Analytic moment and Laplace transform formulae for the quasi-stationary distribution of the Shiryaev diffusion on an interval
aa r X i v : . [ s t a t . M E ] M a y Statistical Papers manuscript No. (will be inserted by the editor)
Analytic moment and Laplace transform formulae forthe quasi-stationary distribution of the Shiryaev diffusionon an interval
Aleksey S. Polunchenko · Andrey Pepelyshev
Received: date / Accepted: date
Abstract
We derive analytic closed-form moment and Laplace transform formulae for thequasi-stationary distribution of the classical Shiryaev diffusion restricted to the interval [ , A ] with absorption at a given A > Keywords
Laplace transform · Markov diffusions · Quasi-stationarity · Shiryaev process · Special functions · Stochastic processes
Mathematics Subject Classification (2000) · This work is an investigation into quasi-stationarity of the classical Shiryaev diffusion re-stricted to an interval. Specifically, the focus is on the solution ( R rt ) t ≥ of the stochasticdifferential equation dR rt = dt + R rt dB t with R r : = r ≥ , (1)where ( B t ) t ≥ is standard Brownian motion in the sense that E [ dB t ] = E [( dB t ) ] = dt , and B =
0. The time-homogeneous Markov process ( R rt ) t ≥ is an important particular version The effort of A.S. Polunchenko was partially supported by the Simons Foundation via a Collaboration Grantin Mathematics under Award of the so-called generalized Shiryaev process. The latter has been first arrived at and stud-ied by Prof. A.N. Shiryaev—hence the name—in his fundamental work [29,30] on quickestchange-point detection. While interest to the Shiryaev process in the context of quickestchange-point detection has never weakened (see, e.g., [20,31,10,5,22,25,23,24]), the pro-cess has received a great deal of attention in other areas as well, notably in mathematicalfinance (see, e.g., [12,8,15]) and in mathematical physics (see, e.g., [18,7]). It has also beenconsidered in the literature on general stochastic processes (see, e.g., [35,36,8,9,28,19,27,26]).The particular version of the Shiryaev process ( R rt ) t ≥ governed by equation (1) is ofspecial importance and interest because it is the only version with probabilistically nontrivialbehavior in the limit as t → + ∞ , exhibited in spite of the distinct martingale property E [ R rt − r − t ] = t ≥ r ≥
0. Moreover, the process is convergent (as t → + ∞ ) regardlessof whether the state space is (I) the entire half-line [ , + ∞ ) with no absorption on the interior;or (II) the interval [ , A ] with absorption at a given level A >
0; or (III) the shortened half-line [ A , + ∞ ) also with absorption at A > r wastouched upon in [19]. Cases (I), (II), and (III) have all been considered in the literature,which we now briefly review.Case (I) is the easiest case. The asymptotic (as t → + ∞ ) distribution of ( R rt ) t ≥ in thiscase is known as the stationary distribution. Formally, the latter is defined as H ( x ) : = lim t → + ∞ P ( R rt ≤ x ) and h ( x ) : = ddx H ( x ) , (2)and it has already been found, e.g., in [29,30,20,10,5,27], to be the momentless distribution H ( x ) = e − x { x ≥ } and h ( x ) = x e − x { x ≥ } , (3)which is an extreme-value Fr´echet-type distribution. Exact closed-form formulae for thedistribution of R rt for any given t ≥ r ≥ t → + ∞ )distributions are quasi -stationary distributions, i.e., stationary but conditional on extendedsurvival. Formally, consider the stopping time S rA : = inf { t ≥ R rt = A } such that inf { ∅ } = + ∞ , where R r : = r ≥ A > Q A ( x ) : = lim t → + ∞ P ( R rt ≤ x | S rA > t ) and q A ( x ) : = ddx Q A ( x ) , (4)and it does depend on whether r ∈ [ , A ] , which is case (II), or r ∈ [ A , + ∞ ) , which is case (III),but the specific value of r inside the state space of choice is irrelevant.Case (III) is arguably the least understood case. To the best of our knowledge, the firstattempt to treat this case was made in [6, Section 7.8.2] where the authors proved that notonly does the quasi-stationary distribution exist for any A >
0, but also that there is a wholeparametric continuum of quasi-stationary distributions when A is not sufficiently large. Fur-ther progress on this case was recently made in [26] where Q A ( x ) and q A ( x ) were, for thefirst time, found analytically for any A >
0. It was also shown in [26] that the quasi-stationarydistribution is unique whenever A ≥ A ∗ ≈ . A ∗ is the solution of a certain n quasi-stationarity of the Shiryaev diffusion 3 transcendental equation. While case (III) may be the least understood case, the focus of thiswork is entirely on case (II), which is discussed next along with the motivation.Case (II) is of importance in quickest change-point detection, and in this context, it wasinvestigated in, e.g., [20,5,25]. See also, e.g., [21,15] and [6, Section 7.8.2]. For example,it is known from [20,21] that, expectedly, the limit of Q A ( x ) , defined in (4), as A → + ∞ is H ( x ) , defined in (2) and given by (3); the convergence is from above, and is pointwise, atevery x ∈ [ , + ∞ ) , i.e., at all continuity points of H ( x ) . Moreover, analytic closed-form for-mulae for Q A ( x ) and q A ( x ) were recently obtained in [25], apparently for the first time in theliterature; see formulae (10) and (11) below. To boot, the distribution of R rt conditional on noextinction prior to time t >
0, for any given t > r ∈ [ , A ) has been derived explicitlyas well (see, e.g., [22,15]); this conditional distribution becomes the quasi-stationary distri-bution in the limit, as t → + ∞ . Due to its connection to quickest change-point detection, itis case (II) that is of interest to this work, which is also motivated by quickest change-pointdetection. Notwithstanding all the headway made lately on case (II), gaps do remain, andthis work seeks to fill some of these gaps in.More precisely, the contribution of this work in relation to case (II) is two-fold: (a) obtainexact closed-form moment formulae for the quasi-stationary distribution; and subsequentlyuse the moment formulae to (b) derive an exact formula (in different forms) for the Laplacetransform of the quasi-stationary distribution. The moment formulae are obtained as an ex-tension of the effort made earlier in [25] where the moment sequence was shown to satisfy acertain recurrence whose closed-form solution, at the time, seemed out of reach. This work“runs that leg” and solves the recurrence explicitly. This is done in the first half of Section 3,which is the main section of the present paper. The second half of Section 3 is devoted tothe computation of the Laplace transform in two different ways: first using the obtained mo-ment formulae, and then also by solving a certain order-two ordinary differential equationthat the Laplace transform of interest can be easily shown (see [25]) to satisfy. Since nearlyall of the formulae involve special functions, we conveniently preface Section 3 and thederivations therein with Section 2 which introduces the relevant special functions. Lastly,Section 4 wraps up the entire paper with a few concluding remarks. For convenience we shall adapt the standard notation employed uniformly across mathe-matical literature. In particular, this applies to a host of special functions we shall deal withthroughout the sequel. These functions, in their most common notation, are:1. The Gamma function Γ ( z ) , z ∈ C , frequently also referred to as the extension of thefactorial to complex numbers, due to the property Γ ( n ) = ( n − ) ! exhibited for n ∈ N .See, e.g., [2, Chapter 1].2. The Pochhammer symbol, or the rising factorial, often notated as ( z ) n and defined for z ∈ C and n ∈ N ∪ { } as ( z ) n : = ( , for n = z ( z + ) · · · ( z + n − ) , for n ∈ N , and it is of note that ( ) n = n ! for any n ∈ N ∪ { } . See, e.g., [33, pp. 16–18]. Also,observe that ( z ) n = Γ ( z + n ) Γ ( z ) for n ∈ N ∪ { } and z ∈ C \ { , − , − , . . . } , A.S. Polunchenko, A. Pepelyshev and if z is a negative integer or zero, i.e., if z = − k and k ∈ N ∪ { } , then ( − k ) n = ( − ) n k ! ( k − n ) ! , for n = , , . . ., k ;0 , for n = k + , k + , . . . ; (5)cf. [33, p. 16–17].3. The special case of the generalized hypergeometric function (see, e.g., [2, Chapter 4])with two numeratorial and two denominatorial parameters. The function, denoted as F [ z ] , is defined via the power series F a , a b , b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z : = ∞ ∑ n = ( a ) n ( a ) n ( b ) n ( b ) n z n n ! , (6)where b , b
6∈ { , − , − , . . . } and | z | < + ∞ . See [33, p. 20]. It is of note that whenonly one of the numeratorial parameters a i , i = ,
2, is a negative integer or zero, then, inview of (5), the power series on the right of (6) terminates, thereby turning the function F [ z ] into a polynomial in z of degree − a i .4. The Whittaker M and W functions, traditionally denoted, respectively, as M a , b ( z ) and W a , b ( z ) , where a , b , z ∈ C . These functions were introduced by Whittaker [34] as thefundamental solutions to the Whittaker differential equation. See, e.g., [32,4].5. The modified Bessel functions of the first and second kinds, conventionally denoted,respectively, as I a ( z ) and K a ( z ) , where a , z ∈ C ; the index a is referred to as the function’sorder. See [3, Chapter 7]. These functions form a set of fundamental solutions to themodified Bessel differential equation. The modified Bessel K function is also known asthe MacDonald function.6. The particular case of the generalized bivariate Kamp´e de F´eriet function F −−−− : a , a ; 1 b , b : −−−−− ; −− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) xy , x : = ∞ ∑ i = ∞ ∑ j = ( a ) i ( a ) i ( ) j ( b ) i + j ( b ) i + j ( xy ) i x j i ! j ! , (7)which is well-defined for b , b
6∈ { , − , − , . . . } and | x | < + ∞ and | y | < + ∞ . See [33,p. 27]. The above F [ x , y ] function was introduced in [14], and is slightly more gen-eral than the original Kamp´e de F´eriet function proposed by Prof. J. Kamp´e de F´erietin [11]. As was mentioned in the introduction, the quasi-stationary distribution defined in (4) wasrecently expressed analytically in [22] through the Whittaker W function. Specifically, itcan be deduced from [22, Theorem 3.1] that if A > λ ≡ λ A > W , ξ ( λ ) (cid:18) A (cid:19) = , (8) n quasi-stationarity of the Shiryaev diffusion 5 where ξ ( λ ) : = p − λ so that λ = (cid:16) − (cid:2) ξ ( λ ) (cid:3) (cid:17) , (9)then the quasi-stationary probability density function (pdf) is given by q A ( x ) = e − x x W , ξ ( λ ) (cid:18) x (cid:19) e − A W , ξ ( λ ) (cid:18) A (cid:19) { x ∈ [ , A ] } , (10)and the respective cumulative distribution function (cdf) is given by Q A ( x ) = , if x ≥ A ; e − x W , ξ ( λ ) (cid:18) x (cid:19) e − A W , ξ ( λ ) (cid:18) A (cid:19) , for x ∈ [ , A ) ;0 , otherwise , (11)and q A ( x ) and Q A ( x ) are each a smooth function of x ∈ [ , A ] and A >
0; observe that q A ( A ) = q A ( x ) and Q A ( x ) is due to the fact thatthe Whittaker W function on the right of (10) and (11) is an analytic function of its argumentas well as of each of its two indices. Remark 1
The definition (9) of ξ ( λ ) can actually be changed to ξ ( λ ) : = −√ − λ withno effect whatsoever on either equation (8), or formulae (10) and (11), i.e., all three areinvariant with respect to the sign of ξ ( λ ) . This was previously pointed out in [25], and thereason for this ξ ( λ ) -symmetry is because equation (8) and formulae (10) and (11) eachhave ξ ( λ ) present only as (double) the second index of the corresponding Whittaker W function or functions involved, and the Whittaker W function in general is known (see,e.g., [4, Identity (19), p. 19]) to be an even function of its second index, i.e., W a , b ( z ) = W a , − b ( z ) .It is evident that equation (8) is a key ingredient of formulae (10) and (11), and conse-quently, of all of the characteristics of the quasi-stationary distribution as well. As a transcen-dental equation, it can only be solved numerically, although to within any desired accuracy,as was previously done, e.g., in [15,22,25,23], with the aid of Mathematica developed byWolfram Research:
Mathematica ’s special functions capabilities have long proven to be su-perb. Yet, it is known (see [15,22]) that for any fixed A >
0, the equation has countably manysimple solutions 0 < λ < λ < λ < · · · , such that lim k → + ∞ λ k = + ∞ . All of them dependon A , but since we are interested only in the smallest one, we shall use either the “short”notation λ , or the more explicit λ A to emphasize the dependence on A . Also, it can be con-cluded from [22, p. 136 and Lemma 3.3] that λ A is a monotonically decreasing function of A such that lim A → + ∞ λ A =
0, and more specifically λ A = A − + O ( A − / ) . Remark 2
Since λ ≡ λ A is monotonically decreasing in A , and such that lim A → + ∞ λ A = ξ ( λ ) is either (a) purely imaginary (i.e., ξ ( λ ) = i α wherei : = √− α ∈ R ) if A is sufficiently small, or (b) purely real and between 0 inclusiveand 1 exclusive (i.e., 0 ≤ ξ ( λ ) <
1) otherwise. The borderline case is when ξ ( λ ) = λ A = /
8, and the corresponding critical value of A is the solution ˜ A > A.S. Polunchenko, A. Pepelyshev the equation W , ( / ˜ A ) =
0. A basic numerical calculation gives ˜ A ≈ . A < ˜ A ≈ . λ A > / ξ ( λ ) is purely imaginary; otherwise, if A ≥ ˜ A ≈ . λ A ∈ ( , / ] so that ξ ( λ ) is purely real and such that ξ ( λ ) ∈ [ , ) withlim A → + ∞ ξ ( λ A ) = λ A = A − + O ( A − / ) was first established (in a more general form)in [25] with the aid of Jensen’s inequality applied to ascertain that the variance of thequasi-stationary distribution (10)–(11) is strictly positive. This is an example of potentialapplications of the quasi-stationary distribution’s low-order moments. We now recover thedistribution’s entire moment series.3.1 The moment seriesLet Z be a random variable sampled from the quasi-stationary distribution given by (10)and (11). Let M n : = E [ Z n ] denote the n -th moment of Z for n ∈ N ∪{ } ; it is to be understoodthat M ≡ A >
0, and that all other M n ’s actually do depend on A . For every fixed A >
0, the series { M n } n ≥ can be inferred from [25, Theorem 3.2, p. 136] to satisfy therecurrence (cid:18) n ( n − ) + λ (cid:19) M n + n M n − = λ A n , n ∈ N , (12)with M ≡
1; recall that λ ≡ λ A and A are interconnected via equation (8). While recur-rence (12) may seem easy to iterate forward on a computer, a general closed-form expressionfor M n for any n ∈ N ∪ { } would be more convenient, especially for analytic purposes. Tothat end, it was lamented in [25] that although the recurrence is possible to solve explicitly,the solution is too cumbersome. We now show that the solution can be expressed compactlythrough the hypergeometric function F [ z ] defined in (6). Lemma 1
For every A > fixed, the solution { M n } n ≥ to the recurrence (12) is given by M n = λ A n n ( n − ) + λ F , − n + ξ ( λ ) − n , − ξ ( λ ) − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A , n ∈ N ∪ { } , (13) where λ ≡ λ A ( > ) is determined by (8) while ξ ( λ ) is defined in (9) ; recall also that F [ z ] denotes the generalized hypergeometric function (6) .Proof The idea is to first rewrite (12) equivalently as (cid:2) n ( n + ) + λ (cid:3) M n + + ( n + ) M n = λ A n + , and then substitute M n of the form M n = λ A n n ( n − ) + λ m ( n , A ) , where m ( n , A ) is the new unknown. After some elementary algebra this gives − ( n + ) A m ( n , A ) + (cid:2) n ( n − ) + λ (cid:3)(cid:2) − m ( n + , A ) (cid:3) = , n quasi-stationarity of the Shiryaev diffusion 7 which can be recognized as a particular case of the contiguous function identity ( b − a ) z F a + , b + c + , d + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + cd F a , b + c , d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z − F a + , bc , d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = , that the function F [ z ] defined in (6) is known to satisfy: it suffices to set a : = , b : = − n − , c : = − + ξ ( λ ) − n , d : = − − ξ ( λ ) − n , and z : = A , and observe directly from (6) that F , a b , b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = , for any appropriate a , b and b . ⊓⊔ It is clear that the obtained formula (13) is symmetric with respect to ξ ( λ ) , as it shouldbe, by Remark 1. More importantly, since one of the numeratorial parameters of the function F [ z ] on the right of (13) is from the set { , − , − , − , . . . } , the power series buried insidethe generalized hypergeometric function terminates, so that M n ends up being a polynomialof degree n in A . However, the coefficients of the polynomial do depend on λ ≡ λ A , andsince the latter is connected to A via the transcendental equation (8), the actual nature ofdependence of M n on A is more complicated than polynomial. Specifically, from (5), (6),and the identity ( z ) n − k = ( − ) k ( z ) n ( − z − n ) k , k = , , , . . ., n , as given, e.g., by [33, Formula (10), p. 17], we readily obtain F , − na − n , b − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = n ∑ k = ( ) k ( − n ) k ( a − n ) k ( b − n ) k z k k ! = n ∑ k = ( − n ) k ( a − n ) k ( b − n ) k z k = n ! n ∑ k = ( a − n ) k ( b − n ) k ( − ) k ( n − k ) ! z k = n ! n ∑ k = ( − ) k ( − a ) n − k ( − b ) n − k ( − a ) n ( − b ) n ( − z ) k ( n − k ) ! = n ! ( − z ) n ( − a ) n ( − b ) n n ∑ k = ( − a ) k ( − b ) k ( − z ) − k k ! , A.S. Polunchenko, A. Pepelyshev whence F , − n + ξ ( λ ) − n , − ξ ( λ ) − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A == ( − ) n n ! A − n (cid:18) − + ξ ( λ ) (cid:19) n (cid:18) − − ξ ( λ ) (cid:19) n ×× n ∑ k = (cid:18) − + ξ ( λ ) (cid:19) k (cid:18) − − ξ ( λ ) (cid:19) k k ! (cid:18) − A (cid:19) k , (14)and subsequently, in view of (13), we finally find M n = ( − ) n n ! (cid:18) + ξ ( λ ) (cid:19) n (cid:18) − ξ ( λ ) (cid:19) n ×× n ∑ k = (cid:18) − + ξ ( λ ) (cid:19) k (cid:18) − − ξ ( λ ) (cid:19) k k ! (cid:18) − A (cid:19) k , n ∈ N ∪ { } , (15)where again λ ≡ λ A ( > ) is determined by (8) and ξ ( λ ) is defined in (9); this formula isalso invariant with respect to the sign of ξ ( λ ) .Let us now briefly contrast the two obtained formulae (13) and (15). To this end, observefirst that formula (15) is more explicit than formula (13): unlike the latter, the former isfree of special functions, and can thus provide more insight into the relationship between M n and A . A better understanding of this relationship can, in turn, shed more light on therelationship between λ ≡ λ A and A , an important question difficult to answer by directanalysis of the transcendental equation (8) connecting the two. For example, from (15) andthe trivial observation that M n > n we readily obtain M = A − λ A > [ Z ] = M − M = λ A − ( A λ A − ) λ A ( + λ A ) > , whence 1 A < λ A < A + + √ A + A for any A > , (16)so that λ A = A − + O ( A − / ) ; cf. [25]. For applications of this result in quickest change-point detection see [24,23]. Similarly, since the quasi-stationary distribution is supported onthe interval [ , A ] , we may further deduce that M n ≤ A i M n − i for any i ∈ { , , . . ., n } and n ∈ N ∪ { } . For n = i =
1, after some elementary algebra, this leads to the lower-bound 1 A + A + A < λ A for any A > , which clearly improves the left half of the double inequality (16). By “playing around” withthe moments more, one can tighten up the lower- and upper-bounds for λ A even further,although every such improvement will come at the price of increased complexity of thebounds. That said, the bounds will remain fully amenable to numerical evaluation. See [25]for very accurate high-order bounds. n quasi-stationarity of the Shiryaev diffusion 9 On the other hand, formula (13) is more convenient than formula (15) to implement insoftware, especially in Wolfram
Mathematica with its excellent special functions capabili-ties. To illustrate this point, we implemented formula (13) in a
Mathematica script, and usedthe script to produce Figures 1 and 2 which show the behavior of M n as a function of A with n fixed and as a function of n with A fixed, respectively; note the different ordinate scalesin the figures. Figures 1(a)–1(f) make it clear that if n is fixed, then M n is an increasingfunction of A , concave for n = M n , the in-creasing nature of its dependence on A is in alignment with one’s intuition. The concavity ofthe M n -vs- A curve for n = n ≥ λ A = A − + O ( A − / ) , implying lim A → + ∞ (cid:0) λ A A (cid:1) = A → + ∞ (cid:0) λ + κ A A (cid:1) = κ >
0; cf. [25,23]. The dependence of M n on n for a fixed A has its nuances too: ascan be seen from Figures 2(a)–2(f), if A is sufficiently small (as in around 1 or even less),then M n is a decreasing function of n , and otherwise M n is an increasing function of n .This is essentially because f ( x ) : = a x with a > x for a > a ∈ ( , ) . It is also noteworthy that the rate of growth (or,correspondingly, the rate of decay) of M n as a function of n with A fixed or as a function of A with n fixed (at 2 or higher) is rather steep: an eye examination of Figures 1(b)–1(f) andFigures 2(a)–2(f) suggests that it is at least exponential, and the rate is the higher, the higherthe (fixed) value of n or A .However, as we shall see below, should one wish to compute the Laplace transformof the quasi-stationary distribution (10)–(11), either of the two formulae is instrumental,although one may find formula (15) to be of greater help than formula (13). The details aswell as the actual computation of the Laplace transform are offered in the next subsection.3.2 Laplace transformWe now use the moment formulae obtained above to recover the Laplace transform of thequasi-stationary distribution (4). Specifically, recall that, for each A > q A ( x ) is given explicitly by (10), and since it is supported on the interval [ , A ] , its Laplace transform can be defined as the integral L Q ( s ) ≡ L Q { q A ( x ) ; x → s } ( s , A ) : = Z A e − sx q A ( x ) dx , s ≥ , (17)and it is connected to the quasi-stationary distribution’s moment sequence { M n } n ≥ , giveneither by (13) or by (15), via the standard identity M n = ( − ) n (cid:20) d n ds n L Q ( s ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) s = , (18)leading to the classical power series representation of the Laplace transform L Q ( s ) = ∞ ∑ n = ( − s ) n n ! M n , (19)which is nothing but the Taylor expansion of L Q ( s ) around the origin. It is this expansion,rather than definition (17), that we intend to employ shortly to compute L Q { q A ( x ) ; x → s } ( s , A ) , although with some restrictions on s and A . The reason to prefer (19) along with (13)and (15) over (17) and (10) is the presence of the Whittaker W function on the right of the (a) n = (b) n = × × × × × (c) n = × × × × × (d) n = × × × × × × × (e) n = × × × × × (f) n = Fig. 1: Quasi-stationary distribution’s n -th moment M n as a function of A for A ∈ [ , ] and n ∈ { , , , , , } .quasi-stationary pdf formula (10): the Whittaker W function is a special function direct inte-gration of which as in (17) is unlikely an option, for existing handbooks of special functionsappear to offer no suitable integral identities. By contrast, the power series (19) and the ex-plicit moment formulae (13) and (15) provide a more straightforward way to recover L Q ( s ) .However, one should keep in mind that the domain of convergence of the series need not beas large as the region of convergence of the integral (17) defining L Q ( s ) . n quasi-stationarity of the Shiryaev diffusion 11 (a) A = × × × × × × × (b) A = × × × × × × (c) A = × × × × × × (d) A = × × × × × (e) A = × × × × × (f) A = Fig. 2: Quasi-stationary distribution’s n -th moment M n as a function of n for n ∈{ , , , , , , , , , } and A ∈ { , , , , , } . Lemma 2
For every A > fixed and finite, the Laplace transform L Q { q A ( x ) ; x → s } ( s , A ) of the quasi-stationary distribution (10) – (11) is given by L Q { q A ( x ) ; x → s } ( s , A ) == F −−−−−−−−−−−−−−−− : − − ξ ( λ ) , − + ξ ( λ ) − ξ ( λ ) , + ξ ( λ ) −−−−−−−−−−−−−−−−−− ; −− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − sA , s , (20) where s ∈ [ , + ∞ ) , and λ ≡ λ A ( > ) is determined by (8) while ξ ( λ ) is defined in (9) ; recallalso that F [ x , y ] denotes the Kamp´e de F´eriet function (7) . Proof
If we tentatively set a : = + ξ ( λ ) b : = − ξ ( λ ) L Q { q A ( x ) ; x → s } ( s , A ) == ∞ ∑ n = ( ( s ) n ( a ) n ( b ) n n ∑ k = ( − a ) k ( − b ) k k ! (cid:18) − A (cid:19) k ) = ∞ ∑ k = ( ( − a ) k ( − b ) k k ! (cid:18) − A (cid:19) k ∞ ∑ n = k ( s ) n ( a ) n ( b ) n ) = ∞ ∑ k = ∞ ∑ n = ( − a ) k ( − b ) k ( ) n ( a ) n + k ( b ) n + k ( − sA ) k ( s ) n k ! n ! = F −−− : 1 − a , − b ; 1 a , b : −−−−−−−− ; −− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − sA , s , and the desired result is now apparent. ⊓⊔ The obtained Laplace transform formula (20) was arrived at through the transform’spower series expansion (19) and the quasi-stationary distribution’s n -th moment formula(15). However, since the n -th moment also has the alternative but equivalent representa-tion (13), the latter, too, by virtue of the power series expansion (19), can be used to obtaina (different, but equivalent) expression for the Laplace transform. Lemma 3
For every A > fixed and finite, the Laplace transform L Q { q A ( x ) ; x → s } ( s , A ) of the quasi-stationary distribution (10) – (11) is given by L Q { q A ( x ) ; x → s } ( s , A ) = λ s ×× F −−−−−−−−−−−−−−−−−− : − − ξ ( λ ) , − + ξ ( λ ) − − ξ ( λ ) , − + ξ ( λ ) −−−−−−−−−−−−−−−−−− ; −− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − sA , s −− e − sA , (21) where s ∈ [ , + ∞ ) , and λ ≡ λ A ( > ) is determined by (8) while ξ ( λ ) is defined in (9) ; recallalso that F [ x , y ] denotes the Kamp´e de F´eriet function (7) .Proof The idea is to multiply equation (12) through by ( − s ) n / n ! to obtain (cid:18) n ( n − ) + λ (cid:19) ( − s ) n n ! M n − s ( − s ) n − ( n − ) ! M n − = λ ( − sA ) n n ! , n quasi-stationarity of the Shiryaev diffusion 13 which, in conjunction with (19), readily gives s L Q { q A ( x ) ; x → s } ( s , A ) = s ∞ ∑ n = ( − s ) n − ( n − ) ! M n − = ∞ ∑ n = (cid:18) n ( n − ) + λ (cid:19) ( − s ) n n ! M n − λ ∞ ∑ n = ( − sA ) n n ! = ∞ ∑ n = (cid:2) n ( n − ) + λ (cid:3) ( − s ) n n ! M n − λ ! − λ (cid:0) e − sA − (cid:1) = ∞ ∑ n = (cid:2) n ( n − ) + λ (cid:3) ( − s ) n n ! M n − λ e − sA , so that if we could now show that12 ∞ ∑ n = (cid:2) n ( n − ) + λ (cid:3) ( − s ) n n ! M n == λ F −−−−−−−−−−−−−−−−−− : − − ξ ( λ ) , − + ξ ( λ ) − − ξ ( λ ) , − + ξ ( λ ) −−−−−−−−−−−−−−−−−− ; −− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − sA , s , (22)then the proof would be complete. To show (22), introduce a : = + ξ ( λ ) b : = − ξ ( λ ) ∞ ∑ n = (cid:2) n ( n − ) + λ (cid:3) ( − s ) n n ! M n == λ ∞ ∑ n = ( − sA ) n n ! F , − na − n , b − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A = λ ( ∞ ∑ n = ( − s ) n ( − a ) n ( − b ) n n ∑ k = ( − a ) k ( − b ) k k ! (cid:18) − A (cid:19) k ) = λ ( ∞ ∑ k = ( − a ) k ( − b ) k k ! (cid:18) − A (cid:19) k ∞ ∑ n = k ( − s ) n ( − a ) n ( − b ) n ) = λ ∞ ∑ k = ∞ ∑ n = ( − a ) k ( − b ) k ( ) n ( − a ) n + k ( − b ) n + k ( − sA ) k ( s ) n k ! n ! = λ F −−−−−−−− : 1 − a , − b ; 11 − a , − b : −−−−−−−− ; −− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − sA , s , which, in view of (23), can be recognized to be exactly (22). The proof is now complete. ⊓⊔ We now return to the point made earlier about the domain of convergence of the se-ries (19) potentially being narrower than the region of convergence of the integral (17)defining L Q ( s ) . This is, in fact, the case, for the obtained Laplace transform formulae (20)and (21) both break down in the limit, as either A → + ∞ or s → + ∞ . The reason is becausethe Kamp´e de F´eriet function involved in either formula is well-defined only when both ofits two arguments are finite. That said, except for the two limiting cases—one as A → + ∞ and one as s → + ∞ —formulae (20) and (21) are valid.At this point one may rightly remark that the Kamp´e de F´eriet function in general isa somewhat “exotic” special function, although its importance appears to have been well-understood in the literature on mathematical physics. To that end, an interesting question iswhether the function F [ x , y ] on the right of formula (20) permits an alternative expressioninvolving either no special functions at all, or, in the worst case, only “less exotic” specialfunctions. While it is very unlikely that our particular function F [ x , y ] can be reduced toa form completely free of special functions, it may be possible to express it in terms of fairlywidespread modified Bessel functions of the first and second kinds, conventionally denotedas I a ( z ) and K a ( z ) , respectively. This possibility is indicated by [17, Identity (4.2a), p. 184]which states that F −−−−−−−−−−−−−−− : a + b + , a − b +
12 ; 1 a + b + , a − b +
32 : −−−−−−−−−−−−−−− ; −− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x y , y == ( a + b + )( a − b + ) y a + ( I b ( y ) Z y e x u u a K b ( u ) du − K b ( y ) Z y e x u u a I b ( u ) du ) , (24)valid so long as ℜ ( + a ± b ) >
0; the condition ℜ ( + a ± b ) > I function I b ( z ) ∼ Γ ( b + ) (cid:16) z (cid:17) b , as z → , provided b
6∈ {− , − , − , . . . } , (25)as given, e.g., by [1, Property 9.6.7, p. 375], and that of the modified Bessel K function K b ( z ) ∼ Γ ( b ) (cid:16) z (cid:17) − b , as z → , provided ℜ ( b ) > , (26)as given, e.g., by [1, Property 9.6.9, p. 375], are such that the two integrals on the rightof (24), i.e., the integrals Z y e x u u a I b ( u ) du and Z y e x u u a K b ( u ) du , are convergent, for any y ∈ [ , + ∞ ) ; cf. [16]. Incidentally, the foregoing two integrals areexamples of incomplete Weber integrals, which arise in mathematical physics and in certainareas of probability theory; see, e.g., [16,17].It is plain to see that the Kamp´e de F´eriet function on the left of identity (24) with a = − b = ξ ( λ ) is of precisely the same form as the Kamp´e de F´eriet function onthe right of the Laplace transform formula (20). However, identity (24) with a = − b = ξ ( λ ) , which is the case we are interested in, does not hold true. This is due to tworeasons. First, the condition ℜ ( + a ± b ) > a = − b = ξ ( λ ) , because n quasi-stationarity of the Shiryaev diffusion 15 ξ ( λ ) , as was explained in Remark 2, is either purely imaginary (so that ℜ ( b ) =
0) or purelyreal and between 0 inclusive and 1 exclusive (so that 0 ≤ b < b = ξ ( λ ) happens to be connected (and in very specific manner!)to the first argument of the Kamp´e de F´eriet function; the connection is through equation (8).Yet, although not directly applicable in our case, identity (24) is still of value: observe thatits right-hand side resembles the variation of parameters formula for a particular solutionto a second-order nonhomogeneous ordinary differential equation. Moreover, this equationis not too difficult to “reverse engineer”. To this end, it can be deduced from [25] that, forevery A > L Q ( s ) ≡ L Q { q A ( x ) ; x → s } ( s , A ) defined in (17)is the solution L ( s ) ≡ L ( s , A ) of the equation s ∂ ∂ s L ( s ) − ( s − λ ) L ( s ) = λ e − sA , s ≥ , (27)where recall that λ ≡ λ A ( > ) and A are coupled together via equation (8). As we shallsee shortly, the right-hand side of identity (24) with a = − b = ξ ( λ ) is preciselywhat the method of variation of parameters yields as a particular solution to the foregoingequation (27). However, this particular solution is not the solution, because it does not satisfythe appropriate boundary conditions, which are lim s → + L ( s ) =
1, lim s → + ∞ L ( s ) =
0, and (cid:20) d n ds n L ( s ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) s = = ( − ) n M n , n ∈ N , (28)where M n is the n -th moment of the quasi-stationary distribution; recall formulae (13)and (15) we established for M n in the preceding subsection. The first two of the boundaryconditions come from the definition (17) of the Laplace transform, and the third conditionis due to (18).To solve equation (27) directly, observe that the change of variables s u ≡ u ( s ) : = √ s and the substitution L ( s ) L ( u ) : = u ℓ ( u ) together convert the equation into u ∂ ∂ u ℓ ( u ) + u ∂∂ u ℓ ( u ) − (cid:16) u + (cid:2) ξ ( λ ) (cid:3) (cid:17) ℓ ( u ) = λ u e − A u , (29)which is a nonhomogeneous version of the modified Bessel equation. Hence, by definition,the two fundamental solutions, ℓ ( ) ( u ) and ℓ ( ) ( u ) , to the homogeneous version of the equa-tion are ℓ ( ) ( u ) : = I ξ ( λ ) ( u ) and ℓ ( ) ( u ) : = K ξ ( λ ) ( u ) , which can be used to construct a particular solution, ℓ ( p ) ( u ) , to the nonhomogeneous equa-tion via variation of parameters. Specifically, since the Wronskian between I a ( z ) and K a ( z ) is W { K a ( z ) , I a ( z ) } : = K a ( z ) ddz I a ( z ) − I a ( z ) ddz K a ( z ) = z , as given, e.g., by [1, Formula 9.6.15, p. 375], the basic variation of parameters formulaasserts, after some calculation, that the function ℓ ( p ) ( u ) : = λ ( I ξ ( λ ) ( u ) Z u e − A x K ξ ( λ ) ( x ) dxx − K ξ ( λ ) ( u ) Z u e − A x I ξ ( λ ) ( x ) dxx ) , when defined, solves the nonhomogeneous equation (29). Parenthetically, it is worth nothingthat, just as the Laplace transform L Q ( s ) should be, by definition (17) and Remark 1, theabove function ℓ ( p ) ( u ) is, too, an even function of ξ ( λ ) , because K a ( z ) = π I − a ( z ) − I a ( z ) ( π a ) , (30)as given, e.g., by [1, Identity 9.6.2, p. 375].The problem now is to understand whether the two indefinite integrals involved in theabove function ℓ ( p ) ( u ) can be turned into convergent definite integrals, so that the result is awell-defined function that still satisfies the nonhomogeneous equation (29). To that end, itcan be gleaned, e.g., from [2, p. 99], that I a ( z ) ∼ √ π z e z , as | z | → + ∞ , and K a ( z ) ∼ r π z e − z , as | z | → + ∞ , which, in conjunction with Remark 2, enables one to see that the integrals Z + ∞ z e − A x I ξ ( λ ) ( x ) dxx and Z + ∞ z e − A x K ξ ( λ ) ( x ) dtx (31)are both convergent for any z >
0, but divergent for z =
0. As a result, one can conclude thatthe function ℓ ( p ) ( u ) : = λ ( I ξ ( λ ) ( u ) Z + ∞ u e − A x K ξ ( λ ) ( x ) dxx − K ξ ( λ ) ( u ) Z + ∞ u e − A x I ξ ( λ ) ( x ) dxx ) , is a well-defined, valid particular solution to equation (29); note the similarity of ℓ ( p ) ( u ) tothe right-hand side of identity (24).We are now in a position to claim that the general solution to equation (27) is of the form L ( s ) = C √ s I ξ ( λ ) ( t ) + C √ s K ξ ( λ ) ( √ s )++ λ ( √ s I ξ ( λ ) ( √ s ) Z + ∞ √ s e − A t K ξ ( λ ) ( t ) dtt −− √ s K ξ ( λ ) ( √ s ) Z + ∞ √ s e − A t I ξ ( λ ) ( t ) dtt ) , s ≥ , (32)where C and C are arbitrary constants, each independent of s , but possibly dependent on A . The only question left to be considered is that of “pinning down” the two constants C and C so as to make the foregoing L ( s ) satisfy the necessary boundary conditions.With regard to fitting the boundary conditions, let us first examine the behavior of L ( s ) given by (32) in the limit as s → + . To that end, from the small-argument asymptotics (25)of the modified Bessel I function, and the derivative formula ddz I a ( z ) = I a + ( z ) + az I a ( z ) , (33) n quasi-stationarity of the Shiryaev diffusion 17 as given, e.g., by [13, Identity 8.486.4, p. 937], we obtainlim u → + ( u I ξ ( λ ) ( u ) Z + ∞ u e − A x K ξ ( λ ) ( x ) dxx ) == lim u → + ( Z + ∞ u e − A x K ξ ( λ ) ( x ) dxx !, u I ξ ( λ ) ( u ) !) ( ∗ ) = lim u → + ( − e − A u K ξ ( λ ) ( u ) u !, − (cid:2) u I ξ ( λ ) ( u ) (cid:3) ddu (cid:2) u I ξ ( λ ) ( u ) (cid:3)!) = lim u → + ( e − A u (cid:2) I ξ ( λ ) ( u ) (cid:3) K ξ ( λ ) ( u ) (cid:2) + ξ ( λ ) (cid:3) I ξ ( λ ) ( u ) + (cid:2) ξ ( λ ) (cid:3) u I ξ ( λ )+ ( u ) ) = ξ ( λ ) (cid:2) + ξ ( λ ) (cid:3) , (34)where equality ( ∗ ) is due to L’Hˆopital’s rule, applicable because the corresponding integralof the modified Bessel K function is divergent when the lower limit of integration is zero.Likewise, from the small-argument asymptotics (26) of the modified Bessel K function,its symmetry with respect to the order, i.e., K a ( z ) = K − a ( z ) , trivially implied by (30), andthe derivative formula ddz K a ( z ) = − K a − ( z ) − az K a ( z ) = − K − a ( z ) − az K a ( z ) , (35)as given, e.g., by [13, Identity 8.486.12, p. 938], we obtainlim u → + ( u K ξ ( λ ) ( u ) Z + ∞ u e − A x I ξ ( λ ) ( x ) dxx ) == lim u → + ( e − A u (cid:2) K ξ ( λ ) ( u ) (cid:3) I ξ ( λ ) ( u ) (cid:2) − ξ ( λ ) (cid:3) K ξ ( λ ) ( u ) − (cid:2) ξ ( λ ) (cid:3) u K − ξ ( λ ) ( u ) ) = ξ ( λ ) (cid:2) − ξ ( λ ) (cid:3) , (36)where we again used L’Hˆopital’s rule, applicable because the corresponding integral of themodified Bessel I function is divergent when the lower limit of integration is zero.Next, from the foregoing two limits (34) and (36), and (9) we obtainlim s → + ( √ s K ξ ( λ ) ( √ s ) Z + ∞ √ s e − A x I ξ ( λ ) ( x ) dxx −− √ s I ξ ( λ ) ( √ s ) Z + ∞ √ s e − A x K ξ ( λ ) ( x ) dtx ) == ξ ( λ ) ( − ξ ( λ ) − + ξ ( λ ) ) = λ , whence, recalling again (25) and (26), one finds that L ( s ) given by (32) converges to unityas s → + , whatever C and C be. Put another way, it turns out that lim s → + L ( s ) =
1, forany choice of C and C . Let us switch attention to the behavior of L ( s ) for large values of s . To that end, from (31)and (35) we obtainlim u → + ∞ ( u I ξ ( λ ) ( u ) Z + ∞ u e − A x K ξ ( λ ) ( x ) dxx ) == lim u → + ∞ ( e − A u (cid:2) I ξ ( λ ) ( u ) (cid:3) K ξ ( λ ) ( u ) (cid:2) + ξ ( λ ) (cid:3) I ξ ( λ ) ( u ) + (cid:2) ξ ( λ ) (cid:3) u I ξ ( λ )+ ( u ) ) = u → + ∞ ( u K ξ ( λ ) ( u ) Z + ∞ u e − A x I ξ ( λ ) ( x ) dxx ) = , so that the limit of L ( s ) given by (32) as s → + ∞ can now be seen to be infinite if C =
0, or0 if C =
0. Hence, with C set to 0, our function L ( s ) simplifies down to L ( s ) = C √ s K ξ ( λ ) ( √ s )++ λ ( √ s K ξ ( λ ) ( √ s ) Z + ∞ √ s e − A t I ξ ( λ ) ( t ) dtt −− √ s I ξ ( λ ) ( √ s ) Z + ∞ √ s e − A t K ξ ( λ ) ( t ) dtt ) , s ≥ , (37)where C is still to be found.To “pin down” C one may invoke (28) for any one value of n ∈ N . The easiest choiceis n =
1, so that, in view of (15), we obtain (cid:20) dds L ( s ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) s = = − M = λ − A , (38)which is what we now intend to make L ( s ) given by (37) satisfy so as to get an equation tosubsequently recover C from.To find dL ( s ) / ds , first recall the symmetry K a ( z ) = K − a ( z ) , and then devise (33) and (35)and integration by parts to establish the indefinite integral identities Z u e − A x I a ( x ) dxx k = a + − k ( u − k e − A u I a ( u )++ A Z u x e − A x I a ( x ) dxx k − Z u x e − A x I a + ( x ) dxx k ) , and Z u e − A x K a ( x ) dxx k = a − + k ( u + k e − A t K a ( u )++ A Z u x e − A x K a ( x ) dxx k + Z u x e − A x K − a ( x ) dxx k ) , n quasi-stationarity of the Shiryaev diffusion 19 so that Z + ∞ u e − A x I ξ ( λ ) ( x ) dxx = − ξ ( λ ) u − e − A u I ξ ( λ ) ( u )++ λ e − A u I ξ ( λ )+ ( u ) −− λ (cid:18) + ξ ( λ ) − A (cid:19) u e − A u I ξ ( λ ) ( u )++ λ (cid:18) + ξ ( λ ) − A (cid:19) A Z + ∞ u x e − A x I ξ ( λ ) ( x ) dx −− λ + ξ ( λ ) Z + ∞ u x e − A x I ξ ( λ )+ ( x ) dx , and Z + ∞ u e − A x K ξ ( λ ) ( x ) dxx = + ξ ( λ ) u − e − A u K ξ ( λ ) ( u ) −− λ e − A u K − ξ ( λ ) ( u ) −− λ (cid:18) − ξ ( λ ) − A (cid:19) u e − A u K ξ ( λ ) ( u )++ λ (cid:18) − ξ ( λ ) − A (cid:19) A Z + ∞ u x e − A x K ξ ( λ ) ( x ) dx ++ λ − ξ ( λ ) Z + ∞ u x e − A x K − ξ ( λ ) ( x ) dx , which is sufficient to compute the limit of dL ( s ) / ds as s → + . Specifically, from (38), afterquite a bit of algebra involving repeated use of (25) and (26), we find that C = + ξ ( λ ) Z + ∞ x e − A x I ξ ( λ )+ ( x ) dx − (cid:18) + ξ ( λ ) − A (cid:19) A Z + ∞ x e − A x I ξ ( λ ) ( x ) dx , which can be brought to a more explicit form by appealing to [13, Identity 6.643.2, p. 716],i.e., the definite integral Z + ∞ x κ e − cx I a ( b √ x ) dx √ x = Γ ( κ + a + / ) Γ ( a + ) e b c bc κ M − κ , a (cid:18) b c (cid:19) , valid for ℜ ( κ + a + / ) >
0; recall that M a , b ( z ) here denotes the Whittaker M function. Theforegoing definite integral immediately gives Z + ∞ x e − A x I ξ ( λ ) ( x ) dx = e A Γ ( + [ ξ ( λ ) + ] / ) Γ ( ξ ( λ ) + ) A M − , ξ ( λ ) (cid:18) A (cid:19) , and Z + ∞ x e − A x I ξ ( λ )+ ( x ) dx = e A Γ ( + [ ξ ( λ ) + ] / ) Γ ( ξ ( λ ) + ) r A M − , ξ ( λ )+ (cid:18) A (cid:19) , so that C = e A ξ ( λ ) + Γ ( ξ ( λ ) + ) Γ (cid:18) ξ ( λ ) + (cid:19)( (cid:2) + ξ ( λ ) (cid:3) r A M − , ξ ( λ )+ (cid:18) A (cid:19) −− (cid:18) + ξ ( λ ) − A (cid:19) M − , ξ ( λ ) (cid:18) A (cid:19)) , where we also used the factorial property of the Gamma function Γ ( z + ) = z Γ ( z ) . Now,from [1, Identity 13.4.28, p. 507], i.e., the identity2 b M a − , b − ( z ) − √ zM a , b ( z ) − b M a + , b − ( z ) = , we find at once that r A M − , ξ ( λ )+ (cid:18) A (cid:19) = (cid:2) + ξ ( λ ) (cid:3)( M − , ξ ( λ ) (cid:18) A (cid:19) − M , ξ ( λ ) (cid:18) A (cid:19)) , whence C = e A Γ ( ξ ( λ ) + ) Γ (cid:18) ξ ( λ ) + (cid:19) A ( + ξ ( λ ) M − , ξ ( λ ) (cid:18) A (cid:19) − A M , ξ ( λ ) (cid:18) A (cid:19)) , which is equivalent to C = e A + ξ ( λ ) Γ ( ξ ( λ ) + ) Γ (cid:18) ξ ( λ ) + (cid:19) A M , ξ ( λ ) (cid:18) A (cid:19) , because of [1, Identity 12.4.29, p. 507], i.e., the recurrence ( + b + a ) M a + , b ( z ) − ( + b − a ) M a − , b ( z ) = ( a − z ) M a , b ( z ) , whereby1 + ξ ( λ ) M − , ξ ( λ ) (cid:18) A (cid:19) − A M , ξ ( λ ) (cid:18) A (cid:19) = + ξ ( λ ) M , ξ ( λ ) (cid:18) A (cid:19) . Next, since the Wronskian between M a , b ( z ) and W a , b ( z ) is W { M a , b ( z ) , W a , b ( z ) } : = M a , b ( z ) ddzW a , b ( z ) − W a , b ( z ) ddz M a , b ( z ) = − Γ ( + b ) Γ ( / + b − a ) , as given, e.g., by [32, Formula (2.4.27), p. 26], and because W a − , b ( z ) = z − a ( / + b − a )( / − b − a ) W a , b ( z ) + z ( / + b − a )( / − b − a ) ddzW a , b ( z ) , as given, e.g., by [32, Formula (2.4.21), p. 25], it follows that λ A Γ (cid:18) ξ ( λ ) − (cid:19) W , ξ ( λ ) (cid:18) A (cid:19) M , ξ ( λ ) (cid:18) A (cid:19) = − Γ ( ξ ( λ ) + ) , where we also appealed to equation (8). n quasi-stationarity of the Shiryaev diffusion 21 Putting all of the above together, we can finally conclude that C = ,( e A W , ξ ( λ ) (cid:18) A (cid:19)) , which is precisely the normalizing factor in the quasi-stationary distribution’s formulae (10)and (11).We have now solved the differential equation (27) and obtained yet another representa-tion of the Laplace transform L Q { q A ( x ) ; x → s } ( s , A ) of the quasi-stationary distribution (4). Lemma 4
For every A > fixed, the Laplace transform L Q { q A ( x ) ; x → s } ( s , A ) of thequasi-stationary distribution (10) – (11) is given by L Q { q A ( x ) ; x → s } ( s , A ) == √ s K ξ ( λ ) ( √ s ) ,( e − A W , ξ ( λ ) (cid:18) A (cid:19)) ++ λ ( √ s K ξ ( λ ) ( √ s ) Z + ∞ √ s e − A x I ξ ( λ ) ( x ) dxx −− √ s I ξ ( λ ) ( √ s ) Z + ∞ √ s e − A x K ξ ( λ ) ( x ) dxx ) , (39) where s ≥ , and λ ≡ λ A ( > ) is determined by (8) while ξ ( λ ) is defined in (9) ; recallalso that W a , b ( z ) denotes the Whittaker W function, and I a ( z ) and K a ( z ) denote the modifiedBessel functions of the first and second kinds, respectively. Yet again, from the symmetry of the Whittaker W with respect to the second index, i.e., W a , b ( z ) = W a , − b ( z ) , one can see that, just like formulae (20) and (21) obtained earlier, thenew formula (39) is also symmetric with respect to ξ ( λ ) , as it should be, by definition (17)and Remark 1. However, unlike formulae (20) and (21), the new formula (39) is not onlyfree of the Kamp´e de F´eriet function, but more importantly, it is valid even in the limit,as A → + ∞ or as s → + ∞ . While the (trivial) limit as s → + ∞ is of little interest, the(nontrivial) limit as A → + ∞ does merit some consideration, especially in the context ofquickest change-point detection [20]. Sincelim A → + ∞ ( e − A W , ξ ( λ ) (cid:18) A (cid:19)) = , which was observed previously in [25, p. 139] as an implication of the limits lim A → + ∞ λ A = A → + ∞ ξ ( λ A ) =
1, it can be shown directly from (39) with the aid of (25) thatlim A → + ∞ L Q { q A ( x ) ; x → s } ( s , A ) = √ s K ( √ s ) = : L H ( s ) , for every s ≥ K a ( bz ) = Z + ∞ (cid:18) bx (cid:19) a e − z (cid:0) x + b x (cid:1) dxx , valid for ℜ ( z ) > ℜ ( b z ) > , the function L H ( s ) : = √ sK ( √ s ) can be recognized to be the Laplace transform ofthe Shiryaev diffusion’s stationary distribution defined in (2) and given explicitly by (3).That is, for every s ≥ L Q { q A ( x ) ; x → s } ( s , A ) as A → + ∞ is precisely L H ( s ) , and, therefore, the stationary distribution (3) is the limit of the quasi-stationary dis-tribution (10)–(11) as A → + ∞ . This convergence of distributions (for a more general familyof stochastically monotone processes) was previously established by Pollak and Siegmundin [20,21], although through an entirely different approach and with no explicit formulae.We conclude with an admission that, in our derivation of the Laplace transform for-mula (39), we actually had to “cut some corners”. Strictly speaking, by Remark 2, we shouldhave considered separately three different cases: (1) A < ˜ A ≈ . λ A > / ξ ( λ ) is purely imaginary; (2) A = ˜ A ≈ . λ A = / ξ ( λ ) =
0; and (3) A > ˜ A ≈ . λ A < / ξ ( λ ) is purely real and strictly between 0 and 1.However, for lack of space, we only attended to the third case. The reason to distinguish thethree cases is because the asymptotics of the modified Bessel I and K functions are highlyorder-dependent, and, in our specific situation, the order of either function is determinedentirely by ξ ( λ ) . For example, the limits (34) and (36) are clearly false when ξ ( λ ) = It is generally rare that quasi-stationary distributions and their characteristics lend them-selves to explicit analytic evaluation. Furthermore, in the rare cases one can recover thedistribution itself or its characteristics analytically, the result is usually of limited use, forthe corresponding formulae, though explicit, are typically rather complex and involve spe-cial functions (or, worse yet, exotic special functions). This work, as a continuation of [25]and a spin-off of [26], provided an example of a situation where the distribution itself, itsLaplace transform as well as the entire moment series are all obtainable analytically andin closed-form, despite the presence of special functions in all of the calculations. It is ourhope that the special functions calculus heavily used in this work will aid further researchon stochastic processes, an area where special functions (including those dealt with in thethis paper) arise routinely.
Acknowledgements
The authors would like to thank the two anonymous referees for the careful reading ofthe manuscript and pertinent comments; the referees’ constructive feedback helped substantially improve thequality of this work and shape its final form.
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