On the dimension reduction in the quickest detection problem for diffusion processes with exponential penalty for the delay *

The problem of the quickest detection of a change in the drift of a time-homogeneous diffusion process is considered under the assumption that the detection delay is exponentially penalized. In this framework, the past literature has shown that a two-or three-dimensional optimal stopping problem needs to be faced. In this note, we show how a change of measure signiﬁcantly simpliﬁes the setting by reducing the dimension of the optimal stopping problem to one or two, respectively. We illustrate this result in the well known Brownian motion case analyzed by Beibel [4] and when a Bessel process is observed, generalizing therefore the results for the linear penalty case obtained by Johnson and Peskir [13].


Introduction
The quickest detection problem for a diffusion process X := (X t ) t≥0 aims at determining the moment θ, known as disorder time or change-point, at which X changes its drift. A good detection strategy should not raise an alarm too early to avoid false alarms and should be able to detect θ by making the detection delay as small as possible. In this work, we assume that X is a time-homogeneous diffusion process and we approach the problem in the so called Bayesian setting with exponential penalty for the delay: θ takes on value 0 with positive probability and is exponentially distributed conditionally to the event θ > 0; the risk of our strategy is given by the linear combination between the probability of raising a false alarm and the exponential delay it takes to detect θ since its occurrence (see Section 2.1).
Problems of quickest detection for the drift of diffusion processes have been mostly studied under the linear penalty for the delay: [20] and [10] considered a Brownian motion within the infinite and finite horizon formulation, respectively; [13] studied the problem for a Bessel process; [12] focused on time-inhomogeneous Gaussian processes and fractional Brownian motion. The introduction of the exponential penalty is due to [18] who studied the discrete-time case for independent and identically distributed random variables; he motivated the exponential penalty for the delay as a more suitable measure to capture the true cost of a delayed action in some financial applications and in the health monitoring of components in interconnected systems. Later, [4] extended the Poor's Bayesian problem to the case of a Brownian motion and [11] generalized it to diffusion processes with non-constant drift. Problems of quickest detection with exponential penalty were also studied by [2], [3], [9] for simple and compound Poisson processes and by [6] for certain type of Lévy processes.
The results in [4] showed that, when the signal-to-noise ratio (defined as the difference between the new drift and the old drift divided by the diffusion coefficient) is constant, the initial quickest detection problem boils down to an optimal stopping problem for a two-dimensional Markov diffusion process, having as components the posterior probability Π and the generalized odds Φ (defined in Section 2.1). The dimension further increases when the signal-to-noise ratio is not constant: in this situation, as shown in [11], the original problem can be reduced to an optimal stopping problem for the three-dimensional Markov diffusion process (X, Π, Φ), whose stochastic differential equations are expressed in terms of the innovation process. This makes the triplet (X, Π, Φ) strongly interdependent and, as a result, the analysis of the optimal stopping problem highly involved (see Section 2.2).
The main contribution of this note is thus twofold: (i) we show that a change of measure argument allows us to reduce the dimension of the optimal stopping problem from two to one, when the signal to-noise-ratio is constant, and from three to two, when it is not. In the first case, the optimal stopping problem is for the Markov diffusion process Φ; in the second case, the optimal stopping problem is for the Markov diffusion process (X, Φ), whose stochastic differential equations are now expressed in terms of the original standard Brownian motion and get uncoupled in the first component. Accordingly, the problem becomes significantly simplified (see Section 3). (ii) Under this new setting, we obtain the solution for the prominent case of the Brownian motion, thus providing an alternative approach to that in [4] (see Section 4). Then, we solve the problem for a Bessel process, which is characterized by a non-constant signal-to-noise ratio; to the best of our knowledge, this is the first time that the quickest detection problem with exponential penalty is formulated for such a process (see Section 5). Recalling that the exponential penalty contains the linear penalty as a limit case (see, e.g., [4,Remark 2] and [2]), our results extend those obtained in [13], where the the linear penalty was studied.

Formulation of the problem
In this section we describe the problem of the quickest detection with exponential penalty for the drift of a time-homogeneous diffusion process following the lines of arguments in [11].

Description of the problem
On the probability space (Ω, F , P π ) a standard Brownian motion W := (W t ) t≥0 and a non-negative random variable θ are defined, with W and θ being independent. The probability measure P π is defined by where P s is the probability measure under which P s (θ = s) = 1, s ≥ 0, and π ∈ [0, 1) and λ > 0 are two given and fixed values. From (2.1) it follows that P π (θ = 0) = π and P π (θ > t|θ > 0) = e −λt , for t > 0. The same space hosts a time-homogeneous diffusion process, which solves the stochastic differential equation (2.2) where S is the state space of X, σ(·) > 0 is the diffusion coefficient, while µ 0 (·) and µ 1 (·) are the drift of X before and after the occurrence of θ, respectively. The latter is therefore called disorder time or change-point, after which X changes its drift.
Through the sequential monitoring of X only, the aim is to determine a stopping time of X at which we can optimally interrupt the observation and raise an alarm. More precisely, solving the Bayesian quickest detection problem with exponential penalty for the delay means computing the value function and determining the optimal stopping time τ at which the infimum (taken over all the stopping times τ of X) is attained. In the expression above, P π (τ < θ) is the probability of a false alarm and E π [e α(τ −θ) + − 1] is the expected exponential detection delay associated with a stopping time τ of X; α > 0 is the known value at which the losses due to the detection delay are compounded and c > 0 is a given value which weights their relevance.
The analysis of (2.3) requires the introduction of the following quantities depending on X: where L := (L t ) t≥0 is the likelihood ratio process, Π := (π t ) t≥0 is the posterior probability process and Φ := (Φ t ) t≥0 is the generalized odds process. In the expressions above, F X t is the sigma-algebra generated by X up to t ≥ 0, P ∞ is the probability measure under which the drift of X never changes (i.e., P ∞ (θ = ∞) = 1), and P 0 t and P ∞ t are the restrictions of P 0 and P ∞ to F X t . Girsanov's theorem and the results in [2] also show that L and Φ admits the expressions which will be used in Sections 4 and 5 to disclose the explicit structure of the optimal detection strategy.

A three-dimensional optimal stopping problem
In [11] it was shown that X, Π, and Φ evolve according to the stochastic differential where ρ(·) is the so called signal-to-noise ratio, given by ECP 26 (2021), paper 73.
Dimension reduction in the quickest detection problem with exponential penalty is the innovation process/standard Brownian motion under P π with respect to the filtration F X . The value function V from (2.3) was also reduced to the equivalent optimal stopping where the infimum is taken over all the stopping times τ of X and the superscripts indicate the dependence of the processes Π and Φ on their starting points. When ρ is constant, (Π, Φ) from (2.8)-(2.9) is a strong Markov process and, to approach (2.12), (Π, Φ) must be enabled to start at any point (π, φ) ∈ [0, 1) × [0, ∞) under the probability measure P π,φ , under which (π 0 , Φ 0 ) = (π, φ). In this way, one needs to consider the extended two-dimensional optimal stopping problem where the infimum is taken over all the stopping times τ of (Π, Φ). Moreover, when ρ is not constant, only the triplet (X, Π, Φ) from (2.7)-(2.9) is strongly Markovian and, to tackle (2.12), (X, Π, Φ) must be enabled to start at any point ( In this case, one ends up with the extended three-dimensional optimal stopping problem (2.14) where the infimum is taken over all the stopping times τ of (X, Π, Φ). Sufficient conditions for the existence of a solution to (2.14) were provided in [11, Section 3].

Reduction to an optimal stopping problem of lower dimension
The arguments in [11] are based on the system of stochastic differential equations (2.7)-(2.9) under the probability measure P π ; the fact that X, Π and Φ are strongly interdependent and the high dimensionality of the optimal stopping problem (2.14) make its analysis very complex. Now we see that a change of measure allows us to move from the optimal stopping problem (2.14) for (X, Π, Φ) to an optimal stopping problem for (X, Φ), whose stochastic differential equations get uncoupled in X. These facts simplify considerably the analysis of the problems analyzed in the subsequent sections.
The next proposition derives the stochastic differential equations of X, Π, and Φ under P ∞ . Proposition 3.1. Under P ∞ , X, Φ and Π solve the stochastic differential equations  Now we are going to show that the optimal stopping problem (2.12) admits an equivalent formulation under the measure P ∞ . Denoting by P ∞ τ and P π,τ the restrictions of the measures P ∞ and P π to F X τ , let us recall from [13, Lemma 1] that the Radon-Nikodym derivative of P π,τ with respect to P ∞ τ is given by for all stopping times τ of X. Recalling that Φ starts at π 1−π and that this dependence will be indicated by a superscript to Φ when needed, we can state the following theorem.
for π ∈ [0, 1), with the infimum taken over all the stopping times τ of X.
Proof. By standard localization arguments and the application of the monotone and dominated convergence theorems, it is enough to show that (3.6)-(3.7) hold true for all bounded stopping times τ of X such that (Φ t∧τ ) t≥0 and ρ(X t∧τ ) t≥0 are bounded. Let such a stopping time be given and fixed in the sequel. Recalling that Π and Φ solve (2.8) and (2.9) under P π , it follows that Then, applying Itô's product formula to (1 − π t )Φ t t≥0 we have Using the definition of τ and the fact that the two stochastic integrals in (3.8)-(3.9) are local martingales, according to the optional sampling theorem from (3.8) we have and from (3.9)-(3.10) we obtain Then, from (2.12) and (3.11) we have (3.12) where the second equality is based on the change of measure (3.5).
Motivated by the structure of (3.12), let ϕ t := πt 1−πt for t ≥ 0. Then, by means of (3.3) and the application of Itô's formula to (ϕ t ) t≥0 and e −λt (Φ t − ϕ t ) t≥0 we have (3.14) According to the definition of τ , the fact that the stochastic integral in (3.14) is a local martingale under P ∞ and the application of the optional sampling theorem, we have which completes the proof.
To approach the optimal stopping problem (3.7) for the strong Markov process (X, Φ) we enable it to start at any point (x, φ) ∈ S × [0, ∞), so that (3.7) can be extended tô where the expectation is with respect to the probability measure P ∞ x,φ , under which (X 0 , Φ 0 ) = (x, φ), and the infimum is taken over all the stopping times τ of (X, Φ). In this way we have reduced the three-dimensional optimal stopping problem (2.14) to a twodimensional optimal stopping problem for the strong Markov process (X, Φ) satisfying (3.1)-(3.2), whose first component is uncoupled from the second one. Let us observe that the optimal stopping problem (3.16) is Lagrange formulated; then, consistently to [16, Chapter 1], we may affirm that its continuation and stopping sets are defined by Noting that the integrand in (3.16) is negative when Φ t < λ/(cα), we conclude that {(x, φ) : φ < λ/(cα)} ⊂ C; equivalently, we can say that the least boundary between C and D, defined by b(x) := inf{φ ∈ [0, ∞) : (x, φ) ∈ D}, x ∈ S, (3.19) satisfies b(x) ≥ λ/(cα), for all x ∈ S. It is also immediate to notice that the value function V is bounded and satisfies −(cα) −1 ≤V ≤ 0, where the first inequality is due to the fact that φ − λ/(cα) ≥ −λ/(cα) for φ ≥ 0, while the second equality comes from the fact that τ = 0 is an admissible stopping time and the definition ofV . We conclude this section by deriving the infinitesimal generator L (X,Φ) of (X, Φ) under P ∞ , which will be exploited in the next sections to construct the free-boundary problem that the value functionV and the free-boundary b solve. Standard arguments based on the application of Itô's formula imply that (3.20) Denoted by C i (·) the class of i-times continuously differentiable functions on the specified set ·, L (X,Φ) acts on f ∈ C 2 (S × [0, ∞)); we also observe that L (X,Φ) is of parabolic type (see, e.g. [15,Section 2]) and this is a consequence of the fact that X and Φ are driven by the same original/standard Brownian motion W under P ∞ , as evident from (3.1)-(3.2).

Remark 3.3.
The change of measure from P π to P ∞ in the linear penalty framework analyzed in [13] has the effect to move from a two-dimensional optimal stopping problem for the process (X, Π) solving (2.7)-(2.8) to another two-dimensional optimal stopping problem for the process (X, Φ) solving (3.1)-(3.2), with α = 0 in the latter equation, without leading to any dimensionality simplification. This is a substantial difference with respect to the logic behind our result, whose primary aim is the dimension shrinkage of the problem, while retaining the effect of decoupling the two-dimensional sufficient statistic (X, Φ) from (3.1)-(3.2) in X.

Quickest detection for a Brownian motion
In this section we solve the quickest detection problem (2.3) when X is a Brownian motion changing its drift from one value to another. Then, (2.2) holds with µ 0 , µ 1 , σ being constant values. This problem was studied under the probability measure P π in [4], who treated the optimal stopping problem (2.12) as a generalized parking problem, using an approach similar to [5]. In this case ρ from (2.10) is constant and, therefore, (2.12) is a two-dimensional optimal stopping problem for the process (Π, Φ) leading to (2.13).
However, Theorem 3.2, the same arguments used in the last part of Section 3 and the fact that ρ is constant show that we can equivalently focus on the one-dimensional optimal stopping problem for the strong Markov process Φ, which solves (3.2). The expectation in (4.1) is with respect to P ∞ φ , under which Φ 0 = φ, and the infimum is taken over all the stopping times τ of Φ. The continuation and stopping sets from (3.17)-(3.18) reduce to with the boundary b from (3.19) being now constant. Then, the general theory of optimal stopping for strong Markov processes (see, e.g., [16,Chapter 3 and 4] or [20,Chapter 3]) leads us to formulate the following free-boundary problem forV and b: V (b) = 0 (smooth fit) , (4.5) where L Φ is the infinitesimal generator of Φ and is given by which acts on f ∈ C 2 ([0, ∞)) and follows by discarding in (3.20) the addends whose differential operator depends on x. Let us define and let U (a, b, z) be the Kummer function where the function g(.) and the constant h are given by  Applying then the Itô-Tanaka formula toV from (4.9), which boils down to Itô's formula due to the smooth fit (4.5), and making use of the optional sampling theorem, it is easily verified that the solution to the free-boundary problem (4.3)-(4.5) coincides with that of the optimal stopping problem (4.1) and that the optimal stopping time τ for the latter is given by τ = inf{t ≥ 0 : Φ t ≥ b}. (4.12) In (4.12) Φ t takes the more explicit form which arises from (2.5)-(2.6), upon recalling that µ 0 , µ 1 , σ and ρ are constant. Finally, we have that the value function V of the initial problem (2.3) and (2.12) is given by V (π) = (1 − π) 1 + cαV π 1 − π , π ∈ [0, 1), (4.14) which follows from (3.6)-(3.7) in Theorem 3.2.

Remark 4.1.
Alternatively to the free-boundary approach, optimal stopping problems for Φ when ρ is constant could also be faced by resorting to the theory of optimal stopping for one-dimensional diffusions (see, e.g., [7], [8], [19]), with the latter having the advantage that there is no need to guess the structure of the optimal stopping set.

Quickest detection for a Bessel process
In this section we study the quickest detection problem (2.3) when X is a Bessel process changing its dimension from δ 0 ≥ 2 to δ 1 > δ 0 at time θ. Then, X evolves , with δ 1 > δ 0 ≥ 2, and σ(x) = 1, x > 0. (5.1) This problem has an appealing interpretation when δ 0 = 2 and δ 1 = 3, because it represents the situation where, through the sequential monitoring of the distance that a Brownian particle has from the origin, we want to detect the moment at which the particle moves from the two-dimensional plane to the three-dimensional space. The formulation of this problem is due to [13], where a linear penalty for the detection delay was adopted. As we said in Section 1, the exponential penalty considered in the present paper generalizes the linear penalty. From (2.10) and (5.1) we observe that the signal-to-noise ratio takes the form and is therefore a non-constant function of the observed Bessel process X. Then, according to Section 2.2, solving (2. 3) under P π would lead to face the three-dimensional optimal stopping problem (2.14). However, Theorem 3.2 allows us to move from the measure P π to P ∞ and, consequently, to solve the extended two-dimensional optimal stopping problem (3.16) for (X, Φ), that, from (3.1)-(3.2), satisfy Comparing (3.16) with [13,Eq. (4.19)], we can appreciate the fact that two optimal stopping problems share a similar structure, with the former being the result of a dimension reduction procedure, unlike the latter (see Remark 3.3 above). Let us also notice that in [13,Eq. (4.19)] the process Φ inside the integral stands for the posterior probability ratio process, which is a particular case of the generalized odds process Φ from (2.4) with α = 0, and the constant values λ/c gets replaced by λ/(cα) in our case.

Free-boundary problem
For the optimal stopping problem (3.16), associated to the strong Markov process (X, Φ) from (5.3)-(5.4), we can formulate the following free-boundary problem for findinĝ V and b (see, e.g., [16,Chapter 3 and 4] or [20,Chapter 3]): V (x, φ) = 0, (x, φ) ∈ D (instantaneous stopping) , where the infinitesimal generator L (X,Φ) from (3.20) takes the more specific form and the continuation and stopping sets C and D from (3.17)-(3.18) are more explicitly given by Clearly, the global condition (5.7) can be replaced by the local conditionV (x, b(x)) = 0 for x > 0, so that the free-boundary problem needs to be considered on the closure of C only (by extendingV to zero on D, as evident from (3.18)). Let us now define by A the class of the pairs of functions (F, h) such that: (5.11) h is continuous and decreasing on (0, ∞) with h(x) ≥ λ/(cα) for x > 0, With this in mind, one can easily extend [13,Theorem 17] to show that the free-boundary problem (5.6)-(5.8) has a unique solution (V , b) in the class A from (5.11)-(5.12), whereV and b are given by (3.16) and (3.19), respectively.
Proof. The proof requires minor modifications to the proof in [13,Theorem 19]; this is due to the the similar structure between the optimal stopping problems (3.16) and [13,Eq. (4.19)] obtained by means of Theorem 3.2.