1. Introduction
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., [
3]). For a continuous, increasing boundary
such that
let
be 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 [
7]) and in diffusion models for neural activity ([
9]).
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]).
The paper is organized as follows:
Section 2 contains the main results, in
Section 3 some explicit examples are reported;
Section 4 is devoted to conclusions and final remarks.
2. Main Results
The following holds:
Theorem 1. 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: Proof. 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., [
3]), 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 [
14]). 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 [
14]), and the dominated convergence theorem, we obtain that
Thus, if
is the Laplace transform of the density of the initial position
we finally get
that is Equation (
2). ☐
Remark 1. If one takes in place of a process of the form with that is, a special case of continuous Gauss-Markov process ([15]) 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]). Remark 2. 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 [2], 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 Remark 3. 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
Remark 4. 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 [8] for where 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 [8], 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 get
and 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 [
2]. 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 [
2]). As for the question of the existence of solutions to the IFPT problem, we have:
Proposition 1. 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 Remark 5. If f is given by Equation (20), that is is the Gamma density, the compatibility condition in Equation (14) becomes which is satisfied under the assumption required by Proposition 1. In the special case when then η has the same distribution as where are independent and exponential with parameter The following result also follows from Proposition 2.5 of [
2].
Proposition 2. 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 every
namely
is enveloped from above and below by the functions
and
Then, by using Proposition (3.13) of [
16] (see also [
1]), we obtain the following:
Proposition 3. 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: - (i)
If and the function for some its Laplace transform must satisfy: - (ii)
If then Equation (25) holds without any further assumption on g (and the term vanishes).
Remark 6. 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., [
17]). For
we denote by
the FPT of
below the boundary
The following holds:
Proposition 4. If there exists a solution to the IFPT problem of below with then its Laplace transform is given by Proof. Taking the derivative, one obtains the FPT density of
where
f is the density of
Then, by the same arguments used in the proof of Theorem 1, we obtain
that is Equation (
26). ☐
Remark 7. - (i)
For namely when no jump occurs, Equation (26) becomes Equation (4). - (ii)
If τ is exponentially distributed with parameter then Equation (26) provides: - (iii)
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 [16] one getswhere, for simplicity of notation we have denoted again with the Laplace transform of of course, if then is the Laplace transform of Notice that, if the last equation is nothing but Equation (5) with in place of
3. Some Examples
Example 1. If with and examples of solution to the IFPT problem, for and various FPT densities can be found in [2]. Example 2. 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.
Example 3. 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 that
where, 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 form
where
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 [
2]). 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 [
2]; two examples of diffusions conjugated to BM are the Feller process, and the Wright–Fisher like (or CIR) process, (see e.g., [
2]). 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 SDE
where the drift
and diffusion coefficients
satisfy the usual conditions (see e.g., [
18]) 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 obtains
that 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., [
3]) 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.