < y, we find the λn such that ψ(y, λn ) = 0 in the following manner, let Jv,n be the positive zeros of the Bessel function Jv , where v := 1 − 2µ σ 2 . Then by (28) the values λn must satisfies the equation 2λn (y − S) = Jv,n . (29) 2 σ2 Thus λn =

(30)

2 σ 2 Jv,n . 8(y − S)

Lemma 7.0.3. Let ψ be the function in (28), then √ √ 3− µ 2(y − S) 4 σ2 2 2 y − S (31) ψλ (y, λn ) = − · · J2− 2µ2 (Jv,n ) . σ σ2 σ · Jv,n Proof. We first compute the derivative of ψ with respect to λ, where ψ is (28). Then we arrive at the following expression for ψλ (x, λ): v+2k−1 2λ(x−S) k v ∞ (−1) (v + 2k) σ2 2(x − S) 2 2(x − S) 1 √ (32) . · · σ2 σ2 k! · Γ(k + v + 1) 2 λ k=0

On the other hand, notice that (33)

Jv (x) =

v+2n−1 ∞ 1 (−1)n (v + 2n) x2 . 2 n=0 n!Γ(v + n + 1)

Hitting time for generalized CIR

Then by evaluating function in (33) at (34)

ψλ (x, λ) =

2(x − S) σ2

v2 + 12

2λ(x−S) , σ2

1 · √ · Jv λ

then (32) reads as 2λ(x − S) . σ2

Now we use the following identity found in [10]: v Jv (z) = −Jv+1 (z) + Jv (z). z

(35)

Applying the identity (35) we arrive at v+1 2λ(x − S) 2(x − S) 2 2 1 · √ · Jv 2 ψλ (x, λ) = σ2 σ2 λ v+1 2(x−S) 2 2 2 σ σv 2λ(x − S) √ · Jv 2 = σ2 λ 2 2λ(x − S) 2λ(x − S) − Jv+1 2 . σ2 Using (30), and the fact that Jv (Jv,n ) = 0, we obtain ψλ (y, λn ) v + 1 2(y − S) 2 2 = σ2 v + 1 2(y − S) 2 2 = σ2

2(y − S) σv Jv (Jv,n ) − Jv+1 (Jv,n ) σ · Jv,n 2 2λn (y − S) 2 2(y − S) · [−Jv+1 (Jv,n )] . σ · Jv,n

This completes the proof. Note that applying the Lemma 7.0.3 we have (36)

ψ(x, λn ) −2 = · λn ψλ (y, λn ) Jv,n

x−S y−S

Jv Jv,n x−S y−S J1+v (Jv,n )

By joining everything, we obtain the following theorem. Theorem 7.0.4. Let X be the process that is solution of dXt = µdt + σ Xt − SdBt .

Suppose that X0 = x and S < x < y, for y fixed. If −1 < killing, then (37) Px (τy ≤ t) =

x−S y−S

−2µ σ2

< 0 and S is

x−S v2 ∞ 2J J J σ t v v,n v,n y−S x−S . − 2· · e− 8(y−S) y−S J · J (J v,n 1+v v,n ) n=0

Hitting time for generalized CIR

Remark 7.0.5. When σ = 2, µ = 2v + 2 and S = 0, then the process X is the Squared Bessel process. Also note that if τyR :=inf{t > 0 : Rt = y} where R represents the Bessel process, then Px τyR ≤ t = Px2 inf{s > 0 : Rs2 = y 2 } ≤ t = Px2 inf{s > 0 : Xs = y 2 } ≤ t = Px2 τyX2 ≤ t .

Therefore using the formula (37), we can recover the formula for the first hitting time density of the Bessel process when S = 0 is killing. Indeed, in [10, p.391] one can see such formula: v 2v 2 t ∞ −Jv,n Jv xy Jv,n x x Px (τyR ≤ t) = . −2 e 2y2 y y Jv,n J1+v (Jv,n ) n=0

Spectral expansion for the reflected generalized Cox-Ingersoll-Ross process

In this section we will consider another process Y that is solution of (38) dYt = (β − bYt )dt + σ S − Yt dBt ,

with state space I = (−∞, S] if S is regular reflecting, or I = (−∞, S) if S is not regular reflecting. The infinitesimal generator is given by σ 2 (S − x) f (x) + (β − bx)f (x). 2 We want to find the density of ζy :=inf {t > 0 : Yt = y}. To this end, we will use the formula of the first hitting time density of the process X that is solution of (9). We first present three tables with the results of this section. Similar to the Tables 1, 2, and 3, Table 4 shows the nature of the end-point S and the type of boundary condition, Table 5 shows the solution of equation (11), and Table 6 shows the formula (14).

(39)

Lf (x) =

Table 4. Nature of the end-point S Parameters I . −1 <

2β−2bS σ2

S <0

with b = 0 II . 2β−2bS ≥0 σ2

Regular reflecting Exit

with b = 0

III .

2β−2bS σ2

≤ −1

with b = 0

Entrance

Boundary Condition ψx (x, λ) =0 lim s (x) x→S + lim ψ(x, λ) = 0

x→S +

lim+

x→S

ψx (x, λ) =0 s (x)

Hitting time for generalized CIR

Table 5. Solution ψ Parameters I . −1 < 2β−2bS <0 σ2 with b = 0 II . 2β−2bS ≥0 σ2 with b = 0

ψ(x, λ)=

III . 2β−2bS ≤ −1 σ2 with b = 0

Solution of Lψ + λψ = 0 −λ 2β b , σ2

ψ(x, λ) = F 2b(x−S) 1− σ2

2bS−2β σ2

ψ(x, λ) = F

2b(x−S) σ2

2b(x−S) 2bS−2β −λ +1, 2− 2bS−2β , b − σ2 σ2 σ2

−λ 2β b , σ2

2b(x−S) σ2

Table 6. Spectral decomposition Parameters I . −1 <

2β−2bS σ2

with b = 0 II .

2β−2bS σ2

≥0

with b = 0 III .

2β−2bS σ2

≤ −1

Spectral decomposition for the first hitting time density ∞ F −λb n , 2bS−2β , 2b(S−x) σ2 σ2 −λn t Px (ζy ≤ t) = 1 + e 2b(S−y) λn Fλ −λb n , 2bS−2β , n=1 2 2 σ σ Px (ζy ≤ t) =

2S−x S

Px (ζy ≤ t) = 1 +

with b = 0

s (z)dz

∞

e−λn t

n=1

∞

n=1

λ n Fλ

e−λn t

ψ(2S − x, λn ) λn ψλ (y, λn )

−λn 2bS−2β 2b(S−x) , σ2 , σ2 b

−λn 2bS−2β 2b(S−y) , σ2 , σ2 b

In order to analyze process Y , we use the Itô’s formula to contruct a new process Z, which will allow us to use the results of previous sections. Suppose that Y0 = x and let y be fixed such that S > x > y. Consider the function g(x, t) := −x + 2S. Applying the Itô’s formula we arrive at (40) dZt = (2bS − β − bZt )dt + σ Zt − S dBt , where Zt = −Yt + 2S. This process is (9) with µ := 2bS − β and a := b. Note that Z0 = 2S − x and define y := 2S − y. Then using the Proposition 4.0.1 for the end-point S, we obtain the following proposition. Proposition 8.0.1. The end-point S obeys the following classification i. If −1 < ii. If iii. If

2β − 2bS < 0 then S is regular. σ2

2β − 2bS ≥ 0 then S is exit. σ2 2β − 2bS ≤ −1 then S is entrance. σ2

Hitting time for generalized CIR

Proof. Notice that the nature of the end-point S for process Y is the same as for process Z. Then we can use Proposition 4.0.1. For the cases I and III in Proposition 8.0.1 we have to find a function ψ(x, λ) such that

(41)

σ 2 (x − S) ψ (x) + (2bS − β − bx)ψ (x) + λψ(x) = 0, and 2 ψx (x, λ) = 0. lim s (x) x→S +

In turn we have the following proposition. Proposition 8.0.2. If b = 0, then the function −λ 2bS − 2β 2b(x − S) (42) ψ(x, λ) = F , , , b σ2 σ2 is solution of (41). Proof. Is similar to Proposition 5.0.1. With the same changes of variable to obtain the Kummer equation. If we define ηy := inf{t > 0 : Zt = y}, then Px (ζy ≤ t) = P2S−x (ηy ≤ t).

(43)

Then applying the formula (14) we arrive at (44)

Px (ζy ≤ t) = P2S−x (ηy ≤ t) =1+

∞

e−λn t

n=1

λ n Fλ

−λn 2bS−2β 2b(S−x) , σ2 b , σ2

−λn 2bS−2β 2b(S−y) , σ2 b , σ2

Remark 8.0.3. To find an approximation for λn (with n large), we use the formula (23) 2 bS − β π(bS − β) nπ 3π σ2 × b. − + − (45) λn ≈ 2b(y − S) 2σ 2 2 8 σ2 Remark 8.0.4. For the case II in the Proposition 8.0.1, we have to find a function ψ(x, λ) such that Lψ + λψ = 0 and lim ψ(x, λ) = 0.

x→S +

If b = 0, we obtain that the solution ψ(x, λ) is given by the expression

2b(x − S) σ2

1− 2bS−2β 2 σ

−λ 2bS − 2β 2bS − 2β 2b(x − S) − + 1, 2 − , 2 b σ σ2 σ2

Hitting time for generalized CIR

Then applying the formula (14) and (8) we arrive at 2S−x ∞ s (z)dz −λn t ψ(2S − x, λn ) + (46) Px (ζy ≤ t) = P2S−x (ηy ≤ t) = S y , e λn ψλ (y, λn ) s (z)dz n=1 S 2bz

2β−2bS

where s (z) = e σ2 (z − S) σ2 . To find an approximation for λn (with n large), we use again the formula (23). Then we have 2 bS−β 2bS−2β π π bS−β − −n + −1 . − (47) λn ≈ −b × y 1− 2 σ 8 2 σ2 σ2

Numerical example

In this section, using the package Wolfram Mathematica 10.1, we give a illustration of a numerical approximation for the case III of Tables 1, 2 and 3. Consider the process X that is solution of (48) dXt = (3 − 2Xt )dt + Xt − 1 dBt .

Suppose that X0 = x = 98 and y = 54 . In this case, applying the Proposition 4.0.1, we have that the end-point S = 1 is entrance because (49)

2aS − 2µ 2·2·1−2·3 = −2 < −1. = 2 σ 1

Then the function ψ that we consider is (50)

ψ(x, λ) = F (−4λ, 1, 2x − 1) .

Using the formula (23) we obtain an approximation for the function ψ, and we also have approximations for the eigenvalues λn (ψ(y, λ) ≈ 0 with y = 54 ) for n = 1, 2, 3.... The approximation of the graph of ψ is drawn in Figure 1.

Figure 1: Approximation graph ψ. Now we give an approximation for cn using the formula (25) to have one picture of an approximation the first hitting time density. For our estimation,

Hitting time for generalized CIR

Figure 2: Approximation first hitting time density. we have truncated the serie (21) at the first 100 terms. The graph of this approximation is presented in the Figure 2.

Acknowledgement I thank the University of Costa Rica and CINVESTAV, also to Carlos Pacheco for his comments. Jonathan Gutierrez Pavón Department of mathematics, University of Costa Rica, San José, Costa Rica, jonathan.gutierrez@ucr.ac.cr

References [1] Albanese, C. and Kuznetsov, A. Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions. Arxiv file http://arxiv.org/abs/0710.1596. [2] Bingham, N.H. and Kiesel, R. (2004). Risk-Neutral Valuation SpringerVerlag (2nd Edition). [3] Borodin, A. and Salminen, P. (1996). Handbook Brownian Motion: Facts and Formulae. Birkhauser Verlag, Basel. [4] Giorno, V. Nobile, A. G. Ricciardi, L. M. and Sato, S. (1989). On the evaluation of the first passage time densities via non-singular integral equations. Advances In Applied Probability, 20-36. [5] Gutiérrez, J. (2017). Spectral Decompositions for diffusions. Ph. D. thesis, CINVESTAV. [6] Klebaner, F. C. (2005). Introduction to Stochastic Calculus with applications. Imperial College Press.

Hitting time for generalized CIR

[7] Lando, D. (2004). Credit Risk Modeling. Princeton University Press. [8] Lánský, P. Smith, C. E. and Ricciardi, L. M. (1990). One-dimensional stochastic diffusion models of neuronal activity and related first passage time problems. Trends In Biological Cybernetics, 153-162. [9] Linetsky, V. (2004). The spectral descomposition of the option value. International Journal of Theoretical and Applied Finance 7, 337-384. [10] Linetsky, V. (2004). Lookback options and diffusion hitting time: a spectral expansion approach. Finance and Stochastics 8, 373-398. [11] Linetsky, V. (2004). Computing hitting time densities for CIR and OU diffusions: Applications to mean-reverting models. Journal of Computational Finance 7. [12] Matomäki, P. (2013). On two sided controlls of a linear diffusion. PhD thesis, Turku School of Economics. [13] Polyanin, A. and Zaitsev, V. (1995). Handbook of Exact Solutions for Ordinary differential Equations. CRC Press, Boca Raton. [14] Polyanin, A. and Zaitsev, V. (2002). Handbook of Linear Partial Differential Equations for Engineers and Scientists. CRC Press, Boca Raton. [15] Slater, L. (1960). Confluent hypergeometric functions. Cambridge University Press. [16] Taira, K. (2014). Semigroups, Boundary Value Problems and Markov Processes Springer Monographs in Mathematics.

Morfismos se imprime en el taller de reproducción del Departamento de Matemáticas del Cinvestav, localizado en Avenida Instituto Politécnico Nacional 2508, Colonia San Pedro Zacatenco, C.P. 07360, México, D.F. Este número se terminó de imprimir en el mes de febrero de 2021. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

Apoyo técnico: Omar Hernández Orozco.

Contents - Contenido Alternative to Euler’s formula for Bernoulli numbers

∞

1 n=1 n2k

with k ∈ Z + and for even indexed