ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES UMUT ÇETIN. Abstract. For a squared Bessel process, X, the Laplace transforms of joint laws of U, R y Xs p ds are studied where R y is the first hitting time of y by X and U is a random variable measurable with respect to the history of X until R y. A subset of these results are then used to solve the associated small ball problems for R y Xs p Ry ds and determine a Chung s law of iterated logarithm. Xs p ds is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The findings are then used to price a class of exotic derivatives on interest rates and determine the asymptotics for the prices of some put options that are only slightly in-the-money.. Introduction Let X be a squared Bessel process which is the unique strong solution to dx t = + dt + X t db t, where is a real constant and B is a standard Brownian motion. Letting = +, X is called a -dimensional squared Bessel process. We will denote such a process with X = z by BESQ z and and will be related by = + throughout the text. In this paper we are interested in the integral functional. Σ p,z,y := Ry X p s ds, where p > and R y := inf{t : X t = y} for y, and X is BESQ z. In the sequel, we will write Ry only if we need to specify the dimension to avoid ambiguity. Squared Bessel processes have found wide applications especially in Finance Theory, see Chapter 6 in 8] for a recent account. They can, e.g., be used to model interest rates in a Cox-Ingersoll-Ross framework. In the above setting, if X p models the spot interest rates, then exp Σp,z,y refers to the cumulative interest until the spot rate hits the barrier y p. As such, this random variable is related to certain exotic options on interest rates see 5] for some formulae regarding barrier options in a similar framework. Bessel processes also appear often in the study of financial bubbles since / X is the prime example of a continuous strict local martingale when X is a BESQ 3 see, e.g., 4], 7] and 8] for how strict local martingales, and in particular Bessel processes, appear in mathematical studies of bubbles. In Section we will determine the joint law of U, Σ p,z,y by martingale methods, where U is a random variable measurable with respect to the evolution of X until R y. In particular we will obtain the joint distributions of R y, Σ p,z,y and max t Ry X t, Σ p,z,y. As a by-product of our findings, if =, we have a remarkable characterisation of the conditional law of Σ p,z,y given that the maximum resp. minimum of X at R y is below resp. above a fixed level in terms of the first hitting time distributions of a 3-dimensional Bessel process when z y resp. z y. Date: June 5, 5. Key words and phrases. Bessel processes, modified Bessel functions, first passage times, small deviations, Chung s law of iterated logarithm, non-homogeneous Feller jump process, time reversal, last passage times, subordinator, interest rate derivatives.
UMUT ÇETIN We will use the results of Section in order to study small ball probabilities for Σ p,z,y in Section 3. Solving the small ball problem for Σ p,z,y amounts to finding the asymptotic behaviour of log ProbΣ p,z,y < ε as ε. We will then use this asymptotic form to determine a law of iterated logarithm for Σ p,,y y as y. Section 4 will analyse Σ p,,y y as a Markov process indexed by y and compute its infinitesimal generator when. We will also consider the process Z which is obtained via a time reversal from Σ p,,y y. More precisely, we will find the generator of Z defined by Z x = L L x X p s ds x,, where L x := sup{t : X t = x}. In particular, we will obtain that Z 4 is identical in law to an increasing family of hitting times of a linear Brownian motion. Finally, in Section 5 we will apply our findings to the pricing of some exotic derivatives on interest rates. The small ball probabilities will be used to find asymptotic behaviour of some put options with small strikes, the options that are only slightly in-the-money.. Preinaries Let Ω, F, F t t, P be a stochastic base where F is completed with the P-null sets. R + -valued semimartingale which is the unique strong solution to. dx t = + dt + X t db t, Let X be an where is a real constant, B is a standard Brownian motion. Let Q z be the measure on the path space, i.e. C,,,, induced by X starting at z. It is wellknown see Section of Chapter XI of 5] that for the set {} is polar, otherwise it is reached a.s.. Moreover, the process is transient for > and recurrent otherwise. We will denote the first hitting time of for X with R. The scale function, s, for BESQ is given by s x = x for >, s x = log x, s x = x for,. We refer the reader to 5] and 6] for a comprehensive study of Bessel processes and relevant bibliography. In subsequent computations we will follow a Feynman-Kac type approach as in, e.g., 7]. Lemma.. Let p >, > and suppose that u C, solves the following ordinary differential equation ODE:. x y + xy y + x ] =, and is strictly positive on,. Then, M u t R t is a local martingale where M u t := u X t X t exp t Xs p ds. In particular, t<r] dm u t = t<r] M u u X t t u X t db t. Xt Proof. If we let wx := u xx, it is easily seen that w solves.3 xw + + w xp w =. Thus, on t < R] dm u t = M u w t w X t X t db t = M u u X t t u X t db t. Xt
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES 3 The solutions to. can easily be determined via the modified Bessel functions, I α and K α, of the first and second kind. We next summarise some properties of these functions and we refer the reader to Section 3.7 of 9] or Section 9.6. of ] for further results and proofs. For α, I α resp. K α is a positive and increasing resp. decreasing solution to.4 x y + xy x + α y = when x is restricted to,. Moreover, I α has the following series representation: x m+α.5 I α x =. m!γm + α + m= The above expansion can be used to define I α for α < when α is not an integer less than since the Gamma function is well defined and finite at non-integer negative values. For α Z, one can define it by taking its and it turns out that for such an α I α = I α. It is also a simple matter to check that I α satisfies.3 for α < as well. Moreover,.6 K α x = π I α x I α x. sinαπ The above identity in particular entails K α = K α. The asymptotic behaviour of the modified Bessel functions can be described in terms of known functions:.7 I α x x α Γα+ as x, α,,... ; I αx ex πx as x. K α x Γα x α as x, α > ; Kα x π x e x as x. K x log x as x. Then, it is easy to check that the solutions to. is of the form.8 C K x + C I p + where C and C are arbitrary constants. p + x, Remark. It is worth to observe some further monotonicity properties regarding modified Bessel functions that will be useful in the sequel. First consider x K x for R and p >. We will now see that this function is decreasing. Indeed, differentiation with respect to x yields x K p + x + x +p K When, utilising the recurrence elation cf. Section 3.7 in 9] K αx = α x K αx K α+ x, α R, x. p + one can see that the above derivative is negative since K α is positive for all α R. If <, the other recurrence relation, i.e. K αx = α x K αx K α x, α R, yields the conclusion. Similarly, the function x I p > and >. This follows from the recurrence relation and the positivity of I α for α >. Note that x, has a positive derivative, i.e. is increasing, whenever I αx = α x I αx + I α+ x, α R, + > under our hypotheses. We now return to determining the joint law of U, Σ p,z,y for arbitrary positive F Ry -measurable random variables U, where Σ p,z,y is as defined in.. We will analyse the cases of negative and positive separately.
4 UMUT ÇETIN.. The case,. Theorem.. Suppose that,, p > and let u x := K < x u xx <, x u zx <. Consequently, M u t R t is a strictly positive bounded martingale with t<r] dm u t = t<r] M u u X t t u X t db t. Xt Proof. Note that u xx = x p + x x K x = x. Then, Γ p + in view of the asymptotic relations from.7. Asymptotic behaviour of K for large x yields u xx has a finite it at infinity, hence the boundedness of M u t R t in view of Lemma.. Strict positivity of K α on, for all α completes the proof of that this martingale is strictly positive. It is well-known that see, e.g. Section.8 in 3] for > Q z exp ] R y = z K.9 z y K y, y z;. = z I z y I y, y z. The above formulae are still valid when and note that z I z resp. z K z is increasing resp. decreasing in view of Remark. Since R <, a.s. when <, the following is a straightforward corollary to the theorem above for < and y < z. Corollary.. Let u be the function defined in Theorem. and suppose that,, p > and y < z. If U is F Ry -measurable, then for r, Q z exp ru ] Σ p,z,y = u z z u P,u z exp ru], y y where Pz,u is defined by dp,u z dq z. dx t = for some P,u z -Brownian motion β. = M u R u X t X t u X t,u. Moreover, under P, X solves z + dt + X t dβ t, t R, Proof. Since M u t R t is a strictly positive martingale due to Theorem., Q z is an equivalent probability measure. The fact that X solves. follows from an application of Girsanov s theorem. Remark. By taking U Corollary. yields the law of Σ p,z,y for y < z. Comparing this Laplace transform with.9 shows that Σ p,z,y d = R y where = +p +, y = y+p +p and R y is the first hitting time of y for some BESQ z +p +p. One can check by comparing the Laplace transforms that we will obtain later in this section that this equality in law would be valid for y z, too. Moreover, the same identities in distribution will hold for, too. These facts also follow from the time-change result given in Proposition XI.. in 5].
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES 5 Recall that if X is a one-dimensional regular diffusion on an interval l, l defined by its infinitesimal generator, A, where A = σ x d dx + bx d dx for some locally bounded functions σ and b such that σ > on the open interval l, l, then for any r > there exists a positive, and strictly decreasing resp. increasing function Φ resp. Ψ, which solves Au = rφ resp. rψ on l, l satisfying certain boundary requirements depending on the nature of the behaviour of the diffusion near l and l. Moreover, any other solution of this equation with the above positivity and monotonicity assumptions is a fixed multiple of Φ resp. Ψ see, e.g., Proposition V.5.3 in 6]. Then, if R y is the first hitting time of y, P z exp rr y ] = Φz Φy, y z; = Ψz Ψy, y z, where P z is the law of the diffusion that started at z at t =. Thus, if U = R y, we obtain the following in view of the above discussion. Corollary.. Let u be the function defined in Theorem. and suppose that, and y < z. Then for r, Q z exp rr y ] Σ p,z,y = u z z u Φz y y Φy, where Φ is a positive and decreasing solution of u. xv + x x u x + v = rv. Proof. The ODE in. corresponds to the diffusion, X, which follows u.3 dx t = X t X t u + dt + X t dβ t, X t where β is a Brownian motion, if a solution exists. Using the recursive relation we obtain u x x u x K αx = α x K αx K α x, = x K p K x x. This shows that the drift of the above SDE is less than that of the SDE solved by a BESQ. Thus, when a solution exists, it never explodes due to the standard comparison results for SDEs. Moreover, using the asymptotic relations for K as x tends to, we can directly verify that u x x x u =. x Thus, the solution is pushed towards the interior of, as soon as it hits since the diffusion coefficient vanishes at. Finally, the existence of a solution taking values in, follows easily since the drift term is locally Lipschitz as K α is strictly positive on, and K α is twice continuously differentiable for all α. The solution is also unique due to drift coefficient being locally Lipschitz. Recall that in view of Corollary. X solves.3 until the first time it hits. Since it cannot hit before hitting y, the formula is a direct consequence of the discussion preceding the corollary.
6 UMUT ÇETIN Corollary. also allows us to compute the law of Σ p,z,y on the event that a certain boundary is yet to be reached via the intimate relationship between the scale functions of diffusions and their exit probabilities see Sect.3 of Chap. VII in 5] for details. Corollary.3. Let u be the function defined in Theorem. and suppose that,, p > and y < z. Then, a scale function of the diffusion defined by. is.4 s x = Thus, for any a > x Q z x yu dy, x. y Ra>R y] exp ] Σ p,z,y = u x u y x s x s a y s y s a. Proof. The representation of the scale function is due to the well-known formulas for the solutions of SDEs, see, e.g., Exercise VII.3. in 5]. Note that the function is well-defined at x =. Indeed, it follows from Theorem. that y u yy >. Thus, yu y > Cy + for some C > for sufficiently small y. Since y + is integrable for,, the claim holds. The second assertion follows from Corollary. after taking r = and U = log Ra>R y] via the defining property of scale functions, see Definition VII.3.3 in 5]. Remark 3. The above result in fact gives us the joint law of max t Ry X t, Σ p,z,y. Indeed, for any a z > y max t R y X t < a] = R a > R y ]. Since K x = π x e x, we have more explicit formulas when =. Corollary.4. Suppose that = and p >. Then, for y z we have the following: i Q z exp rr y ] z y Φz Σ p,z,y = exp Φy, where Φ is a positive and decreasing solution of.5 xv + + x v = rv.6 on,. ii The function s is, up to an affine transformation, given by exp iii For a > z, Q z Ra>R y] exp ] Σ p,z,y x z y exp z = exp exp y sinh a z = sinh. a y. exp exp Note that the expression in.6 yields Q z exp ] Σ p,z,y R a > R y = a y sinh a z a z sinh a y a a using the scale function of X under Q z. Comparing this with. for = / gives the following since I x = π x sinhx.
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES 7 Corollary.5. Suppose that =,,, and p >. Then, for y z < a we have that the law of Σ p,z,y conditioned on the event R a > R y ] is that of the first hitting time of a y /4 by a 3-dimensional squared Bessel process started at a z /4. Note that, since a = xa a = log x, when y >, we obtain that the above conditional laws converge as and, thus, as p to that of the first hitting time of log a log y by a 3-dimensional squared Bessel process started at log a log z. This can be viewed as the analogous statement of the above corollary when = and p =. Next we look at the case when y > z. Observe that the function u as defined in the theorem below is still well defined and finite at z = in view of, e.g., the series representation of I α in.5. Theorem.. Suppose that p, y z, and,. M u t R y t is a bounded martingale with for any y. t<ry]dm u t = t<ry]m u t u X t u X t Let u x := I Xt db t, x. Then, Proof. Note that R < R y ] has a positive probability. Thus, we have to pay attention to the behaviour of wx = u xx at x =. Observe that under our assumptions, >, thus it follows from the series representation of I α that w > and is finite since I α < for any α. Next, we will show that w has an absolutely continuous derivative over,. Using the recurrence relations see Section 3.7 in 9] we obtain that I α x + I α+ x = I αx, and x I α x I α+ x = αi α x,.7 I αx = I α+ x + α x I αx. Using this identity it follows from direct calculations that w x = x p I γ p + x, where γ = +. Since the leading term of I γ x as x is x +, we see that x w x = when p >. Therefore, we obtain immediately from the ODE.3 that when p > x xw x = for,. On the other hand, when p = and >, w x wx = I v+ x x I v x by another application of.7. Thus, in view of the asymptotics of I α as x w x x wx = I v+ x x xi v x = Γ + 4 Γ + = 4 +. Consequently, x + w w = since wx > for all x. Again, it follows from the ODE.3 that x xw x =. However, this condition implies that x w y dy exists and is finite. Since this integral equals w x w for any x,, we conclude that w is absolutely continuous on,
8 UMUT ÇETIN and w x = w + x w y dy for any x,. Then, in view of Problem 3.7.3 in 9] we immediately deduce that wx t = wx + = t t w X s X s db s + w X s X s db s + t t X p s wx s ds. {w X s + + w X s X s } ds A simple application of integration by parts formula now shows that M u is a martingale with the claimed representation. Remark 4. Observe that u xx is still monotone in view of Remark since under our hypothesis. > Corollary.6. Let u be the function defined in Theorem. and suppose that the hypotheses therein hold. Then, we have the following for all z y: i If r and U is F Ry -measurable, Q z exp ru ] Σ p,z,y where P,u z,y is defined by dp,u z,y dq z = M u R y u.8 dx t = X t X t u + X t for some Pz,y,u -Brownian motion β. ii For all r Q z = u z u y z P,u z,y exp ru], y. Moreover, under Pz,y,u exp rr y ] Σ p,z,y = u z u y where Ψ is a positive and increasing solution of u.9 xv + x x u x, X solves dt + X t dβ t, t R y, + v = rv on,. iii A scale function of the diffusion defined in.8 is given by. s x = Thus, for any a < z Q z Ra>R y] exp x yu dy, x. y ] Σ p,z,y = u z u y z Ψz y Ψy, z s z s a y s y s a. Proof. The proof follows the similar lines as in the proofs of analogous results for y z. The only difference is that contrary to the previous case the drift term of the diffusion corresponding to the ODE in.9 is only locally bounded. In particular, it is now larger than +, which in turn implies that the solution is immediately pushed to, as soon as it hits by comparison results for SDEs. Moreover, being only locally bounded causes no concern for our purposes since the computations involve the law of the diffusion until the first time it reaches y. Again, since I x = π x coshx we have Corollary.7. p,, and =. Then, for y z we have the following:
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES 9 i Q z exp rr y ] Σ p,z,y = where Ψ is a positive and increasing solution of. xv + + + x tanh cosh cosh z y Ψz Ψy, x v = rv on,. ii The function s is, up to an affine transformation, given by tanh iii For a < z,. Q z Q z Ra>R y] exp ] Σ p,z,y = Ra>R y] exp ] Σ p,z,y = Proof. Only part iii needs proof. Note that cosh On the other hand, = = = cosh cosh z y { z tanh sinh z cosh sinh z cosh cosh sinh a z cosh, a which yields the claimed representation. x sinh sinh a cosh cosh a + cosh tanh cosh tanh a a x z a y a.. x tanh y tanh } a z sinh a z sinh a a a. Note that in fact we do not need to assume p > for part iii of the above result to hold, since X is never before R a for < a < z. Moreover,. and.6 are the same. Thus, Corollary.8. Suppose that =,, and p >. Then, for y z > a we have that the law of Σ p,z,y conditioned on the event R a > R y ] is that of the first hitting time of a y /4 by a 3-dimensional squared Bessel process started at a z /4. We end this section with a scaling property which will be useful in the subsequent section. It is a direct consequence of the scaling property of BESQ applied to the definition of Σ p,,y. Proposition.. Suppose that p,,. Then, we have the following identity in law for any y : y Σ d p,, = Σ p,,y.
UMUT ÇETIN.. The case. Recall that when the point is polar for X. Thus, one can prove without any difficulty that M u, where u is as defined in Theorem., is a martingale stopped at R y. Note that for the random variable Σ p,,y is well-defined and finite even in the case p,. Indeed, Corollary XI.. in 5] show that +p Xs p ds = d Y p p +p ds, where X is a BESQ + and Y is a BESQ + +p. Since p +p > whenever p, we easily deduce that the random variable on the left-hand side of the above identity is finite. Since a Bessel process with a positive dimension never comes back to again this yields the finiteness of ε Xp s ds for all ε >, which in turn implies the finiteness of Σ p,,y. Theorem.3. Suppose that p >, y z, and. Let u be the function defined in Theorem.. Then, M u t R y t is a bounded martingale. Recall that BESQ is transient when >, thus Q zr y < = whenever y z. Consequently, we can deduce the following. Corollary.9. Let u be the function defined in Theorem. and suppose that p >,. Then, we have the following for all z y: i If r and U is F Ry -measurable, exp ru ] Σ p,z,y where Pz,y,u ii For all r Q z is defined by Q z dp,u z,y dq z s = u z u y z P,u z,y exp ru], y = M u R y. Moreover, under Pz,y,u, X satisfies.8. exp rr y ] Σ p,z,y = u z u y z Ψz y Ψy, where Ψ is a positive and increasing solution of.9 on,. iii For any a < z Q z Ra>R y] exp ] Σ p,z,y = u z z u s z s a y y s y s a, where s is as defined in.. Analogous to Proposition. we have the following scaling property. Proposition.. Suppose that p >,. Then, we have the following identity in law for any y : We now return to the case y z. y Σ p,, d = Σ p,,y. Proposition.3. Let u be the function defined in Theorem. and suppose that, p >. Then, we have the following for all z y: i If r and U is F Ry -measurable, exp ru ] Σ p,z,y Q z = u z u y z P,u z,y exp ru], y where P,u z,y is defined by dp,u z,y dq z = M u R y. Moreover, under Pz,y,u, X satisfies. until R y.
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES ii For all r Q z exp rr y ] Σ p,z,y = u z u y z Φz y Φy, where Φ is a positive and decreasing solution of. on,. iii For any a > z Q z Ra>R y] exp ] Σ p,z,y = u z z u s z s a y y s y s a, where s is as defined in.4. Proof. We will only give the details for the parts of the proof that differ from the analogous results of the previous section. Observe that M u t R y t is a uniformly integrable martingale when X starts at a value larger than y. Thus, using the Optional Stopping Theorem, we obtain u ] zz P,u z,y exp ru] = Q z exp ru M u R y = Q z Ry< ]u yy exp ru ] Σ p,z,y ]. +Q z Ry= ]M u For =, R y is finite a.s., hence the claim. In case of >, we still obtain the formula since, on the set R y = ], Σ p,z,y = as well as u = due to the transience of X. Using the recurrence relations for K α as in Corollary., we can deduce that u x x u x, and the left-hand side of the above inequality converging to the right-hand side as x tends to. solution of the SDE given by. until the first time it hits is less than that of dx t = + dt + X t dβ t. Thus, Note that if > the drift of the above SDE is negative hence its solution corresponds to a squared Bessel process of negative dimension. This SDE has a unique strong solution and it stays, ] as soon as it hits see Section 3 of 6]. Thus, we conclude by means of comparison results for SDEs that there exists a solution to. until R y for every y >. The ODE. has increasing and decreasing solutions which can be determined as before via the Laplace transforms of the first hitting times of the the diffusion on, with the generator which is killed as soon as it reaches. A = x d dx + As before, using the explicit form of K, one gets Corollary.. Suppose that = u x x u x d + dx, and p >. Then, for y z we have the following: i Q z exp rr y ] Σ p,z,y = y z exp z y where Φ is a positive and decreasing solution of.3 xv + + x v = rv on,. Φz Φy,
UMUT ÇETIN.4 x ii The function s is, up to an affine transformation, given by exp iii For a > z Q z Ra>R y] exp ] Σ p,z,y As in the case with and hence Q z = we get exp ] Σ p,z,y R a > R y = y z exp z y exp z a exp y exp a exp a = y sinh z z. a sinh y. = y y a a sinh z z z a a sinh y = y a sinh z a z a a sinh y = a y a sinh z a z, a sinh y Corollary.. Suppose that = and p >. Then, for y z < a we have that the law of Σ p,z,y conditioned on the event R a > R y ] is that of the first hitting time of a y /4 by a 3-dimensional squared Bessel process started at a z /4. Similarly, since I x = π x sinhx we have Corollary.. p > and i Q z =. Then, for y z we have the following: exp rr y ] y sinh Σ p,z,y = z sinh where Ψ is a positive and increasing solution of.5 xv + + + x coth z y Ψz Ψy, x v = rv on,. x ii The function s is, up to an affine transformation, given by coth iii For a < z,.6 Q z Ra>R y] exp ] z Σ p,z,y = y sinh a z. y sinh a Remark 5. Comparing parts i of Corollary.4 and. immediately gives us that, for z y, the distributions of Σ p,z,y are different when has different signs. On the other hand, Corollaries.5 and. imply that they have the same distribution once they are conditioned on the event that the maximum of the underlying squared Bessel process is less than a by time R y. Same conclusion holds when z y..
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES 3 3. Small ball problem and Chung s law of iterated logarithm The small ball problem also called small deviations for a stochastic process Z = Z t t T finding the probability P Z < ε] as ε, consists in is a given norm, usually L p or L. It is connected to many other questions, such as the law of the iterated logarithm of Chung s type Chung s LIL for short, strong it laws in statistics, metric entropy properties of linear operators and several approximation quantities for stochastic processes. The determination of the above probability is not feasible other than in a very few cases and one is inclined to consider the asymptotic behaviour of log P Z < ε] as ε. The solution to the latter problem is also not available in full generality. However, one can get this asymptotic behaviour for Gaussian processes see, e.g., ] and ] or real-valued Lévy processes see ]. There is a large amount of literature on small ball probabilities in the Gaussian setting and one can consult the survey article ]. As one can expect from the computations made in the previous section, we will be interested in the small ball probabilities for the stochastic process X t t, and the norm Ry Z p,y = Z t p dt where p, and R y = inf{t > : X t = y}. Observe that the above definition is not a real norm unless p, however, as the results in this section does not depend on whether p,y is a true norm, this is not a problem. Our results and proofs are close in nature to the results of ]. Interestingly, the small ball probabilities for X under the above norm does not depend on its index,, as seen from the next theorem. Theorem 3.. Let X be a BESQ as defined by. with >, and R y = inf{t > : X t = y}. Then, one has, for z and y, ] log Q z exp X p p,y = z y p + ε εp log Q z X p,y < ε] = p + z y. Proof. Let wx = u xx where u = u resp. u = u for y z resp. y > z and u and u are as defined in Theorems. and., respectively. Then, it follows from the results of the previous section that log Q z exp ] X p p,y = log wz log wy. Moreover, when u = u, log wx log u x = p = x log K p + = x p + log K, x p +. However, using the asymptotic expansions in.7 we obtain that for any α, log K α =. x
4 UMUT ÇETIN This shows that when y z ] log Q z exp X p p,y = z y. p + log I In order to show the above it when y > x, it suffices to show that x x x =, which again follows from.7. This completes the proof of the first assertion of the theorem. The second assertion follows by applying de Bruijn s exponential Tauberian theorem see Theorem 4..9 in 4] to α = and β = z y. Observe that Σ p,,y is an increasing process when indexed by y. We will next use the above theorem to obtain Chung s LIL for Σ p,,y y. Theorem 3.. Let φy := y log log y. Then, for any > and p > one has inf y Proof. It follows form Theorem 3. that ε ε log Q Σ p,,y φy = p +, Q -a.s.. Σ p,, < ε ] = p +, thus, for sufficiently small ε, Q Σ p,, < ε ] exp K, ε where K is a fixed, but arbitrary, constant in,. Fix C > and set y n = C n. Next choose k > so that kc < K, In view of Propositions. and., we get for all large n Q Σ p,,yn < kφy n+ ] ] = Q Σ p,, < kc log log y n+ exp K log n + log C = log C K kc kc n + K kc, which is summable in n. Therefore, by the first Borel-Cantelli lemma, we have that, a.s. for large n, Σ p,,yn φy n+ k. On the other hand, for y y n, y n+ ], which shows that Σ p,,y Σ p,,y n kφy n+ kφy, inf y Σ p,,y φy k, Q -a.s., and thus Σ p,,y inf y φy p +, Q -a.s., by the arbitrariness of C, K and k. We now turn to prove the reverse inequality. First, let s observe that 3. inf y Σ p,,y y <, Q -a.s.. The above claim follows from a direct application of Fatou s lemma since Σ d p,,y = y Σ p,,. Next, fix an ε >, let y n = n n and consider the events ] Ryn E n := X p s ds + ε R yn p + φy n.
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES 5 It follows from the strong Markov property of X that E n s are independent, and we will now see that E n s occur infinitely often due to the second Borel-Cantelli lemma. Indeed, by the definition of Σ p,,y n, we obtain ] Q E n Q Σ p,,y n + ε p + φy n ] = Q Σ p,, + ε p + log log y n exp + ε + ε log log y n, log y n where the second inequality is due to the fact that, for a given ε >, Q Σ p,, η ] exp + ε η for sufficiently small η in view of the convergence result of Theorem 3.. Since n log n is not summable, it follows from the Borel-Cantelli lemma that E n occurs infinitely often. As ε was arbitrary this allows us to conclude, a.s., Thus, Q -a.s., 3. inf n On the other hand, Σ p,,y n φy n inf n inf n Σ p,,y n φy n Ryn R yn X p s ds + φy n p +. Σ p,,y = inf n y n p + n y φy n n + p +. y n log log nn φy n n, which converges to as n. Therefore, in view of 3. and 3., we obtain inf n Σ p,,y n φy n p +, Q -a.s.. Remark 6. We can in fact extend the above result so that for all z inf y Σ p,z,y φy = p +, Q z-a.s.. Indeed, using the strong Markov property of X we have the decomposition Σ p,,y = Σ p,,z + Σ p,z,y where Σ p,,z and Σ p,z,y are independent. Dividing both sides by φy and letting y yields the result. Another formal check to this result can be performed by observing Σ p,z,y Σ d = y Σ d p, z y = y, Σ p, z y, p,, Σ p,, d = Σ p,,y Σ p, z y, Σ p,,, and that Σ p, z y, Σ p,, converges to as y once we identify Σ p, z y, with R R z y X p s ds, where X is BESQ.
6 UMUT ÇETIN 4. Feller property and time reversal In the previous section we have proved a law of iterated logarithm for Σ p,,y by considering it as a process indexed by y. In this section we will see, for, that it is in fact an inhomogeneous Feller process and find its infinitesimal generator. First of all, it immediately follows from the strong Markov property of X that Σ p,,y, F Ry y is Markov. Suppose P z,y is the associated semigroup, i.e. P z,y fa = Q fσ p,,y Σ p,,z = a]. Since the increments of Σ p,,y y are independent, we have for any bounded measurable f 4. P z,y fa = fa + bq zσ p,z,y db. Let C denote the class of continuous functions on R + that vanish at. 4. readily implies that when f C, P z,y f C as well. Moreover, it follows from Corollary.9, and the observation that u is finite at, that for each z the measure Q zσ p,z,y db converges weakly to the Dirac point mass at as y z since its Laplace transform converges to. Therefore, y z P z,y fa = fa and consequently Σ p,,y, F Ry y is Feller. The form of the infinitesimal generator of Σ p,,y, F Ry y will follow from the following theorem. Theorem 4.. Suppose that. Then, for every x there exists a decreasing function πx, satisfying where wx := x I Q 4. y z x e b πx, bdb = w x wx,. Moreover, exp Σ p,,y Σ p,,z = a y z where πz, db := πz, db for b. In particular, Q ] e a Proof. In view of the strong Markov property of X, exp ] Σ p,,y Σ p,,z = a Thus, it follows from Corollary.9 that exp p,,y ] Σ Σ p,,z = a e a 4.3 Q y z y z = bπz, db <. On the other hand, using integration by parts, we obtain Q z exp ] Σ p,z,y = 4.4 = e a Q z = It is well-known see Corollary 3.8 in Chap. VII of 5] that Q zσ p,z,y] = y,, { e a+b e a} πz, db, exp ] Σ p,z,y. = e a Q z exp p,z,y] Σ y z y z = e a y z wz wy = e a w z y z wz. e b Q zσ p,z,y db x p Gz, xmdx, e b Q zσ p,z,y > b db.
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES 7 where m and G are the associated speed measure and Green s function, respectively. In our case, these are given by mdx = x dx; Gz, x = y + x z. Consequently, the above formula yields Q zσ p,z,y] = y z p + p + +, which in particular implies that Πz, y, db := p++ y z Q zσ p,z,y > bdb is a probability measure on, for each z, y. However, 4.3 and 4.4 imply that Πz, y, converges weakly as y tends to z to some probability measure, Πz, on, which satisfies 4.5 L := e b Πz, db = 4p + + z p w z wz. Moreover, since Q zσ p,z,y > b is decreasing in b for each z, y, the iting measure Π is necessarily of the form cε db + p + + z p πz, b db, where ε is the Dirac point mass at, c a nonnegative constant, and πz, is a decreasing function for each x. In particular, πz, =. In order to find the constant c, it suffices to check the value of the function L at. However, using the explicit form of w, since c = due to.7. Thus, we have shown that 4p + + z p w z wz = w z wz = Q z y z x x I αx I α x = x p + + z I I log I α x x = exp p,z,y] Σ = y z z z e b πz, b db. = Since π is decreasing with πz, =, we obtain by integrating by parts Q exp p,,y ] Σ Σ p,,z = a e a { } = e a e b πz, db y z y z where πz, db = πz, db. Finally, note that one necessarily has bπz, db <, since otherwise L, as defined in 4.5, would have been infinite. The above theorem yields that the sequence of measures Q z Σ p,z,y >b y z db converges vaguely to a finite measure on, as y z. Thus, for any f CK R +, R, i.e. the space of continuously differentiable functions with a compact support, we have Q f Σ p,,y Σ p,,z = a ] fa 4.6 = {fa + b fa} πz, db. y z y z Using the integral representation of Iα for α >, it is tedious but straightforward to check that this representation holds for z = as well by taking the it as z and showing that L < for >.
8 UMUT ÇETIN In other words, letting BR + denote the bounded Borel functions defined on R +, if we define the operator A z : BR + BR + by setting A z fa := then we see that the process M f y y defined by {fa + b fa} πz, db M f y := fσ p,,y y A z fσ p,,z dz for f C KR +, R, is a martingale with respect to the filtration F Ry y whenever f belongs to the domain of A z for all z. The form of the infinitesimal generator also reveals the fact that the increasing process Σ p,,y y is purely discontinuous, i.e. there is no interval a, b in which it is continuous. Remark 7. It follows from the fact that R y y is left continuous that Σ p,,y y is a left-continuous process. However, in view of the above Feller property one can obtain a càdlàg version of it when we augment the filtration with the null sets. Existence of a right-continuous version can also be independently verified by observing that Q ] ] exp Σ p,,y = Q exp Σ p,,z. y z We will end this section by analysing a specific time reversal example. To this end let L x := sup{t : X t = x} and suppose that > so that Q L x < = for all x. We will consider the process Z defined by 4.7 Z x := L L x X p s ds x,. In view of the well-known time reversal results for diffusions, see, e.g., Exercise.3 in Chap. XI of 5], the law of the process X L t, t < L under Q is identical to that of X t, t < R under Q. Recall that Q R < =. Thus, we can write 4.8 Z x = R x X p s ds, X = BESQ. Note that the above equality is to be understood in the sense of equality between the laws of the processes. Due to the strong Markov property of X we again have that Z x x, is a Markov process with respect to the filtration F x x, where F x := σx s ; L x s L. Observe, more easily in view of 4.8, that Z, too, has independent increments rendering its Feller property in view of the arguments that led to the Feller property of Σ p,,z z at the beginning of this section. The next theorem will yield the form of the infinitesimal generator. Its proof will follow similar lines of the proof of Theorem 4. so we will only give the details when it differs. Theorem 4.. Let Z be as defined in 4.7 and suppose, ]. Then, for every x, there exists a decreasing function πx, satisfying where wx := x K Q 4.9 y x x e b πx, bdb =. Moreover, exp Z y Z x = a ] e a = y x where ρx, db := πx, db for b. In particular, w x,,, w x bρx, db <. { e a+b e a} ρx, db,
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES 9 Proof. First observe that in view of 4.8 and the aforementioned independent increments property Q exp ] Z y Zx = a = e a Q β x exp ] Σβ p, x, y, where β =. Therefore, the same arguments in the beginning of the proof of Theorem 4. yields that the measures Qβ x Σβ p, x, y >bdb y x converges vaguely to some probability measure, Πx, on, which satisfies 4. L := e b w Πx, x db = w x. Moreover, the iting measure is necessarily of the form cε db + πx, b db, where ε is the Dirac point mass at, c a nonnegative constant, and πx, is a decreasing function for each x. Observe that since Q β x Σβ p, x, y =, the measures Qβ x Σβ p, x, y >bdb y x are not finite and neither is their vague it. On the other hand, we can still conclude that c = since, in view of.7, one has x x Next, integrating 4. by parts yields 4. πx, K αx K α x = x log K α x x =. e b πx, db = w x w x. Thus, due to the dominated convergence theorem, taking the it of the above as tends to yields πx, = w x w x. Also note that using the asymptotic behaviour of K near.7 one has that for any α. This in turn yields that αx x xk K α x = log K α x = α. x log x w x wx = x + p + K x and thus πx, =. As in the proof of Theorem 4. e a w x w x = Q y x K x exp Z y x p + x =, Z x = a ] e a. y x Thus, combining above with 4. and plugging in the value of πx, yield Q exp Z y Z x = a ] e a { = e a+b e a} ρx, db. y x y x Example 4.. As an application of the above theorem consider the case when, ] and the associated π is defined by e b πx, bdb = x in view of the explicit form for K. Thus, by inverting the above transform, we have πx, b = x. πb =. Then,
UMUT ÇETIN This reveals that the infinitesimal generator, à x of Z is defined by à x = x {fa + b fa} πb db 3 for any f in C with a compact support. Consequently, fz x is an F-martingale for such f. x y { fz y + b fzy } db dy πb3 Example 4.. Observe that although Z, or Σ p,,y, is an increasing process with independent increments, it is not a subordinator see 3] for a definition and further properties since the increments are not stationary. However, in the above example, if one takes =, then one sees that Z becomes time homogeneous, i.e. Z is a subordinator. Observe that = implies p = in this framework. More precisely, Z x = Z 4 x = L L x X s ds, where X is BESQ 4, is a subordinator. Moreover, Corollary.4 and 4.8 yield Q exp Z 4 x = exp x. Thus, Z 4 x d = T x, where T x is the first hitting time of x for a Brownian motion starting at. 5. Applications to finance Our aim in this section is to give some examples arising from some financial models and discuss how the results from previous sections can be used to obtain prices of certain financial products. As explained in Introduction the process X is commonly used in the finance literature to model interest rates. Suppose the spot interest rate is given by X p where p > and X is BESQ z and consider the following exotic derivative on interest rates which pays one unit of a currency at time R y if the accumulated interest is less than k, i.e. Σ p,z,y log k. This is an example of a digital option and its price, as usual in the Finance Theory, is given as an expectation of its discounted payoff: Dk;, p, z, y = Q x Σ p,z,y k] exp Σ p,z,y ]. On the other hand, if one computes the Laplace transform of Dk;, p, z, y, one obtains e µk Dk;, p, z, y dk = Q z e µk exp ] Σ p,z,y dk = Q ] z exp µ + Σ p,z,y, Σ p,z,y which is at our disposal due to the results from Section. In particular, the above identity implies hence 5. u e µk e k Dk;, p, z, y dk = e µu Q z Σ p,z,y du ], e k Dk;, p, x, y dk = Q z Σ p,z,y u ], u.
ON CERTAIN INTEGRAL FUNCTIONALS OF SQUARED BESSEL PROCESSES Moreover, once the function D is determined by inverting the corresponding Laplace transform, one can also obtain the prices for put options written on the accumulated interest. Indeed, for any K > K P ; z, y, K := Q z exp Σ + ] p,z,y exp Σ p,z,y K = Q z Σ p,z,y log k] dk exp Σ ] p,z,y 5. = K Dlog k;, p, z, y dk = log K e u Du;, p, z, y du. Note that for K =, i.e when the option is at-the-money since the cumulative interest at t = is defined to be, the put option is worthless. We can in fact get how fast the option becomes worthless as K approaches to. Indeed, comparing 5. to 5. yields the following asymptotics for the option value in view of Theorem 3.: 5.3 log K log P ; z, y, K = K p + z y. The above expression tells us how small the option price becomes when the option is slightly in-the-money, i.e. when K is very close to. Next, assume y < z and consider another type of a put option on the maximum of the short rate with maturity R y and payoff K max t Ry X t + for K > z. The price of this option equals Q z exp Σ ] K p,z,y maxt Ry X t<a] da = Q z exp Σ ] K p,z,y Ra>R y] da = u z u y x z K y x s z sa s y s a da, in view of Corollary.3 and Proposition.3, where the pair u, s is computed by setting =. Finally, if one is interested in pricing an Asian option on the short rate until time R y, it suffices to use the identity Q z K Σ p,z,y R y + exp Σ p,z,y = Q z exp Σ ] K p,z,y Σ p,z,y <kr y] dk and invert the joint Laplace transform of R y and Σ p,z,y obtained in Section. References ] Abramowitz, M. and Stegun, I. A. 964: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55. ] Aurzada, F. and Dereich, S. 9: Small deviations of general Lévy processes. The Ann. Prob., 375, pp. 66 9. 3] Bertoin. J. 999. Subordinators: Examples and Applications. Ecole d été de Probabilités de St-Flour XXVII, pp. -9, Lect. Notes in Maths 77, Springer, Berlin. 4] Bingham, N. H., Goldie, C. M. and Teugels, J. L. 987: Regular variation. Cambridge University Press, Cambridge. 5] Davydov, D., Linetsky, V. : The valuation and hedging of barrier and lookback options under the CEV process. Manag. Sci., 47, pp. 949-965. 6] Göing-Jaeschke, A. and Yor, M. 3: A survey and some generalizations of Bessel processes. Bernoulli, 9, pp. 33 349. 7] Jeanblanc, M., Pitman, J. and Yor, M. 997: The Feynman-Kac formula and decom- position of Brownian paths. Mat. Apl. Comput. 6, pp. 7 5. 8] Jeanblanc, M., Yor, M. and Chesney, M. 9: Mathematical Methods for Financial Markets. Springer, London. 9] Karatzas, I., and Shreve, S. E. 99: Brownian motion and stochastic calculus, nd Edition. Springer, New York. ] Khosnevisan, D. and Shi, Z. 998: Chung s law for integrated Brownian motion. Trans. Amer. Math. Soc., 35, pp. 453 464. ] Li, W. V. : Small ball probabilities for Gaussian Markov processes under the L p -norm. Stoc. Proc. App., 9, pp. 87.
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