Asymptotics for fixed transaction costs

Abstract

An investor with constant relative risk aversion trades a safe and several risky assets with constant investment opportunities. For a small fixed transaction cost, levied on each trade regardless of its size, we explicitly determine the leading-order corrections to the frictionless value function and optimal policy.

Appendices

Appendix A: Pointwise estimates

Proposition A.1

There exists K=K(β,μ,r,σ,γ)>0 such that for k=0,1,2 and j=0,1,2,

$$|D^k_{\xi}\partial^j_z w(z,\xi)|\leq K z^{-j-3k/4-\gamma} \quad \textit{for all }(z,\xi) \leftrightarrow(x,y)\in\mathrm {NT}^{\epsilon}. $$

Proof

This follows from tedious but straightforward computations since all the functions and domains involved are known explicitly (cf. [43, Sect. 4.2] for a similar calculation). □

Remark A.2

Proposition A.1 yields the expansion

where the remainder satisfies the bound

$$|{\mathcal{R}}_w(z,\xi)|\leq K \sum_{k=0}^3\epsilon^{k}z^{(1-k)/4-\gamma }\quad \mbox{for all }(z,\xi) \leftrightarrow(x,y)\in\mathrm{NT}^{\epsilon}. $$

In particular, over the same region, we have

We can also expand

with a bound on the remainder of

$$|{\mathcal{R}}_u(z,\xi)|\leq K(\epsilon z^{1/4-\gamma} + \epsilon^2 z^{-\gamma})\quad \mbox{for all }(z,\xi) \leftrightarrow(x,y)\in\mathrm {NT}^{\epsilon}. $$

Proposition A.3

Let ϵ>0 be given and consider $S:=[z_{0}-r_{0},z_{0}+r_{0}]\times \mathbb {R}^{d}\subset\mathrm{K}_{\epsilon}$ for some z ₀>r ₀>0. Then, given any Ψ∈C ²(S) for which D _ξ Ψ has a compact support, there exists K>0, independent of ϵ, such that

$$\|\varPsi\|_{C^2(S)} \leq K $$

and

Proof

This again follows from a tedious but straightforward calculation. □

Appendix B: Proof of Theorem 2.1

In this section, we prove that for each fixed λ, the value function v ^λ is a viscosity solution of the corresponding dynamic programming equation (3.7) on the domain

$${\mathcal{O}}_\lambda= \{ (x,y) \in\mathrm{K_\lambda}\ :\ x+ y \cdot{\mathbf{1}}_d > 2\lambda\}. $$

As observed by Bouchard and Touzi [8], a “weak version” of the dynamic programming principle is sufficient to derive the viscosity property. For the convenience of the reader, we present a direct proof of the weak dynamic programming principle in our specific setting using the techniques of [8]. Then we use it to prove that v ^λ is indeed a viscosity solution of (3.7).

2.1 B.1 Weak dynamic programming principle for v ^λ

Fixing $(x,y) \in {\mathcal{O}}_{\lambda}$ and δ>0, let $B_{\delta }(x,y)\subset \mathbb{R}^{d+1}$ denote the ball of radius δ centered at (x,y) and set

$$K(x,y;\delta)_\lambda:= \{ (x^\prime,y^\prime)\ :\ x+ y \cdot{\mathbf{1}}_d -\delta-\lambda\le x^\prime+ y^\prime\cdot{\mathbf{1}}_d \le x+ y \cdot{\mathbf{1}}_d +\delta\}. $$

Take δ>0 sufficiently small so that $K(x,y;2\delta)_{\lambda}\subset {\mathcal{O}}_{\lambda}$. For any investment–consumption policy ν and initial endowment (x′,y′)∈B _δ/2(x,y), define θ:=θ ^ν as the exit time of the state process (X,Y)^ν,x′,y′ from B _δ/2(x,y). Following the standard convention, our notation does not explicitly show the dependence of θ on ν. It is then clear that

$$(X_{\theta^-},Y_{\theta^-}) \in\overline{B_{\delta/2}(x,y)}\quad {\mbox{and}} \quad (X_{\theta},Y_{\theta}) \in K(x,y,\delta)_\lambda. $$

The following weak version of the DPP is introduced in [8]. Let φ be a smooth and bounded function on K(x,y,2δ) _λ satisfying

$$v^{\lambda} \leq\varphi\quad\mbox{on } K(x,y,2\delta)_\lambda. $$

Then we have

$$ v^\lambda(x,y) \le\sup_{\nu\in\varTheta_\lambda(x,y)}\mathbb{E}\left[ \int_0^\theta e^{-\beta t} U(c_t)\,dt + e^{-\beta\theta} \varphi\big( X_{\theta},Y_{\theta}\big) \right]. $$

(B.1)

(The restriction to bounded test functions φ is possible since by (4.12) v ^λ is bounded on K(x,y;2δ) _λ .) Conversely, let φ be a smooth function bounded on K(x,y,2δ) _λ , satisfying

$$v^{\lambda} \geq\varphi\quad\mbox{on } K(x,y,2\delta)_\lambda. $$

Then we have

$$ v^\lambda(x,y) \ge\sup_{\nu\in\varTheta_\lambda(x,y)}\mathbb{E}\left[ \int_0^\theta e^{-\beta t} U(c_t)\,dt + e^{-\beta\theta} \varphi\big( X_{\theta},Y_{\theta}\big) \right]. $$

(B.2)

For proving (B.1) and (B.2), without loss of generality, let $\varOmega= C_{0}([0,\infty), \mathbb{R}^{d})$ be the space of continuous functions starting at zero, equipped with the Wiener measure $\mathbb{P}$, a standard Brownian motion W, and the completion $({\mathcal{F}}_{t})_{t\geq0}$ of the filtration generated by W. Given a control ν∈Θ _λ (x,y) and the exit time θ:=θ ^ν from above, fix ω∈Ω and define

$$\nu^{\theta,\omega}(\omega',t) := \nu\big(\omega\stackrel {\theta}{\oplus }\omega', t + \theta(\omega)\big)\qquad\forall\omega'\in\varOmega ,\ t\geq0, $$

where

$$(\omega\stackrel{\theta}{\oplus}\omega')_t := \left\{ \begin{array}{ll} \omega_t & \quad\mbox{if }t\in[0,\theta(\omega)), \\ \omega'_{t-\theta(\omega)} + \omega_{\theta(\omega)} & \quad \mbox{if }t\geq\theta(\omega). \end{array} \right. $$

We start with the proof of (B.1). By construction,

$$\nu^{\theta,\omega}\in\varTheta_{\lambda}\Big(X_{\theta(\omega )}^{\nu ,x},Y_{\theta(\omega)}^{\nu,y}\Big); $$

in particular, ν ^θ,ω is a well-defined impulse control. Moreover, note that

$$(X_{\theta(\omega)}^{\nu,x},Y_{\theta(\omega)}^{\nu,y}) \in K(x,y,\delta )_\lambda $$

lies in the set K(x,y,2δ) _λ on which φ dominates v ^λ by definition. Therefore,

$$\begin{aligned} &\left.\mathbb{E}\left[\int_0^{\infty} e^{-\beta t}U(c^{\nu}_t)\,dt \right |{\mathcal{F}}_{\theta} \right](\omega)\\ &\qquad= \int_0^{\theta(\omega)}e^{-\beta t}U\big(c^{\nu }_t(\omega) \big)\,dt + e^{-\beta\theta(\omega)}\int_{\varOmega}\int_0^{\infty} e^{-\beta t}U\big(c_t^{\nu^{\theta,\omega}}(\omega')\big)\,dt \,d\mathbb{P}(\omega ')\\ & \qquad\leq\int_0^{\theta(\omega)}e^{-\beta t}U\big(c^{\nu }_t(\omega) \big)\,dt + e^{-\beta\theta(\omega)}v^{\lambda}\big(X_{\theta (\omega )}^{\nu,x},Y_{\theta(\omega)}^{\nu,y}\big)\\ &\qquad\leq\int_0^{\theta(\omega)}e^{-\beta t}U\big(c^{\nu }_t(\omega) \big)\,dt + e^{-\beta\theta(\omega)}\varphi\big(X_{\theta(\omega )}^{\nu ,x},Y_{\theta(\omega)}^{\nu,y}\big). \end{aligned}$$

As a result, for any ν∈Θ _λ (x,y),

$$\begin{aligned} \mathbb{E}\left[\int_0^\infty e^{-\beta t} U(c^{\nu}_t)\,dt\right] \le \mathbb{E}\left[\int_0^\theta e^{-\beta t} U(c^{\nu}_t)\,dt + e^{-\beta\theta}\varphi(X_{\theta}^{\nu,x},Y_{\theta}^{\nu ,y})\right]. \end{aligned}$$

By taking the supremum over all policies ν we arrive at (B.1).

To prove (B.2), set $\mathbb{V}$ to be the right-hand side of (B.2), that is,

$$\mathbb{V}:= \sup_{\nu\in\varTheta_\lambda(x,y)}\mathbb{E}\left[\int_0^\theta e^{-\beta t} U(c^{\nu}_t)\,dt + e^{-\beta\theta}\varphi(X_{\theta}^{\nu,x},Y_{\theta}^{\nu ,y})\right]. $$

For any η>0, we can choose ν ^η∈Θ _λ (x,y) satisfying

$$ \mathbb{V}\le\eta+\mathbb{E}\left[ \int_0^\theta e^{-\beta t} U(c^{\nu^\eta}_t)\,dt + e^{-\beta\theta}\varphi\big(X_{\theta}^{\nu{^\eta},x},Y_{\theta }^{\nu ^{\eta},y}\big) \right]. $$

(B.3)

We have already argued that $(X_{\theta}^{\nu^{\eta},x} ,Y_{\theta}^{\nu ^{\eta},y} )\in K(x,y,\delta)_{\lambda}$. The next step is to construct a countable open cover of K(x,y,δ) _λ . For every point $\zeta= (\tilde{x}, \tilde{y})$ in K(x,y,2δ) _λ , set

$$\begin{aligned} R(\zeta):=\ & R_\eta(\tilde{x}, \tilde{y})\\ =\ &\{ (x^\prime,y^\prime) \in K(x,y,2\delta)_\lambda\ :\ x^\prime>\tilde{x}, \ y^\prime> \tilde{y}, \ \varphi(x^\prime,y^\prime) < \varphi(\tilde{x}, \tilde{y}) + \eta\}. \end{aligned}$$

By monotonicity of the value function,

$$ v^\lambda(\zeta) \le v^\lambda(x^\prime,y^\prime) \quad\forall\ (x^\prime,y^\prime) \in R(\zeta). $$

Also, since φ is smooth, each R(ζ) is open, and

$$K(x,y,\delta)_\lambda\subset\bigcup_{\zeta\in K(x,y,2\delta )_\lambda } R(\zeta). $$

Hence, by the Lindelöf covering lemma [26, Theorem 15], we can extract a countable subcover

$$K(x,y,\delta)_\lambda\subset\bigcup_{n\in \mathbb{N}}\ R(\zeta_n). $$

Now define a mapping ${\mathcal{I}}: K(x,y,\delta)_{\lambda}\to \mathbb{N}$ that assigns to each point one of the neighborhoods in the subcover to which it belongs by

$${\mathcal{I}}(x^\prime,y^\prime):= \min\{n: (x^\prime,y^\prime) \in R(\zeta _n)\} $$

and set

$$\zeta(x^\prime,y^\prime):= \zeta_{{\mathcal{I}}(x^\prime,y^\prime)}. $$

By definition,these constructions imply

$$ \varphi(x^\prime,y^\prime) \le\varphi\big(\zeta(x^\prime ,y^\prime)\big) + \eta\quad \forall\ (x^\prime,y^\prime) \in K(x,y,\delta)_\lambda. $$

(B.4)

As a final step, for each positive integer n, we choose a control ν ⁿ∈Θ _λ (ζ _n ) so that

$$ v^\lambda(\zeta_n) \le \mathbb{E}\left[ \int_0^\infty e^{-\beta t} U(c^{\nu^n}_t)\,dt \right] + \eta. $$

(B.5)

By monotonicity, ν ⁿ∈Θ _λ (x′,y′) for every (x′,y′)∈R(ζ _n ). We now define a composite strategy ν ^∗ that follows the policy η satisfying (B.3) until the corresponding state process $(X,Y)^{\nu ^{\eta },x,y}$ leaves B _δ/2(x,y) at time $\theta= \theta^{\nu ^{\eta }}$. It then switches to the policy ν ⁿ corresponding to the index n that the state process is assigned to by the mapping $\mathcal{I}$, that is,

$$\nu^*(\omega\stackrel{\theta}{\oplus}\omega', t):= \left\{ \begin{array}{ll} \nu^\eta(\omega,t) &\quad{\mbox{if}}\ t \in[0,\theta(\omega)],\\ \nu^{{\mathcal{N}}(\omega)}(\omega', t-\theta(\omega))&\quad{\mbox{if}}\ t >\theta(\omega),\ \end{array} \right. $$

with ${\mathcal{N}}(\omega)= {\mathcal{I}}(X_{\theta(\omega)}^{\nu^{\eta },x},Y_{\theta (\omega)}^{\nu^{\eta},y})$. This construction ensures that we have ν ^∗∈Θ _λ (x,y). Hence, it follows from the definitions of the value function and ν ^∗, inequalities (B.5), v ^λ≥φ (which holds on K(x,y,2δ) _λ by the definition of φ), (B.4), and (B.3) that

$$\begin{aligned} v^\lambda(x,y) &\geq \mathbb{E}\left[\int_0^{\infty}e^{-\beta t}U(c_t^{\nu ^*})\,dt \right]\\ &= \mathbb{E}\left[ \int_0^\theta e^{-\beta t} U(c^\eta_t)\,dt + e^{-\beta\theta} \int_0^{\infty}e^{-\beta t}U(c_t^{{\mathcal{N}}})\,dt \right ]\\ &\ge \mathbb{E}\left[ \int_0^\theta e^{-\beta t} U(c^\eta_t)\,dt + e^{-\beta\theta} \bigg(\varphi\Big(\zeta\big(X_{\theta}^{\nu ^{\eta },x},Y_{\theta}^{\nu^{\eta},y}\big)\Big) - \eta\bigg)\right]\\ &\ge \mathbb{E}\left[ \int_0^\theta e^{-\beta t} U(c^\eta_t)\,dt + e^{-\beta\theta} \Big(\varphi\big(X_{\theta}^{\nu^{\eta },x},Y_{\theta }^{\nu^{\eta},y}\big) -2 \eta\Big)\right]\\ &\ge \mathbb{V}- 3 \eta. \end{aligned}$$

Since η was arbitrary, this establishes (B.2), thereby completing the proof.

2.2 B.2 v ^λ Is a viscosity solution of (3.7)

We first state and prove some facts about the intervention operator M from (3.8), which are needed in the subsequent proofs. Similar observations appear in [37] for the case of proportional and fixed transaction costs. The modifications below are required to extend them to the case of pure fixed costs, for which the set of attainable portfolios at a fixed wealth level is no longer compact.

Throughout, $\underline{\psi}$ and $\overline{\psi}$ will denote the lower- and upper-semicontinuous envelopes of a locally bounded function ψ, respectively.

Lemma B.1

Suppose that $\varphi:\mathrm{K}_{\lambda}\to \mathbb{R}$ satisfies sup _z∈K ∥φ(z,⋅)∥_∞<∞ for every nonempty compact set $K\subset \mathbb{R}_{+}$.

• (i) If φ is lower-semicontinuous, then M φ is lower-semicontinuous. In particular, if φ≥M φ, then $\underline{\varphi} \geq\mathbf{M}\underline {\varphi}$.

• (ii) Let φ∈C ¹(K _λ ). If (z,ξ)↦D _ξ φ(z,ξ) is compactly supported on $C\times \mathbb{R}^{d}$ for any compact set C⊂R ₊, then M φ is upper-semicontinuous.

Proof

(i) Assume to the contrary that there exist ζ ₀=(z ₀,ξ ₀)∈K _λ , a constant η>0, and a sequence K _λ ∋ζ _n =(z _n ,ξ _n )→ζ ₀ for which

$$\mathbf{M}\varphi(\zeta_0) > \liminf_{n\to\infty}\mathbf {M}\varphi(\zeta _n) + 2\eta. $$

Choose $\zeta_{0}^{*} = (z_{0}-\lambda, \hat{\xi}_{0})$ such that $\varphi (\zeta _{0}^{*}) + \eta/2 \geq\mathbf{M}\varphi(\zeta_{0})$ and $\zeta_{n}^{*} = (z_{n}-\lambda, \hat{\xi}_{n})$ with $\varphi(\zeta_{n}^{*})+1/n \geq \mathbf {M}\varphi(\zeta_{n})$. Then, up to choosing a subsequence,

$$\varphi(\zeta_0^*) \geq\lim_{n\to\infty}\varphi(\zeta_n^*) + \eta. $$

By the lower-semicontinuity of φ, there exist an open neighborhood O of $\xi_{0}^{*}$ and an integer N>0 such that for all n≥N and all ζ∈O,

$$ \varphi(\zeta)\geq\varphi(\zeta_n^*) + \eta. $$

(B.6)

Observe that since z _n →z ₀ as n→∞, we have

$$ A_n:=\{z_n-\lambda\}\times \mathbb{R}\cap O \neq\emptyset. $$

(B.7)

Combining (B.7) and (B.6) yields

$$\varphi(\zeta_n^*) + \frac{1}{n} \geq\mathbf{M}\varphi(\zeta_n) \geq \sup_{\zeta'\in A_n}\varphi(\zeta')\geq\varphi(\zeta_n^*)+ \eta, $$

which is a contradiction for n large enough. Finally, observe that if φ is lower-semicontinuous and φ≥M φ, then $\underline{\varphi} \geq\underline{\mathbf{M}\varphi} = \mathbf {M}\underline{\varphi}$ by the previous discussion.

(ii) This follows similarly as in the proof of [37, Lemma 3.2(i)] because the requirements we place on the gradient D _ξ φ ensure the existence of optimizers and accumulation points also in our setting. □

We are now ready to tackle the proof of Theorem 2.1, which we split into two lemmas.

Lemma B.2

The value function v ^λ is a viscosity supersolution of the dynamic programming equation (3.7) on ${\mathcal{O}}_{\lambda}$.

Proof

Let $(x_{0},y_{0}){\,\in\,}{\mathcal{O}}_{\lambda}$, and let φ be a smooth and bounded function on K(x ₀,y ₀,δ) _λ satisfying

$$0 = (\underline{v}^{\lambda} - \varphi)(x_0,y_0) = \min\{ (\underline {v}^{\lambda} - \varphi)(x',y'):\ (x',y')\in K(x_0,y_0,\delta )_\lambda \}. $$

Using Lemma B.1 and the inequality $\underline{v}^{\lambda }\geq\varphi$ on K(x ₀,y ₀,δ) _λ , we obtain

$$\varphi(x_0,y_0) = \underline{v}^{\lambda}(x_0,y_0)\geq\mathbf {M}\underline{v}^{\lambda}(x_0,y_0) \geq\mathbf{M}\varphi(x_0,y_0). $$

Therefore, it remains to show that

Assume to the contrary that (βφ−U(c ^∗)+c ^∗ φ _x −ℒφ)(x ₀,y ₀)<0 for some c ^∗>0 and set ϕ(x,y):=φ(x,y)−ϵ(|x−x ₀|⁴+∥y−y ₀∥⁴). Then for ϵ>0 and r>0 small enough, continuity yields

Select a convergent sequence of points $(x_{n},y_{n},v^{\lambda }(x_{n},y_{n}))\to(x_{0},y_{0},\underline{v}^{\lambda}(x_{0},y_{0}))$ and denote by $(X^{n}_{t},Y^{n}_{t}):=(X_{t}^{x_{n}},Y_{t}^{y_{n}})$ the portfolio process starting at (x _n ,y _n ) under the consumption-only strategy c _t ≡c ^∗. Define

$$H^n: = \inf\{t\geq0:\ (X^n_t, Y^n_t) \notin\overline{B}_r(x_0,y_0) \} $$

and note that $\liminf_{n\to\infty} \mathbb{E}[H^{n}] > 0$. Hence, there exists δ>0 with $\mathbb{E}[e^{-\beta H^{n}}] > \delta$ for all n sufficiently large. Itô’s formula gives

By construction of ϕ there exists η>0 with φ≥ϕ+η on $K(x_{0},y_{0},\delta)\backslash\overline {B}_{r}(x_{0},y_{0})$. Hence,

$$\phi(x_n,y_n) \leq \mathbb{E}\bigg[ e^{-\beta H^n}\varphi(X^n_{H^n}, Y^n_{H^n}) + \int_0^{H^n}e^{-\beta s}U(c^*)\,ds \bigg] -\delta\eta. $$

Taking into account (v ^λ−ϕ)(x _n ,y _n )→0, we note that for n large enough,

$$v^{\lambda}(x_n,y_n) \leq \mathbb{E}\bigg[ e^{-\beta H^n}\varphi(X^n_{H^n}, Y^n_{H^n}) + \int_0^{H^n}e^{-\beta s}U(c^*)\,ds \bigg] -\frac{\delta \eta }{2}. $$

This contradicts the weak dynamic programming principle (B.2) for v ^λ, thereby completing the proof. □

The image of an arbitrary smooth function under M is upper-semicontinuous only under additional assumptions (cf. Lemma B.1(ii)). As is customary in the theory of viscosity solutions (see, e.g., Sect. 9 of [10]), the viscosity subsolution property in the following lemma is therefore formulated in terms of the lower-semicontinuous envelope of the DPE.

Lemma B.3

The value function v ^λ is a viscosity subsolution of

Proof

Step 1. Throughout this proof, C>0 denotes a generic constant that may vary from line to line. We argue by contradiction. Let $(x_{0},y_{0})\in {\mathcal{O}}_{\lambda}$, and let φ be a smooth and bounded function on K(x ₀,y ₀,δ) _λ satisfying

$$0 = (\overline{v}^{\lambda} - \varphi)(x_0,y_0) = \max\{(\overline {v}^{\lambda} - \varphi)(x',y'):\ (x',y')\in K(x_0,y_0,\delta )_\lambda \}. $$

Suppose that for some η>0, we have

By continuity, there is a small rectangular neighborhood

$$N = N(x_0,y_0,\rho) := \Big\{ (x,y)\in \mathbb{R}\times \mathbb{R}^d:\ \max _{i=1,\ldots, d}\big(|x-x_0|, |y^{i} - y^i_0|\big)< \rho\Big\} $$

such that

(B.8)

for all c>0 and (x,y)∈N.

Step 2. Choose a sequence N∋(x _n ,y _n )→(x ₀,y ₀) for which v ^λ(x _n ,y _n ) converges to $\overline{v}^{\lambda}(x_{0},y_{0})$. At each of these points, choose a $\frac{1}{n}$-optimal control ν ⁿ in Θ _λ (x _n ,y _n ). We denote by $(c_{t}^{n})$ and τ ⁿ the consumption process and first impulse time of ν ⁿ, respectively, and write $(X^{n}_{t},Y^{n}_{t}):= (X_{t}^{\nu^{n}, x_{n}}, Y_{t}^{\nu^{n},y_{n}})$ for the corresponding controlled process. Define the stopping times

$$H^{n} := \inf\{t\geq0: (X^n_t,Y^n_t)\notin N \}\wedge1 + \infty1_{\{ \theta^n=\tau^n\}} $$

and

$$\theta^n := H^n\wedge\tau^n. $$

We can further decompose $H^{n} = \underline{H}^{n}\wedge\overline{H}^{n} \wedge1$, where

$$\underline{H}^n:= \inf\{t\geq0:\ (X^n_t,Y^n_t)\in\partial N\cap\{x_0 - \rho\}\times \mathbb{R}^d \} + \infty1_{\{\theta^n=\tau^n\}} $$

and

$$\overline{H}^n:= \inf\{t\geq0:\ (X^n_t,Y^n_t)\in\partial N\cap\{x: x > x_0-\rho\}\times \mathbb{R}^d \}+ \infty1_{\{\theta^n=\tau^n\}}. $$

Then there exists δ>0 such that $\mathbb{E}[\overline{H}^{n}] > \delta$ for all n sufficiently large.

Step 3. Write

$$h(c,x,y) := I(c,x,y) - \sup_{\hat{c} > 0}I(\hat{c},x,y), $$

where

Note that I(c,x,y)<0 for all $c\in \mathbb{R}_{+}$ and (x,y)∈N by (B.8). If we now set c ^∗(x,y)=(U′)⁻¹(φ _x (x,y)), then it follows that

$$h(c,x,y)= I(c,x,y) - I\big( c^*(x,y),x,y\big)\leq0. $$

By smoothness of φ and c ^∗ and compactness of N, there exists a constant L _ρ >0 with |I(c ^∗(x,y),x,y)|≤L _ρ for all (x,y)∈N. On the other hand, there is α>0 such that I(c,x,y)≤−αc for all c>0. This leads to the upper bound

$$ h(c,x,y) \leq(-\alpha c + L_{\rho})\wedge0\qquad\mbox{for all } c>0, (x,y)\in N. $$

(B.9)

Since we only consider times t up to θ ⁿ, we can assume without loss of generality that $c^{n}_{t} = c^{*}(X^{n}_{t},Y^{n}_{t})$ for t∈(θ ⁿ,H ⁿ]. Together with (B.9), we obtain

$$\begin{aligned} \mathbb{E}\bigg[ \int_0^{\theta^n}-e^{-\beta t}h(c_t,X_t,Y_t)\,dt\bigg] =& \mathbb{E}\bigg[\int_0^{H^n}-e^{-\beta t}h(c_t,X_t,Y_t)\,dt\bigg]\\ \geq& C\alpha \mathbb{E}\bigg[\int_0^{H^n} e^{-rt}c_t\,dt\bigg] - L_{\rho} \mathbb{E}[ H^n]. \\ \geq& C\alpha \mathbb{E}\bigg[\int_0^{\underline{H}^n\wedge1} e^{-rt}c_t 1_{\{\theta^n = \underline{H}^n\}}\,dt\bigg] - L_{\rho} \mathbb{E}[H^n], \end{aligned}$$

(B.10)

where the first inequality uses (B.9) to change the discount factor.

Step 4. Set $\zeta^{n}_{t}:= (X^{n}_{t}, Y^{n}_{t})$. Weak dynamic programming in (B.1) implies

$$\begin{aligned} v^{\lambda}(x_n,y_n) \leq& \frac{1}{n} + \mathbb{E}\bigg[\int_0^{\theta^n} e^{-\beta t} U(c^n_t)\,dt + e^{-\beta\theta^n}\varphi(\zeta^n_t) \bigg]\\ \leq& \frac{1}{n} + \varphi(x_n,y_n) + \mathbb{E}\bigg[\int_0^{\theta^n} e^{-\beta t}I(c^n_t,\zeta^n_t)\,dt\bigg]\\ &{}+ \mathbb{E}\big[ e^{-\beta\theta^n} \big(\varphi(\zeta^n_{\theta ^n}) - \varphi(\zeta^n_{\theta^n-})\big)1_{\{\theta^n=\tau^n\}}\big]\\ \leq& \frac{1}{n} + \varphi(x_n,y_n) + \mathbb{E}\bigg[\int_0^{\theta^n} e^{-\beta t}I\big(c_t^*(\zeta^n_t),\zeta^n_t\big)\,dt\bigg] \\ &{} + \mathbb{E}\bigg[\int_0^{\theta^n} e^{-\beta t}h(c_t^n,\zeta ^n_t)\,dt\bigg] - C\eta \mathbb{P}[\theta^n = \tau^n]\\ \leq&\frac{1}{n} + \varphi(x_n,y_n) - CL_{\rho}\eta \mathbb{E}[\theta^n] - C\eta \mathbb{P}[\theta^n = \tau^n] \\ &{}+ \mathbb{E}\bigg[\int_0^{\theta^n} e^{-\beta t}h(c_t^n,\zeta ^n_t)\,dt\bigg]. \end{aligned}$$

Since $v^{\lambda}(x_{n},y_{n}) - \varphi(x_{n},y_{n}) - \frac{1}{n}\to0$ as n→∞ and since the other terms on the right-hand side are negative, they must each vanish as n tends to infinity.

Step 5. We derive a contradiction using that

$$ \lim_{n\to\infty}\max\bigg(\mathbb{E}[\theta ^n], \mathbb{P}[\theta^n = \tau^n], \mathbb{E}\bigg[\int_0^{\theta^n} -e^{-\beta t}h(c_t^n,\zeta^n_t)\,dt\bigg]\bigg) = 0. $$

(B.11)

Observe that since the first two terms vanish, $\mathbb{E}[{H}^{n}]\to0$ and $\mathbb{P}[H^{n} = \theta^{n}]\to1$. Since $\mathbb{E}[\overline{H}^{n}] > \delta$ for all n sufficiently large, we must therefore have $\mathbb{E}[\underline {H}^{n}]\to 0$ and $\mathbb{P}[\underline{H}^{n} = \theta^{n}]\to1$. As a consequence,

$$\mathbb{E}\bigg[\int_0^{\underline{H}^n}e^{-rt}c^n_t 1_{\{\theta^n = \underline {H}^n\}} \,dt\bigg]\to\rho, $$

which follows from the simple observation that for any fixed n, the term inside the expectation represents the amount of discounted consumption needed for cash in the bank account to decrease from x _n to x ₀−ρ. However, by (B.11) and (B.10) we must have

$$0 = \lim_{n\to\infty} \mathbb{E}\bigg[\int_0^{\theta^n}-e^{-\beta t}h(c_t,X_t,Y_t)\,dt\bigg] \geq C\alpha\rho- L_{\rho}\lim_{n\to \infty} \mathbb{E}[ H^n] = C\alpha\rho> 0, $$

which is a contradiction. □