Stochastic maximum principle with Lagrange multipliers and optimal consumption with Lévy wage

Abstract

We show how a stochastic version of the Lagrange multiplier method can be combined with the stochastic maximum principle for jump diffusions to solve certain constrained stochastic optimal control problems. Two different terminal constraints are considered; one constraint holds in expectation and the other almost surely. As an application of this method, we study the effects of inflation- and wage risk on optimal consumption. To do this, we consider the optimal consumption problem for a budget constrained agent with a Lévy income process and stochastic inflation. The agent must choose a consumption path such that his wealth process satisfies the terminal constraint. We find expressions for the optimal consumption of the agent in the case of CRRA utility, and give an economic interpretation of the adjoint processes.

Appendix: Some results from stochastic analysis

Let $f : \mathbb {R}_+ \times \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}$ and $g: \mathbb {R} \rightarrow \mathbb {R}$ be given, continuous functions. Consider the stochastic optimal control problem

$$\begin{aligned}&\sup _{u \in \mathcal {A}}\quad E\left[ \int _0 ^T f(t,X(t),u(t)) dt + g(X(T))\right] \nonumber \\&\text{ subject } \text{ to } \nonumber \\&dX(t) = b(t,X(t),u(t))dt + \sigma (t,X(t),u(t))dB(t) \nonumber \\&\quad \qquad + \int _{\mathbb {R}} \gamma (t,X(t^-),u(t^-),z) \tilde{N}(dt,dz)\nonumber \\&X(0)=x \end{aligned}$$

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where $\mathcal {U} \subset \mathbb {R}$ is a given set, $b: \mathbb {R}_+ \times \mathbb {R} \times \mathcal {U} \rightarrow \mathbb {R}$, $\sigma : \mathbb {R}_+ \times \mathbb {R} \times \mathcal {U} \rightarrow \mathbb {R}$ and $\gamma : \mathbb {R}_+ \times \mathbb {R} \times \mathcal {U} \times \mathbb {R} \rightarrow \mathbb {R}$. The control $u(t) = u(t,\omega ): \mathbb {R}_+ \times \Omega \rightarrow \mathcal {U}$ is admissible, denoted $u \in \mathcal {A}$, if the dynamics of X has a unique, strong solution for all $x \in \mathbb {R}$ and $E^x[\int _0 ^T f(t,X(t),u(t)) dt + g(X(T))] < \infty$.

In the following theorem, the function $H : [0,T] \times \mathbb {R} \times \mathcal {U} \times \mathbb {R} \times \mathbb {R} \times \mathcal {R} \rightarrow \mathbb {R}$ is the Hamiltonian function, defined by

$$\begin{aligned} H(t,x,u,p,q,r) = f(t,x,u) + b(t,x,u)p + \sigma (t,x,u)q + \int _{\mathbb {R}} \gamma (t,x,u,z)\nu (dz) \end{aligned}$$

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where $\mathcal {R}$ is the set of functions such that the integral above converges.

Theorem 3

(A sufficient maximum principle for stochastic optimal control with jumps, Framstad et al. [4]) Let $\hat{u}$ be an admissible control, i.e. $\hat{u} \in \mathcal {A}$, with corresponding state process $\hat{Y} = Y^{\hat{u}}$ and suppose there exists a solution $(\hat{p}(t), \hat{q}(t), \hat{r}(t,z))$ of the corresponding adjoint equation

$$\begin{aligned} dp(t)= & {} - \nabla _y H(t, Y(t), u(t), p(t), q(t), r(t,\cdot ))dt \nonumber \\&+\,\, q(t)dB(t) + \int _{\mathbb {R}} r(t^-, z)\tilde{N}(dt,dz),\quad t<T, \nonumber \\ p(T)= & {} \nabla g(Y(T)) \end{aligned}$$

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satisfying

$$\begin{aligned} E \bigg [ \int _0^T(\hat{Y}(t)- Y^{u}(t))^2 \big \{ \hat{q}^2(t) + \int _{\mathbb {R}} r^2(t,z)\nu (dz) \big \} dt \bigg ] < \infty \end{aligned}$$

and

$$\begin{aligned}&E \bigg [ \hat{p}^2(t) \bigg \{ \sigma ^2(t, Y^u(t),u(t)) + \int _{\mathbb {R}} \gamma ^2(t,Y^u(t),u(t),z) \nu (dz) \bigg \} dt \bigg ] \\&\quad < \infty \quad \text{ for } \text{ all } \,\, u \in \mathcal {A}. \end{aligned}$$

Moreover, suppose that

$$\begin{aligned} H(t,\hat{Y}(t), \hat{u}(t), \hat{p}(t), \hat{q}(t), \hat{r}(t,\cdot )) = \sup _{v \in \mathcal {U}} H(t,\hat{Y}(t), v, \hat{p}(t), \hat{q}(t), \hat{r}(t,\cdot )) \end{aligned}$$

for all t, that g is a concave function of y and that

$$\begin{aligned} \hat{H}(y) := \max _{v \in \mathcal {U}} H(t,y,v,\hat{p}(t), \hat{q}(t), \hat{r}(t,\cdot )) \end{aligned}$$

exists and is a concave function of y, for all $t \in [0,T]$. Then, $\hat{u}$ is an optimal control.

Proof

See Framstad et al. [4].

Lemma 2

Consider the stochastic differential equation,

$$\begin{aligned} dX(t)= & {} \left( \mu (t) + \alpha X(t) \right) dt + \beta X(t)dB(t) \\ X(a)= & {} x_a\nonumber \end{aligned}$$

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where $\mu (t)$ is an adapted stochastic process and $\alpha , \beta \in \mathbb {R}$.

Then,

$$\begin{aligned} X(t)= & {} \exp \left( \alpha t - \frac{1}{2}\beta ^2 t + \beta B(t)\right) \bigg ( x_a\exp \left\{ -\left( \alpha a - \frac{1}{2}\beta ^2 a + \beta B(a)\right) \right\} \nonumber \\&+ \int _a^t \exp \left( -\alpha s + \frac{1}{2} \beta ^2 s - \beta B(s)\right) \mu (s)ds \bigg ). \end{aligned}$$

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Proof

The idea is to get rid of the terms of the SDE involving X(t) by multiplying with the integrating factor

$$\begin{aligned} J(t) :=\exp \left\{ -\bigg (\alpha t + \frac{1}{2}\beta ^2 t - \beta B(t)\bigg )\right\} . \end{aligned}$$

By Itô’s product rule (see Øksendal [11, Exercise 4.3 i]),

$$\begin{aligned} d(X(t)J(t)) = X(t)dJ(t) + J(t)dX(t) + dX(t)dJ(t). \end{aligned}$$

Itô’s formula implies that

$$\begin{aligned} dJ(t)= ( (-\alpha + \beta ^2)dt -\beta dB(t) ) \exp \left( \alpha t + \frac{1}{2}\beta ^2 t - \beta B(t)\right) . \end{aligned}$$

Hence,

$$\begin{aligned} d(X(t)J(t)) = \exp \left( \alpha t + \frac{1}{2}\beta ^2 t - \beta B(t)\right) \mu (t) dt. \end{aligned}$$

So, by integrating from a to t on both sides and multiplying by $\frac{1}{J(t)}$, we find the solution

$$\begin{aligned} X(t)= & {} \exp \left( \alpha t - \frac{1}{2}\beta ^2 t + \beta B(t)\right) \bigg ( x_a\exp \left\{ -\left( \alpha a - \frac{1}{2}\beta ^2 a + \beta B(a)\right) \right\} \\&+ \int _a^t \exp \left( -\alpha s + \frac{1}{2} \beta ^2 s - \beta B(s)\right) \mu (s)ds \bigg ). \end{aligned}$$

This concludes the proof.

Lemma 3

(Solution of linear BSDE, Proposition 1.3, El Karoui et al. [3]) Consider a linear BSDE of the form

$$\begin{aligned} -dY(t)= & {} (\phi (t) + Y(t)\beta (t) + Z(t)\mu (t))dt - Z(t)dB(t) \\ Y(\bar{T})= & {} \xi (\bar{T}) \end{aligned}$$

where $\xi (\bar{T})$ is an $\mathcal {F}_{\bar{T}}$-measurable random variable. This BSDE has a unique solution (Y, Z), where Y is explicitly given by

$$\begin{aligned} Y(t) = E\left[ \xi (\bar{T})\Gamma _{t,\bar{T}} + \int _t^{\bar{T}} \Gamma _{t,s} \phi (s)ds | \mathcal {F}_t\right] \end{aligned}$$

where

$$\begin{aligned} d\Gamma _{t,s}= & {} \Gamma _{t,s}(\beta (s)ds + \mu (s)dB(s)) \\ \Gamma _{t,t}= & {} 1, \Gamma _{t,s} \Gamma _{s,u}= \Gamma _{t,u}\quad \forall t \le s \le u. \end{aligned}$$

Proof

See El Karoui et al. [3].

Finally, we give the proof of Lemma 1.

Proof of Lemma 1   We solve the SDE by multiplying with the integrating factor

$$\begin{aligned} J(t)=\exp \left( [\hat{\pi }+\beta (1-\epsilon )]t+\tilde{\pi }B(t)+\frac{1}{2}\tilde{\pi }^{2}t-\ln (\epsilon )\int _a^t\int _{\mathbb {R}}\tilde{N}_2(du,dz)\right) , \end{aligned}$$

[chosen to get rid of the terms involving W in Eq. (14)]. By the Itô product rule for jump processes (see Øksendal and Sulem [12, Exercise 1.2]),

$$\begin{aligned} d(W(t)J(t))= & {} J(t)\big (\alpha -[\hat{\pi }+\beta (1-\epsilon )]W(t)\big )dt-J(t)\tilde{\pi }W(t)dB(t) \\&+\,\, W(t)J(t)\big (\hat{\pi }+\beta (1-\epsilon )+\tilde{\pi }^{2}\big )dt+W(t)J(t)\tilde{\pi }dB(t) \\&-\,\,\tilde{\pi }^{2}W(t)J(t)dt+\int _{\mathbb {R}}J(t-)z\tilde{N}_1(ds,dz)\\&+\int _{\mathbb {R}}\bigg (-(1-\epsilon )W(t)J(t)\left( \frac{1}{\epsilon }-1\right) +W(t)J(t)\left( \frac{1}{\epsilon }-1\right) \\&-\,\,W(t)J(t)(1-\epsilon )\bigg )\tilde{N}_2(dt,dz)\\= & {} \alpha J(t) dt + \int _{\mathbb {R}}zJ(t-)\tilde{N}_1(dt,dz).\\ \end{aligned}$$

So,

$$\begin{aligned} W(t)J(t)=w_aJ(a)+ \int _{a}^{t}\alpha J(s) ds +\int _{a}^{t}\int _{\mathbb {R}}zJ(s-)\tilde{N}_1(ds,dz), \end{aligned}$$

which gives the solution in Eq. (15).