On the free boundary of an annuity purchase

It is known that the decision to purchase an annuity may be associated to an optimal stopping problem. However, little is known about optimal strategies, if the mortality force is a generic function of time and if the `subjective' life expectancy of the investor differs from the `objective' one adopted by insurance companies to price annuities. In this paper we address this problem considering an individual who invests in a fund and has the option to convert the fund's value into an annuity at any time. We formulate the problem as a real option and perform a detailed probabilistic study of the optimal stopping boundary. Due to the generic time-dependence of the mortality force, our optimal stopping problem requires new solution methods to deal with non-monotonic optimal boundaries.


Introduction
In an ageing world an accurate management of retirement wealth is crucial for financial well being. It is important for working individuals to carefully consider the existing offer of financial and insurance products designed for retirement, beyond the state pension. This offer includes for example occupational pension funds and tax-advantaged retirement accounts (e.g. Individual Retirement Account (US)). Most of these products rely on annuities to turn retirement wealth into guaranteed lifetime retirement income. Life annuities provide a lifelong stream of guaranteed income in exchange for a (single or periodic) premium. The purchase of an annuity helps individuals to manage the longevity risk, i.e. the risk of outliving their financial wealth, but it is usually an irreversible transaction. In fact, most annuity contracts impose steep penalties for partial or complete cancellation by the policyholder, especially in the early years of the contract.
Timing an annuity purchase (so-called annuitization) is a complex financial decision that depends on several risk factors as, e.g., market risk, longevity risk, potential future need of liquid funds and bequest motive. The study of this topic has motivated a whole research field since the seminal contribution of Yaari [15], who showed that individuals with no bequest motive should convert all their retirement wealth into annuities.
After Yaari, several authors have analysed the annuitization decision under the socalled all or nothing institutional arrangement, where a lifetime annuity is purchased in a single transaction (as opposed to gradual annuitization). Initially the individual's wealth is invested in the financial market and, at the time of an annuity purchase, it is converted into a lifetime annuity. The central idea in this literature is to compare the value deriving from an immediate annuitization with the value of deferring it, while investing in the financial market. Therefore, a strict analogy holds with the problem of exercising an American option, and the annuitization decision can be considered as the exercise of a real option.
Milevsky [9] proposed a model where an individual defers annuitization for as long as the financial investment's returns guarantee a consumption flow which is at least equal to the one provided by the annuity payments. In particular [9] adopts a criterion based on controlling the probability of a consumption shortfall.
Other papers study the optimal annuitization time in the context of utility maximisation, and formulate the problem as one of optimal stopping and control. The investor aims at maximising the expected utility of consumption (pre-retirement) and of annuity payments (post-retirement).
Assuming constant force of mortality and CRRA utility, Stabile [14] analytically solves a time homogeneous optimal stopping problem. He proves that if the individual has the same degree of risk aversion before and after the annuitization, then an annuity is purchased either immediately or never (the so-called now or never policy). Instead, in case the individual is more risk averse during the annuity payout phase, the annuity is purchased as soon as the wealth falls below a constant threshold (the optimal stopping boundary).
Constant force of mortality is also assumed in Gerrard et al. [4] and Liang et al. [8]. The model in [4] is analogous to the one studied in [14], but with quadratic utility functions, and authors find a closed-form solution: if (X t ) t≥0 represents the individual's wealth process, then it is optimal to stop when X leaves a specific interval (hence the optimal stopping boundary is formed by the endpoints of such interval). In [8], in contrast to the previous papers, the authors assume that the individual may continue to invest and consume after annuitization. By using martingale methods, explicit solutions are provided in the case of CRRA utility functions. Contrarily to [4], in [8] the optimal annuitization occurs when the wealth process enters a specific interval, whose endpoints form the optimal stopping boundary.
Assuming time-dependent force of mortality, Milevsky and Young [10] analyze both the all or nothing market and the more general anything anytime market, where gradual annuitization strategies are allowed. For the all or nothing market they find that the optimal annuitization time is deterministic as an artifact of CRRA utility. Thus, the annuitization decision is independent from the individual's wealth.
Our work is more closely related to work by Hainaut and Deelstra [6]. They consider an individual whose retirement wealth is invested in a financial fund which eventually must be converted into an annuity. The fund is modelled by a jump diffusion process and pays dividends at a constant rate. The mortality force is a time-dependent, deterministic function and the individual aims at maximising the market value of future cashflows before and after annuitization. According to the insurance practice, it is assumed that the individual can only purchase the annuity by a given maximal age. Authors in [6] cast the problems as an optimal stopping one and write a variational inequality for the value function. They then use Wiener-Hopf factorisation and a time stepping method to solve the variational problem numerically. Hainaut and Deelstra argue that the decision to purchase the annuity should be triggered by either an upper or a lower, time-dependent threshold in the time-wealth plane. Here and in what follows t is the state variable associated to time and x is the one associated to wealth, so that the time-wealth plane is simply referred to as the (t, x)-plane. The threshold discussed in [6] is the optimal stopping boundary in their setting and we denote it by a mapping t → b(t). Numerical examples are provided in [6] where the annuitization occurs when the value of the financial fund is high enough or, in alternative, low enough.
In this paper we perform a detailed mathematical study of the optimal stopping problem associated to an annuitization decision similar to that considered in [6]. In the interest of a rigorous analysis of the optimal stopping boundary we simplify the dynamic of the financial fund by considering a geometric Brownian motion with no jumps. As in [6] we look at the maximisation of future expected cashflows for an individual who joins the fund and has the opportunity to purchase an annuity on a time horizon [0, T ]. Time 0 is the time when the individual joins the fund and time T is the time by which the individual reaches the maximal age for an annuity purchase. The present value of future expected cashflows, evaluated at the optimum, gives us the so-called value function V .
Notice that, a closer inspection of the problem formulation in (2.4) below, shows that at time T the fund is converted into an annuity (the same occurs in [6]). This means that the individual will eventually purchase the annuity at time T , but she also has an option to buy it earlier. One could think of this feature as part of the fund's contract specifications or as commitment of the investor at time 0. It is however important to remark that the methods developed in this paper apply also to the case T = +∞, up to adding obvious integrability assumptions on the processes involved.
One of the key features of the model presented here is the use of a rather general time-dependent, deterministic mortality force. This is a realistic assumption commonly made in the actuarial profession. As in [10], we consider two different mortality forces: a subjective one, used by the individual to weigh the future cashflows (denoted µ S ), and an objective one, used by the insurance company to price the annuity (denoted µ O ). The interplay between these two different mortality forces contributes to some key qualitative aspects of the optimal annuitization decision (see Section 5 for more details). Interestingly the generic time-dependent structure of the mortality force constitutes also the major technical challenge in the mathematical study of the problem.
On the one hand, standard optimal stopping results ensure that the time-wealth plane splits into a continuation region C, where the option to wait has strictly positive value, and a stopping region S, where the annuity should be immediately purchased. Denoting by (X t ) t≥0 the process that represents the fund's value (or equally the individual's retirement wealth), an optimal stopping rule is given by stopping at the first time the two-dimensional process (t, X t ) t≥0 enters the set S. Moreover, under some mild technical assumptions these two sets are split by an optimal boundary (free boundary, in the language of PDE), which only depends on time, i.e. t → b(t).
On the other hand, technical difficulties arise when trying to infer properties of the boundary b. In fact, due to the generic time dependence of the mortality force, it is not possible to establish any monotonicity of the mapping t → b(t). It is well known in optimal stopping and free boundary theory that monotonicity of b is the key to a rigorous study of the regularity of the boundary (e.g. continuity) and of the value function (e.g. continuous differentiability). The interested reader may consult [12], for a collection of examples, and the introduction in [3], for a deeper discussion.
We overcome this major technical hurdle by proving that the optimal boundary is in fact a locally Lipschitz continuous function of time. In order to achieve this goal we rely only on probabilistic methods which are new and specifically designed to tackle our problem. This approach draws from similar ideas in [3], but we emphasise that our problem falls outside the class of problems addressed in that paper (see the discussion prior to Theorem 4.6 below).
Once Lipschitz regularity is proven we then obtain also that the value function V is continuously differentiable in t and x, at all points of the (t, x)-plane and in particular across the boundary of C. This is a stronger result than the more usual smooth-fit condition, which states that z → V x (t, z) is continuous across the optimal boundary. Finally, we find non-linear integral equations that characterize uniquely the free boundary and the value function.
The analysis in this paper is completed by solving numerically the integral equation for some specific examples and studying their sensitivity to variations in model's parameters. It is important to remark that the optimal boundary turns out to be non monotonic in some of our examples, under natural assumptions on the parameters. This fact shows that the new approach developed in this paper is indeed really necessary to study the annuitization problem.
In summary our contribution is at least twofold. On the one hand, we add to the literature concerning annuitization problems, in the all-or-nothing framework, by addressing models with time-dependent mortality force. As we have discussed above, and to the best of our knowledge, such models were only considered in [10] (which produces only deterministic optimal strategies) and in [6] (mostly in a numerical way). We provide a rigorous theoretical analysis of the optimal annuitization strategy, in terms of the optimal boundary b. Our study also reveals behaviours not captured by [6] as, e.g., lack of monotonicity of b. The latter may reflect the change over time in the investor's priorities, due to (deterministic) variations in the mortality force. On the other hand, it is rather remarkable that we started by considering an applied problem, with a somewhat canonical and seemingly innocuous formulation, but we soon realised that its rigorous analysis is far from being trivial. Therefore we developed methods which are new in the probabilistic literature on optimal stopping and of independent interest.
The rest of the paper is organized as follows. In Section 2 we introduce the financial and actuarial assumptions and then the optimal annuitization problem. In Section 3 we provide some continuity properties of the value function and useful probabilistic bounds on its gradient. In Section 4 we illustrate sufficient conditions under which the shape of the continuation and stopping regions can be established, and we study the regularity of the optimal boundary. Moreover, we find non-linear integral equations that characterise uniquely the free boundary and the value function. In Section 5 we present some numerical examples to illustrate the range of applicability of our assumptions. In Section 6 we present some final remarks and extensions.

Problem formulation
In our model we consider an insurance company and an individual whose age η ≥ 0 is fixed at time 0 in the optimisation problem. At time t ≥ 0 the individual uses a subjective mortality force function µ S : [0 + ∞) → [0, +∞) to compute her self-assessed life expectancy z p S η+t , i.e. the subjective probability to survive z years. This may be expressed in terms of µ S as The insurance company instead relies on a so-called objective survival probability function z p O η+t . This is computed as in (2.1) but replacing µ S with the objective mortality force function µ O : [0 + ∞) → [0, +∞). The latter is public information provided by a demographic analysis of the population. The different survival probability functions adopted by the insurer and the individual account for the imperfect information available to the insurer on the individual's risk profile.
In this study we are interested in the portion of the individual's wealth allocated for an annuity purchase. Such wealth is invested in a financial fund which eventually will be converted into an annuity. The dynamic of the wealth invested in the fund 1 is modelled by a stochastic process (X t ) t≥0 on a filtered probability space (Ω, F, (F t ) t≥0 , P). The filtration is generated by a Brownian motion (B t ) t≥0 and it is augmented with P-null sets. For t > 0 the fund's value evolves according to where θ is the average continuous return of the financial investment, α is the constant dividend rate and σ > 0 is the volatility coefficient. Notice that X x s = X t,x t+s in law where X t,x denotes the process started at time t from the initial value x.
The insurance company uses the objective survival probabilities to price annuities. In particular, according to standard actuarial theory, the value at time t > 0 of a life annuity that pays at a rate of one monetary unit per year (purchased by the individual aged η + t) is given by Here ρ is a constant discount rate for the future cash flows. As noticed in [6], insurance companies typically have maximum age limit for the purchase of an annuity. We denote the latter by η + T so that T is the remaining time for the individual before a compulsory annuitization of the fund. In our model the individual has the option to annuitize prior to T : if she decides to annuitize at a time t ∈ [0, T ], with the fund's value equal to X, then the annuity payout rate is constant and it equals where the constant K is either a fixed acquisition fee (K > 0) or a tax incentive (K < 0). The case K = 0 leads to trivial solutions as explained in Remark 3.2 below. The optimization criterion pursued by the individual is the maximization of the present value of future expected cash-flows, via the optimal timing of the annuity purchase. In order to properly define the future expected cash-flows, for t ≥ 0 we denote by Γ t D : (Ω, F) → [0, +∞) the subjective residual life time of the individual of age η + t. We assume that Γ t D is independent of the Brownian motion for all t ≥ 0 and notice that P(Γ t D > z) = z p S η+t . Letting the fund's value at time t ∈ [0, T ] be x > 0, the value function of the optimisation is defined by where the supremum is taken over (F t ) t≥0 -stopping times. Before annuitization, i.e. for s < τ , the individual receives the dividends from the fund at rate α. After annuitization, i.e. for s > τ , she gets the annuity payment at constant rate P η+t+τ per year. In case the individual dies before the time of the annuity purchase, i.e. on the event {Γ t D ≤ τ }, she leaves a bequest equal to her wealth.
Due to the assumed independence between the demographic uncertainty and fund's returns (i.e. Γ D independent of (B t ) t≥0 ), and since the optimisation is over (F t ) t≥0stopping times, the value function can be rewritten by using Fubini's theorem. For t ∈ [0, T ] and x > 0 it reads In particular we will refer to (2.6) as the so-called "money's worth". Notice that there is no loss in generality in assuming that the individual's discount rate is equal to the interest rate used in the annuity pricing formula. In fact, ifρ were the subjective discount rate, then the differenceρ − ρ may be incorporated in the subjective mortality force ( [10], see the comment following equation (6)).
Let L be the second order differential operator associated to the diffusion (2.2), i.e.
Assuming for a moment that V is regular enough, by applying the dynamic programming principle and Itô's formula, we expect that the value function should satisfy the following variational inequality In the rest of the paper we will show that (2.7) holds in the a.e. sense with . Moreover we will study the geometry of the set where V = G, i.e. the so-called stopping region.

Properties of the value function
In this section we provide some continuity properties of the value function and useful probabilistic bounds on its gradient. From now on we denote {t < T } : To study the optimization problem (2.5) we find it convenient to introduce the function which may be financially understood as the value of the option to delay the annuity purchase. We can easily compute An application of Itô's formula gives and therefore it is straightforward to verify (see (2.5)) that Notice that (3.4) includes a deterministic discount rate which is not time homogeneous. Optimal stopping problems of this kind are relatively rare in the literature. They feature technical difficulties which are more conveniently handled by considering a discounted version of the problem. Hence we introduce Since the problem for w is equivalent to the one for v and V , from now on we focus on the analysis of (3.5).
thanks to well known properties of X and to Assumption 3.1.
As usual in optimal stopping theory, we let be respectively the so-called continuation and stopping regions. We also denote by ∂C the boundary of the set C and we introduce the first entry time of (t, X) into S, i.e.
is continuous as well. It follows that w is lower semi-continuous and therefore C is open and S is closed. Moreover, the finiteness of w and standard optimal stopping results (see [12, Cor. 2.9, Sec. 2]) guarantee that (3.8) is optimal for w(t, x).
For future frequent use we introduce here a new probability measure P defined by its Radon-Nikodym derivative and notice that It is well known that P and P are equivalent on F t for all t ∈ [0, T ].
Remark 3.2. In case K = 0 in (2.3), the change of measure above reduces (3.5) to The latter is equivalent to the deterministic problem of maximising the function As a result the optimal annuitization time only depends on t (as it happens in [10]).
The next proposition starts to analyse the regularity of w, and it provides a probabilistic characterization for its gradient which will be crucial for our subsequent analysis of the boundary of C.
and there exists a uniform constant C > 0 such that is linear then it is not difficult to show that for x, y ∈ R + , λ ∈ (0, 1) and for any stopping time τ . Taking the supremum over τ ∈ [0, T − t] the claim follows.

(gradient bounds)
This proof draws from results in Theorem 3.1 of [3] so that we only outline it here for the sake of brevity. First notice that (3.5) falls in the class of optimal stopping problems analyzed in [3] (see eq. (1.1) therein). In our framework the functions h, f and g of [3] become and y = ln(x).
Lipschitz continuity of w follows by direct probabilistic estimates on |w(t, x + ε) − w(t, x)| and |w(t + ε, x) − w(t, x)|, for ε ∈ R. Here we only outline estimates for the x variable and refer the interested reader to [3] for the full proof. Fix (t, x) ∈ [0, T ] × R + and pick ε > 0. Let τ * = τ * (t, x) be optimal in w(t, x), hence admissible and sub-optimal in w(t, x + ε), so that we have where for the last equality we used (3.10). For the upper bound we repeat the above argument with τ ε := τ * (t, x + ε) optimal for w(t, x + ε) and find The latter two estimates imply that w(t, ·) is Lipschitz with a constant which is locally uniform in t. Since the choice of (t, x) was arbitrary, the Lipschitz property implies that for each t ∈ [0, T ) the function w(t, ·) admits derivative at a.e. point in R + . A slightly more involved argument leads to the conclusion that for each x ∈ R the function w(·, x) is continuous in [0, T ] and Lipschitz locally in [0, T ). Hence we conclude that w ∈ C([0, T ] × R + ), locally Lipschitz and differentiable a.e. in [0, T ] × R + .
Finally, to obtain (3.11), we let (t, x) be a point of differentiability of w and notice that same arguments as above also give, for τ * = τ * (t, x), Dividing (3.15) and (3.16) by ε, taking limits as ε → 0 and using that w x (t, x) exists we see that (3.11) holds.
Similar estimates and (3.18) give Continuity of w and standard optimal stopping theory guarantee that, for all t ∈ [0, T ], the process The next technical lemma states some properties of w that will be useful to study the regularity of the boundary ∂C. Its proof is given in Appendix.
It is worth noticing that (iii) does not exclude that there may exists t ∈ (0, T ) such that S ∩ ({t} × R + ) = ∅.

Properties of the optimal boundary
In this section we provide sufficient conditions for the boundary ∂C to be represented by a function of time b. We establish connectedness of the sets C and S with respect to the x variable and finally study Lipschitz continuity of t → b(t). It is worth emphasizing that this study is mathematically challenging because of the lack of monotonicity of the map t → b(t) and falls outside the scope of the existing probabilistic literature on optimal stopping and free boundary problems. In Section 5 we show that the Gompertz-Makeham mortality law (a mainstream model in actuarial science) leads naturally to the set of assumptions that we make below.
An initial insight on the shape of C is obtained by noticing that the set with τ R the first exit time of (t + s, X x s ) from R. For all t ∈ [0, T ] such that g(t) = 0 the boundary ∂R is given by the curve Moreover, for each t ∈ [0, T ], we denote the t-section of R by ( Then if θ ≥ α and K > 0, i.e. K is an acquisition fee, we have that R t = [0, γ(t)) and The latter means that the annuity should not be purchased if the individual's wealth is less or equal than K and the funds value has a positive trend (net of dividend payments).
Motivated by 1 in the remark above we will later assume that γ(·) > 0 on [0, T ].
Now we illustrate sufficient conditions under which the probabilistic representation (3.11) easily provides the shape of the continuation and stopping regions.  In particular Proof. In case (i) w x (t, x) ≥ 0 and this implies (t, x) ∈ C =⇒ (t, x ) ∈ C for all x ≥ x, and the claim follows. A similar argument applies to case (ii).
For each t ∈ [0, T ] we denote S t := {x ∈ R + : (t, x) ∈ S} and C t = R + \ S t . These are the so-called t-sections of the stopping and continuation set, respectively, and clearly S t = [0, b(t)] under (i) and S t = [b(t), +∞) under (ii) of the above proposition.
In the rest of the paper we make the following standing assumption. This will hold in all the results below without explicit mention. In Sections 5 and 6 we discuss its range of applicability and some extensions.  (a) If g(·) > 0 on (0, T ) then for each t ∈ (0, T ) there exists g 0 (t) > 0 such that The next simple lemma will be useful in what follows.
Proof. Recalling (3.2) and using dominated convergence we obtain Next we show that the optimal boundary is locally Lipschitz continuous on [0, T ), hence also bounded on any compact. The idea and method of proof come from Theorem 4.3 in [3], however we cannot directly apply results therein since Condition (D) of [3] is not satisfied in the present setting. Proof. In this proof most of the arguments are symmetric when we consider case (i) and case (ii) of Assumption 4.3.
We start by noticing that, in case (i) of Assumption 4.3, Lemma 4.5 holds and w x > 0 inside C. The free boundary b is the zero-level set of w, so continuity of w implies that for each t ∈ [0, T ), and δ > 0 sufficiently small, there exists b δ (t) such that w(t, b δ (t)) = δ.
By applying the implicit function theorem we get , t ∈ [0, T ).
Next we obtain a bound on b δ , independent of δ, for any bounded interval [t 0 , t 1 ] ⊂ (0, T ). Hence Lipschitz continuity of b will follow from (4.7) and an application of Ascoli-Arzela's theorem (see (4.11) in [3] and subsequent comments therein).
To find such uniform bounds we divide the proof in steps.
Step 1. Let us start by observing that w x (t, b δ (t)) > 0 in case (i) of Assumption 4.3, so that . (4.9) To simplify notation, in what follows we set x δ := b δ (t) and t is fixed. To estimate the numerator in (4.9) we use the bound in (3.12), thus obtaining where τ δ = τ * (t, x δ ), for simplicity.
Plugging (4.10) inside (4.9) we finally obtain . (4.11) A similar estimate but with the denominator replaced by −w x (t, x δ ) can be obtained, up to obvious changes, in the setting of (ii) in Assumption 4.3, i.e.
At this point, in order to make (4.11) and (4.12) uniform in δ, we must look at their limits as δ → 0. For this we need to consider separately case (i) and case (ii) of Assumption 4.3.
Step 2 -Case (i). Here S t = [0, b(t)] and, since b ≤ γ and γ ∈ C(0, T ), then b is locally bounded. Notice that, under P defined in (3.9), the process B s = B s − σs, s ≥ 0 is a Brownian motion, and the individual's wealth (2.2) started from x δ may be written as dX x δ s = (θ − α + σ 2 )X x δ s ds + σX x δ s d B s . Then, for any s ∈ [0, T − t] we have and thus E τ δ ≥ Eτ δ . Substituting this estimate in the numerator of (4.11) allows us to write . (4.14) Recalling Assumption 3.1, (3.11) and (4.5) it is now clear that there exists a constant c > 0 independent of (t, x δ ) such that Hence plugging the latter into (4.14) we conclude that with c > 0 a suitable constant and where we used x δ ≤ γ(t) as δ → 0. Since b(t) ≤ γ(t) for all t ∈ [0, T ], then the uniform bound in (4.16) implies that b is locally Lipschitz as claimed.
Step 2 -Case (ii). Here S t = [b(t), +∞). The analysis in this part is more involved than in the previous case. An argument similar to the one in (4.13) gives E τ δ ≥ E τ δ which unfortunately does not help with the estimate in (4.12). So we need to proceed in a different way.
From (4.12) we see that, in order to bound |b δ (t)|, we must bound E τ δ /|w x (t, x δ )| and E τ δ /|w x (t, x δ )| (recall that w x ≤ 0). The former can be bounded easily by using (3.11) and (4.6), since The other term requires more work because the expectation in the numerator is taken with respect to P whereas the one in the denominator uses P. Although we can still use (4.17) to estimate the ratio E τ δ /|w x (t, x δ )| now we end up with Our next task is to find a bound for the ratio E τ δ / E τ δ .
For this part of the proof it is convenient to think of Ω as the canonical space of continuous paths ω = {ω(t), t ≥ 0} and denote θ s the shifting operator θ s ω = {ω(s + t), t ≥ 0}. Here we denote E t, With the new notation we must be careful that for fixed (t, x δ ) and any s ≥ 0 we have P(τ δ > s) = P t,x δ (τ * − t > s) and P(τ δ > s) = P t,x δ (τ * − t > s), because τ * = inf{u ≥ t : (u, X t,x δ u ) ∈ S} under P t,x δ . Our first estimate gives for a suitable uniform constant c 1 > 0. Next we obtain The last term can be further simplified: where we have used that {τ * < T } ⊆ {X τ * ≥ γ(τ * )} under P t,x δ and γ(τ * ) ≥ 2a. The last term in the above expression may be estimated by using iterated conditioning and the strong Markov property: Notice that the strict positivity above uses γ(T ) ≥ 2a. Moreover (4.24) may be verified by using the known joint law of the Brownian motion and its running supremum in the expression below E s,a X T 1 {τ * =T } ≥ E s,a X T 1 {sup s≤u≤T Xu≤2a} .
From (4.21), (4.22) and (4.23) we get with c 3 = a ∧ c 2 . Now we plug the latter into (4.20) and then back in (4.19) and obtain with c = c · c 1 · c 3 > 0 a uniform constant. The above expression and (4.18) may now be substituted into (4.12) and give where c > 0 is a suitable constant.
We recall that, in case (ii) of Assumption 4.3, Lemma 3.4-(iii) implies that for any 0 ≤ t 0 < t 1 < T we can find t 2 ∈ (t 0 , t 1 ) such that b δ (t 2 ) ≤ b(t 2 ) < +∞. The latter and (4.27) allow to apply Gronwall's inequality to obtain that, for any 0 ≤ t 0 < t 1 < T , there exists a constant c t 0 ,t 1 > 0, independent of δ and such that The same bound therefore holds for the boundary b, and it shows that b is bounded on any compact. Moreover (4.27) and (4.28) also give a uniform bound for b δ on [t 0 , t 1 ] as needed.
The next corollary follows by standard PDE arguments used normally in optimal stopping literature, see, e.g. [7,Thm. 2.7.7].
Corollary 4.7. The function w is C 1,2 inside C and it solves the following boundary value problem: It may appear that (4.29) is given in a slightly unusual form, but one should remember that w(t, x) = e − t 0 r(u)du v(t, x) (see (3.5)) so that for v we obtain the more canonical expression Lipschitz continuity of the boundary has important consequences regarding the regularity of the value function w, which we summarize below. Proof. Corollary (4.7) tells us that w t and w x are continuous at all points in the interior of C and of S, and thus it remains to analyze the regularity of w across the boundary. In order to do that it is crucial to observe that since t → b(t) is locally Lipschitz, the law of iterated logarithm implies that τ * is indeed equal to the first time X goes strictly below the boundary, in case (i), or strictly above the boundary, in case (ii). In other words the first entry time to S is equal to the first entry time to its interior part. This is an important fact that can be used to prove that (t, x) → τ * (t, x) is continuous P-a.s., and it is zero at all boundary points (see for example [2, Lemma 5.1 and Proposition 5.2]).
The final claim regarding continuity of w xx follows from (4.29). We know from Corollary 4.7 that w xx is continuous in C. Moreover for any (t 0 , x 0 ) ∈ ∂C ∩ {t < T } we can take limits in (4.29) as (t, x) → (t 0 , x 0 ) with (t, x) ∈ C and use that w(t 0 , which proves our claim. As it will be discussed in Section 5, for the numerical evaluation of the optimal boundaries it is important to find the limit value of b(t) when t → T . This will be analyzed in the next proposition. For future use we introduce the function (recall g 0 as in Lemma 4.4) Proof. Let us first consider case (i) in Assumption 4.3. Here we recall the notation x δ = b δ (t), τ δ = τ * (t, x δ ) used in the proof of Theorem 4.6.
From (4.8) and the upper bound in (3.12) we get (recall that w where in the last inequality we have used E τ δ ≥ E τ δ which follows from (4.13). Employing (4.15) and letting δ → 0 we find where we have also used that b is bounded from above by γ (see (4.2)) and C = C/c. Recalling u 0 (·) and settingb(t) = b(t)+C t 0 u 0 (s)ds, the mapping t →b(t) is increasing. Thus, lim t↑Tb (t) exists and since u 0 is integrable and positive on [0, T ] then the limit lim t↑T b(t) exists as well.
Notice that b(t) ≤ γ(t) for all t ∈ [0, T ) and therefore b(T −) ≤ γ(T ). If γ(T ) = 0 then clearly b(T −) = 0 and (4.34) is trivial. Let us consider the case γ(T ) > 0. For the proof of (4.34) we follow the approach of [1]. Arguing by contradiction we assume b(T −) < γ(T ). Then we can pick a, b such that b(T −) < a < b < γ(T ) and t < T such that (a, b) × [t , T ) ⊂ C. It is convenient here to work with v rather than w (see (3.5)) and to refer to (4.32).
Let  ∈ (a, b). From the above we also deduce that F ϕ (·) is continuous up to T , thus there exists δ > 0 such that F ϕ (s) > 0, for s ∈ [T − δ, T ], and ∈ (a, b). Hence a contradiction.
We can therefore defineb(t) := b(t) − c 0 t 0 1/g 0 (s)ds and the mapping t →b(t) is non-increasing so thatb(T −) := lim t→Tb (t) exists. Since 1/g 0 (t) is integrable and positive on [0, T ] then b(t) := lim t→T b(t) exists too, it is finite and b(T −) ≥ γ(T ). To prove (4.34) we argue by contradiction, assuming b(T −) > γ(T ). Then following analogous arguments to those employed in the proof of case (i) above we reach the desired contradiction.

4.1.
Characterisation of the free boundary and of the vale function. In the next theorem we will find non-linear integral equations that characterise uniquely the free boundary and the value function. Here we notice that all regularity properties of w obviously transfer to V of (2.5), due to (3.1) and (3.5). In particular we notice that V ∈ C 1 ([0, T ) × R + ) and V xx ∈ L ∞ loc ([0, T ) × R + ), with the only discontinuity of V xx occurring across ∂C. It is important to remark that Corollary 4.7 and the remaining properties of w studied above imply that indeed V solves (2.7) in the almost everywhere sense (more precisely at all points (t, x) / ∈ ∂C).
Theorem 4.11. For all (t, x) ∈ [0, T ] × R + , the value function (2.5) has the following representation In case (i) of Assumption 4.3 (resp. in case (ii)), the optimal boundary b is the unique continuous solution, smaller (resp. larger) than γ, of the following non-linear integral equation: for all t ∈ [0, T ] Proof. Here we only show how to obtain (4.38). Then inserting x = b(t) in (4.38) and using that V (t, b(t)) = G(t, b(t)), we get (4.39). We omit the proof of uniqueness which is standard in modern optimal stopping literature and it dates back to [11] (for more examples see [12]).
Let V (n) n≥0 be a sequence with V (n) ∈ C ∞ ([0, T ) × R + ) such that (see [5,Sec. 7.2]) as n ↑ ∞, uniformly on any compact set, and Let (K m ) m≥0 be an increasing sequence of compact sets converging to [0, T ] × R + and define . Then an application of Itô calculus gives We want to let n ↑ ∞ and use (4.40) and (4.41), upon noticing that (t + s, X s ) lies in a compact for s ≤ τ m , and that its law is absolutely continuous with respect to the Lebesgue measure on [0, T ] × R + . From dominated convergence and (4.38), we thus obtain Therefore, using (2.7) (or equivalently Corollary 4.7) we also find Finally we take m ↑ ∞, use that τ m ↑ (T − t) and V (T, x) = G(T, x), and apply dominated convergence to obtain (4.38).

Numerical findings
Here we apply the results obtained in the previous sections to some situations of practical interest. A standard choice to model the force of mortality is the so-called Gompertz-Makeham law, which corresponds to where A, B and C are real-valued and are estimated by statistical data of the population. For simplicity, here we assume A = 0.00055845, B = 0.000025670, C = 1.1011 as in [6] 2 . Time is measured in years and we consider two different scenarios: (a): f (t) ≡ f > 0 (see (2.6)) and µ S (·) = µ(·); (b): µ O (·) = µ(·) and µ S (·) = (1 +μ)µ O (·) withμ ∈ (−1, +∞) In the first scenario the money's worth function (2.6) is constant. If the individual believes she is healthier than the average in the population, then µ S ( · ) < µ O ( · ) and therefore f > 1. Conversely, for an individual who is pessimistic about her health µ S ( · ) > µ O ( · ) and therefore f < 1. It is important to notice that the the function g is monotonic increasing (decreasing) if f is a constant smaller (greater) than 1. The second scenario, uses the so-called proportional hazard rate transformation introduced in actuarial science by Wang ([16], see also [10]). Ifμ < 0 (resp.μ > 0), the individual considers herself healthier (resp. unhealthier) than the average. The limit caseμ → −1, is not relevant in practice as it corresponds to an individual whose lifeexpectancy is infinite. Similarly, the caseμ → +∞ is also irrelevant in practice as it corresponds to an individual who believes she is about to die. An important difference between scenarios (a) and (b) above is that, in the latter, the money's worth f varies over time. In particular, ifμ < 0 (resp.μ > 0) then f (t) > 1 (resp. f (t) < 1), for all t ∈ [0, T ].
We notice that in all our numerical experiments the function ( · ) of (3.3) is positive on [0, T ], so that the sign of γ in (4.2) only depends on that of K and g. For the reader's convenience we also recall the standard numerical algorithm to compute (4.39). We take a equally-spaced partition 0 = t 0 < t 1 < . . . < t n−1 < t n = T with h := t i+1 −t i . Starting from b(T ) = γ(T ), for i = 1, 2, . . . n we solve Notice that the above formula is intended for S t = [0, b(t)]. To deal with S t = [b(t), +∞) we must change the indicator variable in the last expression in the obvious way.
Let T = 30, η = 50, θ = 4.5%, α = 3.5% σ = 1%, ρ = 4%. In Figure 1 the function g in (3.3) is computed for different values of the constant f . As noticed in [6], it is reasonable to expect that the value of f is close to 1. Notice that if f is high enough (f = 1.2) then g remains always negative even if θ > ρ. We observe that g varies slowly and in most cases it does not change sign, hence meeting the requirement in Assumption 4.3. However, if f changes its sign (at most once, since g is monotonic) we can still apply our methods as described in fuller details in Section 6. Figure 2 shows the optimal annuitization regions and boundaries examining two of the cases considered in Figure 1, where g is either negative (f = 1.2) or positive (f = 0.8) on [0, T ]. We note that in the former case, if K ≤ 0 then an immediate annuitization is optimal for all (t, x) ∈ [0, T ] × R + (see Remark 4.1). In the presence of a fixed acquisition fee K > 0, instead, individuals annuitize as soon as the fund's value exceeds the boundary b(·) (left plot in Fig. 2). Remarkably, we observe a non-monotonic optimal boundary. On the other hand, in the case f = 0.8, if K ≥ 0 then the annuity is never purchased (Remark 4.1). Instead, in presence of a fixed tax incentive K < 0, the annuity is purchased as soon as the fund's value falls below the boundary b(·) (right plot in Fig. 2).
In Figure 3 we look at scenario (b) and the function g is plotted for different values of the constantμ in cases θ < ρ (left plot) and θ > ρ (right plot). We note that, in a right neighborhood of zero g has the same sign of θ − ρ. For most parameter choices, either g does not change sign or it changes it once (we refer to Section 6 for a theoretical discussion). Notice however that the change in sign occurs for t ≈ 30 years, hence Assumption 4.3 is very reasonable.
Optimal annuitization regions and their boundaries are presented in Figure 4. Once again we notice the non-monotonic behaviour of the boundary (left plot). In the case g < 0, an accurate numerical solution of the integral equation for b needs a very fine partition of the interval [0, T ], which results in long computational times. We believe this is due to the steep gradient of b near T and to its lack of monotonicity. To simplify our analysis (which is intended for illustrative purpose) we consider shorter time horizons for the investor than in scenario (a), i.e. T = 5.5 years.
In Figure 5 we also study sensitivity of the annuitization boundary with respect toμ. Recall that asμ increases the individual considers herself increasingly unhealthier than  the average population. As a results, we observe that the boundary b(·) is pushed downward and the continuation region expands. This is intuitively clear because annuities are financially less appealing for individuals with shorter (subjective) life expectancy.

Final remarks and extensions
As discussed above, our main technical assumption (Assumption 4.3) is supported by numerical experiments on the Gompertz-Makeham mortality law. The latter is widely used in the actuarial profession, hence it is a natural choice from the modelling point of view. We notice that allowing the function g to vanish at T prevents us from using results in [13] for the Lipschitz regularity of the optimal boundary. Such condition may be important for applications as it allows us to extend our results to cover examples where g is monotonic on [0, T ] and it changes its sign once. The latter are observed in Figures 1 and 3 (although the change of sign occurs only on rather long time horizons, e.g. T > 20 years). On the other hand it appears that the function (·) (see (3.3)) is positive in all of our numerical experiments.  Figure 5. Scenario (b). Sensitivity of the optimal boundary with respect toμ for θ > ρ and K = −2.
Here we explain how our results can cover extensions to the case of g(·) changing its sign once. We shall consider separately the case of K < 0 and K > 0. From now on we assume that g is monotonic and there exists t 0 ∈ (0, T ) such that g(t 0 ) = 0. We also assume that (t) > 0 for t ∈ [0, T ] and recall R and γ from (4.1) and (4.2).
1. (g(·) decreasing). In this setting we have γ(t) > 0 for t ∈ [0, t 0 ) with γ(t) ↑ +∞ as t ↑ t 0 . Moreover R lies above the curve γ on [0, t 0 ). For t ∈ [t 0 , T ] we have R = ∅ and therefore S ∩ {t ≥ t 0 } = [t 0 , T ] × R + (see Remark 4.1). This implies that t 0 is an effective time horizon for our optimization problem (2.5) since it is optimal to immediately stop for any later time. From a mathematical point of view this means that we can equivalently study (2.5) with T replaced by t 0 . On the effective time horizon [0, t 0 ] part (i) of Assumption 4.3 holds and we can repeat the analysis carried out in Sections 3 and 4.
In summary, S t = [0, b(t)] for all t ∈ [0, T ] and most of the analysis in Section 4 carries over to this setting. However, it should be noted that methods used in Theorem 4.6 only allow to establish Lipschitz continuity of b in [t 0 , T ]. A complete study of the boundary in [0, t 0 ] requires new methods and we leave it for future work.
1. (g(·) decreasing). Here R∩{t ≤ t 0 } = [0, t 0 ]×R + , so that C ∩{t ≤ t 0 } = [0, t 0 ]×R + and it is optimal to delay the annuity purchase at least until t 0 , regardless of the dynamics of the fund's value. On (t 0 , T ] instead we find that γ(·) > 0 with γ(t) ↑ +∞ as t ↓ t 0 and R lies below the curve γ. From a mathematical point of view this means that we only need to study our problem (2.5) on the restricted time horizon (t 0 , T ] where our Assumption 4.3 holds.
This case is much more challenging and we could not cover it with methods developed so far. We leave it for future research but nevertheless we would like to make an observation to highlight a key difficulty.
Take t ∈ [0, t 0 ). The martingale property in (3.21) allows us to rewrite problem (3.4) as follows where v(t 0 , x) may be explicitly calculated. In fact, from (3.3) we get So it is clear that v(t 0 , x) = c 1 +c 2 x with c 1 and c 2 positive constants that depend on t 0 . Due to the geometry of R we would expect that the stopping region lies somewhere above the curve γ for t ∈ [0, t 0 ). However if we now compute v x as in Proposition 3.3 it turns out that v x (t, x) = E τ * 0 e − s 0 r(t+u)du g(t + s)e (θ−α)s ds + e − τ * 0 r(t+u)du+(θ−α)(t 0 −t) c 2 1 {τ * =t 0 −t} .
Since g(·) < 0 on [0, t 0 ) and c 2 > 0 it is no longer obvious that v x is negative on [0, t 0 ) × R + . This would have been a sufficient condition to guarantee that S t and C t are connected for all t ∈ [0, t 0 ). Noticing that H(t, x) and v(t 0 , x) are linear in x, the asymptotic behaviour of v(t, x)/x as x → ∞ and convexity of v(t, ·) (see Proposition 3.3) suggest that, for a fixed t ∈ [0, t 0 ), we should have S t = [b 1 (t), b 2 (t)] with b 2 (t) possibly infinite. This however leaves several open questions concerning the actual shape of S and the regularity of its boundary. A complete answer to such questions requires the use of different methods and we leave it for future work.
Notice now that if there exists t * < T such that [t * , T ) × R + ⊂ C, then for all s ≥ t * one has w(s, γ 0 (s)) = E We now proceed in two steps.
Step 1. Here we prove that S ∩ ([t, T ) × R + ) = ∅ for all t < T . Assume the contrary, i.e. there exists t * < T such that [t * , T ) × R + ⊂ C. Hence for any t ∈ [t * , T ) given and fixed we have τ * (t, x) = T − t P-a.s. for all x ∈ R + . In particular we take x > γ 0 (t). In order to obtain an upper bound for the value function we use the martingale property of (3.21) and get 0 ≤ w(t, x) =E τ 0 0 e − t+s 0 r(u)du H(t + s, X x s )ds + w(t + τ 0 , X x τ 0 ) where c > 0 is a uniform lower bound for the discount factor.
Step 2. Here we prove that S ∩ ((t 1 , t 2 ) × R + ) = ∅ for all t 1 < t 2 in [0, T ]. Let us argue by contradiction and assume that indeed S ∩ ((t 1 , t 2 ) × R + ) = ∅ for a given couple t 1 < t 2 in [0, T ]. With no loss of generality we can set t 2 := sup{t > t 1 : S ∩ ((t 1 , t) × R + ) = ∅} and we know from step 1 above that t 2 < T . The idea is to show that indeed t 2 = t 1 , hence a contradiction.
Taking limits as x → ∞ (3.22) is easily verified