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Mathematics 2018, 6(6), 91; doi:10.3390/math6060091
The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion
Dipartimento di Matematica, Università Tor Vergata, 00133 Rome, Italy
Received: 4 April 2018 / Accepted: 25 May 2018 / Published: 28 May 2018
Let be a continuously time-changed Brownian motion starting from a random position a given continuous, increasing boundary, with and F an assigned distribution function. We study the inverse first-passage time problem for which consists in finding the distribution of such that the first-passage time of below has distribution generalizing the results, valid in the case when is a straight line. Some explicit examples are reported.
Keywords:first-passage time; inverse first-passage problem; diffusion
This brief note is a continuation of [1,2]. Let be a regular enough non random function, and let where is standard Brownian motion (BM) and the initial position is a random variable, independent of Suppose that the quadratic variation is increasing and then there exists a standard BM such that namely is a continuously time-changed BM (see e.g., ). For a continuous, increasing boundary such that letbe the first-passage time (FPT) of below We assume that is finite with probability one and that it possesses a density where Actually, the FPT of continuously time-changed BM is a well studied problem for constant or linear boundary and a non-random initial value (see e.g., [4,5,6]).
Assuming that is increasing, and is a continuous distribution function, we study the following inverse first-passage-time (IFPT) problem:
given a distribution F, find the density g of η (if it exists) for which it results .
The function g is called a solution to the IFPT problem. This problem, also known as the generalized Shiryaev problem, was studied in [1,2,7,8], essentially in the case when is BM and is a straight line; note that the question of the existence of the solution is not a trivial matter (see e.g., [2,7]). In this paper, by using the properties of the exponential martingale, we extend the results to more general boundaries
The IFPT problem has interesting applications in mathematical finance , in particular in credit risk modeling, where the FPT represents a default event of an obligor (see ) and in diffusion models for neural activity ().
Notice, however, that another type of inverse first-passage problem can be considered: it consists in determining the boundary shape S, when the FPT distribution F and the starting point are assigned (see e.g., [10,11,12,13]).
2. Main Results
The following holds:
Let be a continuous, increasing boundary with a bounded, non random continuous function of and let be the integral process starting from the random position we assume that is increasing and satisfies Let F be the probability distribution of the FPT of X below the boundary S ( is a.s. finite by virtue of Remark 3). We suppose that the r.v. η admits a density for we denote by the Laplace transform of
Then, if there exists a solution to the IFPT problem for the following relation holds:
The process is a martingale, we denote by its natural filtration. Thanking to the hypothesis, by using the Dambis, Dubins–Schwarz theorem (see e.g., ), it follows that the process is a Brownian motion with respect to the filtration so the process can be written as and the FPT can be written as For let us consider the process as easily seen, is a positive martingale; indeed, it can be represented as (see e.g., Theorem 5.2 of ). We observe that, for the martingale is bounded, because is non negative and therefore Then, by using the fact that, for any finite stopping time one has (see e.g., Formula (7.7) in ), and the dominated convergence theorem, we obtain that
Thus, if is the Laplace transform of the density of the initial position we finally getthat is Equation (2). ☐
If one takes in place of a process of the form with that is, a special case of continuous Gauss-Markov process () with mean then is still a continuously time-changed BM, and so the IFPT problem for and is reduced to that of continuously time-changed BM and a constant barrier, for which results are available (see e.g., [4,5,6]).
By using Laplace transform inversion (when it is possible), Equation (4) allows to find the solution g to the IFPT problem for the continuous increasing boundary and the distribution F of the FPT Indeed, some care has to be used to exclude that the found distribution of η has atoms together with a density. However, as already noted in [2,7], the function may not be the Laplace transform of some probability density function, so in that case the IFPT problem has no solution; really, it may admit more than one solution, since the right-hand member of Equation (4) essentially furnishes the moments of η of any order but this is not always sufficient to uniquely determine the density g of In line of principle, the right-hand member of Equation (4) can be expressed in terms of the Laplace transform of though it is not always possible to do this explicitly. A simple case is when with and that is, in fact, one obtainswhich coincides with Equation (2.2) of , and it provides a relation between the Laplace transform of the density of the initial position η and the Laplace transform of the density of the FPT
Let be increasing and then τ is a.s. finite; in fact where is increasing and is the first hitting time to of BM starting at since is a.s. finite, also is so. Next, from the finiteness of it follows that is finite, too. Moreover, if one seeks that a sufficient condition for this is that and are both convex functions; indeed, where is the FPT of BM starting from η below the straight line which is tangent to the graph of at Thus, since it follows that is finite, too; finally, being concave, Jensen’s inequality for concave functions implies that and therefore
Theorem 1 allows to solve also the so called Skorokhod embedding (SE) problem:
Given a distribution find an integrable stopping time such that the distribution of is namely
In fact, let be increasing, with first suppose that the support of H is then, from Equation (4) it follows thatand this solves the SE problem with it suffices to take the random initial point in such a way that its Laplace transform satisfies
In the special case when and Equation (7) becomes (cf. the result in  forwhere and denotes the Laplace transform of
In analogous way, the SE problem can be solved if the support of H is now, the FPT is understood as that is, the first hitting time to the boundary from below.
Therefore, the solution to the general SE problem, namely without restrictions on the support of the distribution can be obtained as follows (see , for the case when is a straight line).
The r.v. can be represented as a mixture of the r.v. and
Suppose that the SE problem for the r.v. and can be solved by and and and respectively. Then, we get that the r.v.and the boundary solve the SE problem for the r.v.
If is analytic in a neighbor of then the moments of order n of exist finite, and they are given by By taking the first derivative in Equation (4) and calculating it at we obtain
By calculating the second derivative of at we getand so
Thus, we obtain the compatibility conditions
If a solution to the IFPT problem does not exist. In the special case when and Equation (11) becomes and Equation (13) becomes while Equation (14) coincides with Equation (2.3) of . By writing the Taylor’s expansions at of both members of Equation (4), and equaling the terms with the same order in one gets the successive derivatives of at thus, one can write any moment of in terms of the expectation of a function of for instance, it is easy to see that
2.1. The Special Case
If with from Equation (4) we get
Thus, setting we obtain (see Equation (5)):having denoted by the density of In this way, we reduce the IFPT problem of below the boundary to that of BM below the linear boundary For instance, taking the solution to the IFPT problem of through the cubic boundary and the FPT density is nothing but the solution to the IFPT problem of BM through the linear boundary and the FPT density
Under the assumption that with a number of explicit results can be obtained, by using the analogous ones which are valid for BM and a linear boundary (see ). As for the question of the existence of solutions to the IFPT problem, we have:
Let be with for suppose that the FPT density is given by(namely the density of is the Gamma density with parameters Then, the IFPT problem has solution, provided that and the Laplace transform of the density g of the initial position η is given by:which is the Laplace transform of the sum of two independent random variables, and such that has distribution Gamma of parameters γ and where and
The following result also follows from Proposition 2.5 of .
Let be with for suppose that the Laplace transform of has the form:for some Then, there exists a value such that the solution to the IFPT problem exists, provided that
If and the Laplace transform of has the form:then, the solution to the IFPT problem exists.
2.2. Approximate Solution to the IFPT Problem for Non Linear Boundaries
Now, we suppose that there exist with and such that, for everynamely is enveloped from above and below by the functions and
Let a continuous, increasing boundary satisfying Equation (24) and suppose that the FPT τ of below the boundary has an assigned probability density f and that there exists a density g with support which is solution to the IFPT problem for and the boundary as before, denote by the density of and by its Laplace transform, for Then:
- If and the function for some its Laplace transform must satisfy:
- If then Equation (25) holds without any further assumption on g (and the term vanishes).
The smaller and the better the approximation to the Laplace transform of Notice that, if g is bounded, then the term can be replaced with
2.3. The IFPT Problem for Large Jumps
As an application of the previous results, we consider now the piecewise-continuous process , obtained by superimposing to a jump process, namely we set for where T is an exponential distributed time with parameter we suppose that, for the process makes a downward jump and it crosses the continuous increasing boundary irrespective of its state before the occurrence of the jump. This kind of behavior is observed e.g. in the presence of a so called catastrophes (see e.g., ). For we denote by the FPT of below the boundary The following holds:
If there exists a solution to the IFPT problem of below with then its Laplace transform is given by
For one has:
Taking the derivative, one obtains the FPT density ofwhere f is the density of Then, by the same arguments used in the proof of Theorem 1, we obtainthat is Equation (26). ☐
- If τ is exponentially distributed with parameter then Equation (26) provides:
- In the special case when we can reduce to the FPT of BM + large jumps below the linear boundary then, it is possible to write in terms of the Laplace transform of Really, by using Proposition 3.10 of  one gets
3. Some Examples
If with and examples of solution to the IFPT problem, for and various FPT densities can be found in .
Let be with and suppose that τ has density (that is, the density of is exponential with parameter By using Proposition 1 we get that where are independent random variable, such that has exponential distribution with parameter ( where and Then, the solution g to the IFPT problem for the boundary S and the exponential FPT distribution, is:
In general, for a given continuous increasing boundary and an assigned distribution of it is difficult to calculate explicitly the expectation on the right-hand member of Equation (4) to get the Laplace transform of Thus, a heuristic solution to the IFPT problem can be achieved by using Equation (4) to calculate the moments of (those up to the fifth order are given by Equations (11), (12) and (15)–(17)). Of course, even if one was able to find the moments of of any order, this would not determinate the distribution of However, this procedure is useful to study the properties of the distribution of provided that the solution to the IFPT problem exists.
Let be and suppose that τ is exponentially distributed with parameter we search for a solution to the IFPT problem by using the method of moments, described above. The compatibility condition in Equation (14) requires that (for instance, one can take From Equations (11), (12) and (15)–(17), and calculating the moments of τ up to the eighth order, we obtain:
Notice that, under the condition the first four moments of η are positive, as it must be. However, they do not match those of a Gamma distribution.
An information about the asymmetry is given by the skewness valuemeaning that the candidate η has an asymmetric distribution with a tail toward the left.
4. Conclusions and Final Remarks
We have dealt with the IFPT problem for a continuously time-changed Brownian motion starting from a random position For a given continuous, increasing boundary with and an assigned continuous distribution function the IFPT problem consists in finding the distribution, or the density g of such that the first-passage time of below has distribution In this note, we have provided some extensions of the results, already known in the case when is BM and is a straight line, and we have reported some explicit examples. Really, the process we considered has the form where is standard Brownian motion, and is a non random continuous function of time such that the function is increasing and it satisfies the condition Thus, a standard BM exists such that Our main result states thatwhere, for denotes the Laplace transform of the solution g to the IFPT problem.
Notice that the above result can be extended to diffusions which are more general than the process considered, for instance to a process of the formwhere w is a regular enough, increasing function; such a process U is obtained from BM by a space transformation and a continuous time-change (see e.g., the discussion in ). Since the IFPT problem for the process the boundary and the FPT distribution is reduced to the analogous IFPT problem for starting from instead of the boundary and the same FPT distribution When i.e. the process is conjugated to BM, according to the definition given in ; two examples of diffusions conjugated to BM are the Feller process, and the Wright–Fisher like (or CIR) process, (see e.g., ). The process given by Equation (32) is indeed a weak solution of the SDE:where and denote first and second derivative of
Provided that the deterministic function is replaced with a random function, the representation in Equation (32) is valid also for a time homogeneous one-dimensional diffusion driven by the SDEwhere the drift and diffusion coefficients satisfy the usual conditions (see e.g., ) for existence and uniqueness of the solution of Equation (34). In fact, let be the scale function associated to the diffusion driven by the SDE Equation (34), that is, the solution of where L is the infinitesimal generator of U given by As easily seen, if the integral converges, the scale function is explicitly given by
If by It’s formula one obtainsthat is, the process is a local martingale, whose quadratic variation is
The (random) function is differentiable and if it is increasing to by the Dambis, Dubins–Schwarz theorem (see e.g., ) one gets that there exists a standard BM such that Thus, since w is invertible, one obtains the representation in Equation (32).
Notice, however, that the IFPT problem for the process U given by Equation (32) cannot be addressed as in the case when is a deterministic function. In fact, if given by Equation (37) is random, it results that and the FPT are dependent. Thus, in line of principle it would be possible to obtain information about the Laplace transform of only in the case when the joint distribution of was explicitly known.
This research was funded by the MIUR Excellence Department Project awarded tothe Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
I would like to express particular thanks to the anonymous referees for their constructive comments and suggestions leading to improvements of the paper.
Conflicts of Interest
The author declares no conflict of interest.
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