A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

: Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the ﬁrst passage time density through speciﬁc boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the ﬁrst passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.


Introduction
Due the wide range of applications, from those of the financial context to those in biology, computational neurosciences, genetics, and physics, the Gaussian processes have always been of great interest (see, for instance, [1][2][3]). The theory of diffusion processes has extensively developed and many interesting mathematical results, particularly useful for applications, were obtained (see, for instance, [4][5][6][7]). Indeed, even if accurate models were designed to describe and investigate real phenomena, often these models are too complicated to be treated mathematically; in most of these cases the diffusion approximation provided the solution that was the right compromise between the need to have more realistic models and a simple and effective mathematical description. Some instances can be found in the field of computational neuroscience, such as in [8][9][10][11], or in the field of queueing theory ( [12,13]).
In the last two decades, also the Gauss-Markov (GM) processes ( [14]) are often involved in a similar way to specialize existing models; they constitute a simplified mathematical tool respect to stochastic process that are Gaussian but do not have the Markov property. Under specific hypotheses, the Gauss-Markov processes are also diffusions, hence, they are called Gauss-Diffusion processes; in this case, their transition density function solves the Fokker-Planck partial differential equation (pde), typical of diffusion processes ( [15][16][17]), but with specified coefficient functions. Here, we will focus our attention on these kind of processes.
The central interest is the determination of the probability density function (pdf) of the first passage time (FPT) of these kind of processes through boundaries (constant or time-dependent). Some results were already obtained for diffusion processes, even if most of them were obtained by transformation methods of the Bachelier-Lévy formula (see, for instance, [18][19][20][21][22]) for the Brownian motion and a linear boundary. Then, for GM processes, in [14,23,24] can be found contributions in Let T ⊆ R be a continuous parameter set and (Ω, F , {F t } t∈T , {X(t)} t∈T , P) be a stochastic process with state space S = R. The process {X(t), t ∈ T} is a real continuous Gauss-Markov process if it is normal and it has a continuous mean function m(t) := E[X(t)] in T and a continuous covariance function c(s, t) := E{[X(s) − m(s)][X(t) − m(t)]} in T × T. Moreover, {X(t)} is non-singular process except at the end points of T, i.e., if T = [a, b] (a, b ∈ R), then {X(t)} is non-singularly normal distributed except possibly in t = a or t = b, where X(t) could be X(t) = m(t) with probability 1.
More specifically, the covariance function c(s, t) of a GM process is typically such that c(s, t) = h 1 (s)h 2 (t), s ≤ t, s, t, ∈T with h 1 (t), h 2 (t) we call the covariance factors. The ratio function of the covariance factors, i.e., r(t) = is a monotonically increasing function; note that h 1 (t)h 2 (t) > 0, ∀t ∈ T due the process is non-singular in the interior of T. The transition mean and variance of the process X(t) are for t, τ ∈ T, τ < t, and the normal transition pdf f (x, t|y, τ) remains completly specified by the above quantities. Now, we consider two C 1 (T)-class functions, i.e., S 1 (t) and S 2 (t) such that We call S 1 (t) the lower and S 2 (t) the upper boundary, respectively. We define the following random variables, ∀t ≥ t 0 , t, t 0 ∈ T, x 0 is the lower FPT through the boundary S 1 (t), T x 0 is the upper FPT through S 2 (t) and T x 0 is the FET from the R × R open subset (S 1 (t), S 2 (t)), respectively. Furthermore, the respective pdfs are the following We note that, for X(t 0 ) = x 0 , if we consider the events, , Consequently, we can denote the probability that X(t) firstly attains , and with P(T x 0 < t) we denote the probability that X(t) firstly attains either S 1 (t) or S 2 (t) by time t. We recall that the pdfs g 1 (t|x 0 , t 0 ) and g 2 (t|x 0 , t 0 ) are solutions of the two coupled non-singular second kind Volterra integral equations ( [23]): recalling that f [x, t|y, τ] is the transition pdf of X(t). By solving the system (5) it is possible to evaluate g from (4). Closed form results for (5) are known in only a few cases (cf., [14,23]).

Closed-Forms Results
From [23] we recall that integral equations (5) can be reduced to a single equation under some conditions. Indeed, under specific assumptions on the process and the boundaries, the first-exit time pdf g(t | x 0 , t 0 ) solves a single non-singular Volterra integral equation in place of Equations (5).
In addition to all previous assumptions, if the following conditions are satisfied, i.e., we have: then the system (5) reduces to the following integral equation Finally, if (9) holds for all t ≥ t 0 and the initial state x 0 satisfies the following relation then one has Hence, in this case, the solution g(t | x 0 , t 0 ) of the integral Equation (10) is given by In [23], solutions of (10) are given as series of functions when the two boundaries are specific functions of the mean and covariance of the GM process. Successively, such solutions are specialized for some GM processes and boundaries in [39]. In any other case, the system (5) can be solved by numerical procedures providing reliable approximations of the solutions.

Symmetry Properties
In the state space of the process {X(t), t ∈ T}, consider the following curves: (the symmetric curve of u(t) respect to the mirror y(t)) ∈ R, and the corresponding symmetry functions denoted by ψ 0 (x, t), φ 0 (x, t), (associated with y(t)) with v(t) = ψ 0 (u(t), t).
The symmetry properties of GM processes ( [28]) are such that for a general curve z(t) = m(t) + ah 1 (t) + bh 2 (t), (a, b ∈ R), with the associated symmetry functions the following relations hold We point that the above relations written for the couple of functions (ψ, φ) hold for the symmetry functions (ψ i , φ i ) , for i = 0, 1, 2, for symmetry curves y(t), u(t), v(t) of (14), respectively.

Transition Distribution Function in a Two-Sided Region
Assuming that P{X(τ) = y(τ)} = 1 for a fixed τ ∈ T, we denote by S 1 (t, τ) and S 2 (t, τ) the C 1 (T) functions such that, for t ∈ T and t ≥ τ, verify the following conditions and the transition probability distribution function of X(t) between the two boundaries S 1 (t, τ), S 2 (t, τ), and its density, respectively. Now, in the next theorem, we give our first main result in which we give a closed form expression for β(x, t|y(τ), τ) when the boundaries S 1 (t, τ), S 2 (t, τ) are of Daniels type ( [20,25]). (Note that we can also consider these boundaries as absorbing boundaries.) Theorem 1. For a fixed t 0 ∈ T, let the lower and upper boundaries be Then, the transition probability density function of the process X(t) between the two boundaries S 1 (t, t 0 ) and S 2 (t, t 0 ), for t ≥ t 0 , has the form We will give the proof of the above theorem in Section 4.1, but before we need some preliminary results about some representations and properties of the transition probability density β(x, t|y(t 0 ), t 0 ).

Characterization of the Transition Density in a Two-Sided Region
with and Proof. Note that, referring to (15), y(t 0 ) can also be written as follows Hence, using (27) in the right-hand-side of (23), we havẽ Then, referring to the general symmetry relation (17) and Finally, (24) is obtained by using (27) and (29) in (28). Furthermore, (25) is the second of (16) for the symmetry function Again by using the second of (16) for v(t) and (26) is obtained as follows Remark 1. We can also note that the functionβ(x, t|y(t 0 ), t 0 ) can also be rewritten as The last form will be useful in the next section.

Proposition 1.
The functionβ(x, t|y(t 0 ), t 0 ) solves the Fokker-Planck partial differential equation: where with the initial delta-type condition, i.e., Proof. We note that from (24) of Lemma 1 Hence,β is a linear combination of f (x, t|·, t 0 ). We recall that the transition pdf f (x, t|·, t 0 ) of the GM process X(t) solves the following Fokker-Planck pde with corresponding initial delta-type conditions and A 1 (x, t), A 2 (t) as in (34). Then, taking into account (36), Equation (35) can be explicitly written as Finally, rearranging the last equation, we obtain Equation (33) with the corresponding initial condition. Hence, the thesis holds.
the functionβ(x, t|y(t 0 ), t 0 ) has the following expression: Proof. Coming back to the expression (23) ofβ(x, t|y(t 0 ), t 0 ) and by using the symmetry relation (18) we havẽ Recalling that i.e., 2y(t) − u(t) = v(t), and by using the symmetry property of the symmetry curve z(t) such that . Hence, the last term in (38) becomes: Therefore, from (38) and (39), we havẽ that is the (37).
We finally claim thatβ = β, i.e.,β is the transition pdf of the process X(t) in presence of the two boundaries (21).

Pdf of FET
Here, exploiting the form and the properties of the transition pdf β(x, t|y(t 0 ), t 0 ) in the presence of the boundaries (21), we can obtain the first-exit-time probability density from the above two-sided open set in closed form. First, we provide an expression of the distribution function of the FET density.

Proposition 2.
Under the hypotheses of Theorem 1, for the two-sided region delimited by the boundaries (21), for t > t 0 , we have the following result for the distribution function of FET as the integral of g(t|y(t 0 ), t 0 ):

Proof.
Recalling that the following relation between the transition density β(x, t|y(t 0 ), t 0 ) and the FET density g[t|y(t 0 ), t 0 ] holds Using here Remark 1, and proceeding to integrate β(x, t|y(t 0 ), t 0 ) between the two boundaries (21), the thesis holds, with F[x, t|y, τ] the transition probability distribution function of the process.
We now need to prove a preliminary lemma about an integral representation of FET density.

Lemma 3.
Under the hypotheses of Theorem 1, for the two-sided region delimited by the boundaries (21), taking into account the functions U and R of Lemma 2 , for the GM process X(t), for t > t 0 , we first obtain the FET pdf such as Proof. From (44), by differentiating, we obtain in which we will insert β(x, t|y(t 0 ), t 0 ) such as from Remark 1. Furthermore, from Proposition 1, we know that β(x, t|y(t 0 ), t 0 ) solves the Fokker-Planck pde (33). Hence, we can write: Now, taking into account we obtain Similarly, it can be proved Indeed, taking into account that v(t 0 ) = 2y(t 0 ) − u(t 0 ) − y(t 0 ) = y(t 0 ) − u(t 0 ), one has: Finally, by inserting (47) and (48) in (46), the lemma is proved.

Theorem 2.
Under the hypoteses of Theorem 1, for the two-sided region delimited by the boundaries (21), taking into account the result and the functions U and R of Lemma 2 , for the GM process X(t), for t > t 0 , we have the following closed form for the FET pdf Recalling that: By using the following relation and inserting this in (50) the proof is completed.

Remark 2.
We note that Hence, by using also (51), we can write and finally, by substituting the last relation in (50), we obtain the following expression (comparable with that of [24])

Two Specific Examples
Only for explanatory purposes and comparisons with known results (see, for instance, [28,34]), we give some explicit expressions for the above functions of two GM processes such as Wiener and Ornstein-Uhlenbeck processes, due their central rule in the class of GM processes ( [14]).
As last remark, we note that the Ornstein-Uhlenbeck process X(t) here considered is also solution of the following stochastic differential equation (SDE): with W(t) is a standard Wiener process. The determination of the first passage time density of X(t) from a region is the central problem for very large number of models based on the above SDE. The symmetry strategy and the obtained expressions in presence of Daniels-type boundaries can be useful also in such modeling contexts, because, under specific assumptions, some (piecewise) approximations can be derived.