Consumption optimization for recursive utility in a jump-diffusion model

Abstract

In this paper, we consider a market model with prices and consumption following a jump-diffusion dynamics. In this setting, we first characterize the optimal consumption plan for an investor with recursive stochastic differential utility on the basis of his/her own beliefs, then we solve the inverse problem to find what beliefs make a given consumption plan optimal. The problem is viewed in general for a class of homogeneous recursive utility, and later we choose a logarithmic model for the utility aggregator as an explicitly computable example. When beliefs, represented via Girsanov’s theorem, get incorporated into the model, the change of measure gives rise, up to a transformation, to a backward stochastic differential equation whose generator exhibits a quadratic behavior in the Brownian component and a locally Lipschitz one in the jump component, which is solvable on the basis of some recent results.

Appendix: Existence and uniqueness results

In this appendix, we recall the existence and uniqueness result in Antonelli and Mancini (2016) and we show that we can apply it to solve our equation in Theorem 2.

We assume the same notation as in Sect. 2. We take a bounded $\mathscr {F}_T-$measurable r.v. $\xi$ and a predictable process $l_s(y,z,u,q)$ continuous in (y, z, u, q), locally Lipschitz in y, Lipschitz in z and increasing in q and we set $L_s(y,z,u)=l_s(y,z,u,z^2)$.

We assume that $\mathbb {P}-$a.s.

$$\begin{aligned} -a_s-b_1|y|-\frac{b_2}{2}|z|^2-[[-u]]_{b_2}\le L_s(y,z,u)\le a_s+b_1|y|+\frac{b_2}{2}|z|^2+[[-u]]_{b_2} \end{aligned}$$

where $b_1,b_2>0$, $\{a_t\}_{t\in [0,T]}$ is a strictly positive predictable process bounded below by $\frac{b_1}{b_2}$, so that $\displaystyle ||[a||_{\text {int}}:= ||\alpha ||_{\text {int}}:= \text {essup}_{\omega }\int _0^T \alpha _s \hbox {d}s< +\infty$ and for any $b \in {\mathbb {R}}$, we denoted

$$\begin{aligned}{}[[u]]_b:=\int _E\frac{\mathrm e^{b u(e)}-1-b u(e)}{b}\gamma (\hbox {d}e),\qquad \text { for } u\in L^2(\gamma ). \end{aligned}$$

Finally we suppose that

$$\begin{aligned} L_s(y,z,u)-L_s(y,z,u^{\prime })\le \int _E\tau ^{y,z,u,u^{\prime }}_s(e)(u(e)-u^{\prime }(e))\gamma (\hbox {d}e), \end{aligned}$$

where $\tau ^{y,z,u,u^{\prime }}_s:{\varOmega }\times [0,T]\times E\longrightarrow \mathbb {R}$ is $\mathscr {F}\times \mathcal {B}([0,T])\times \mathcal {B}(E)-$measurable, such that $D_1\le \tau ^{y,z,u,u^{\prime }}_s(e)\le D_2$ with $-1< D_1\le 0$ and $D_2\ge 0$, for all $s,y,z,u,u^{\prime },e$, then exists $(Y,Z,U)\in \mathcal {S}^{\infty }\times \mathcal {L}^2\times \mathcal {L}^2({\tilde{N}})$ solution of

$$\begin{aligned} Y_t=\xi +\int _t^TL_s(Y_{s-},Z_s,U_s)\hbox {d}s-\int _t^TZ_s \hbox {d}W_s-\int _t^T\int _EU_s(e){\tilde{N}}(\hbox {d}s,\hbox {d}e), \end{aligned}$$

so that $||Y||_\infty <K_1, \, \Vert Z\Vert _{\mathcal {L}^2}\le K_1; \,\, \Vert U\Vert _{\mathcal {L}^2({\tilde{N}})}\le K_3$ with positive constants $K_1,K_2, K_3$ depending only on $\Vert \xi \Vert _{\infty },[\alpha ]_{\text {int}} , b_1,b_2, T$.

Uniqueness is achieved if we add that

$$\begin{aligned} L_s(y,z,u)-L_s(y,z^{\prime },u)\le C(k_s+|z|+|z^{\prime }|)|z-z^{\prime }|, \end{aligned}$$

for some constant $C>0$ and some predictable process k such that $\int _0^\cdot k_s \hbox {d}W_s$ is a BMO martingale.

We recall that

Definition 1

A martingale M is said to be in the class of BMO martingales (bounded mean oscillation) if there exists a constant $C>0$, such that

$$\begin{aligned} \text {es}\sup _{\tau \in \mathcal {T}} \mathbb {E}\Big (<M>_T - <M>_\tau |\mathscr {F}_\tau \Big ) \le C \quad \text { and } |{\varDelta }M_\tau |^2\le C,\quad \forall \, \tau \in \mathcal {T}, \end{aligned}$$

where $\mathcal {T}$ is the set of all $\mathscr {F}_t-$stopping times in [0, T] and $<M>$ denotes the continuous part of the quadratic variation of the martingale.

Proof of Theorem 2

To show the existence and uniqueness of the solution of Eq. (19), we first transform it into an equivalent BSDE.

For any $b\in {\mathbb {R}}$ let us denote by $\varphi _b(u):=\frac{\mathrm e^{b u}-1-b u}{b}$ so that $\displaystyle [[u]]_b = \int _E \varphi _b(u(e))\gamma (\hbox {d}e).$ Setting $G_t:=-\ln (V_t)$, $M_t=-\frac{Z_t}{V_{t-}}$ and $J_t(e)=-\ln \Big (1+\frac{U_t(e)}{V_{t-}}\Big )$ and applying formally Itô’s formula, we get

$$\begin{aligned} \begin{aligned} G_t&=-\ln (1+F(c_T))+\int _t^T\Big \{-\beta G_{s-}-\beta \ln c_{s-}+\frac{M_s^2}{2}\\&\quad +\int _E\Big [ [\varphi _1(-J_s)+\varphi _1(J_s)](1-\iota _s)(1+\theta _s)- \varphi _1(-J_s)\Big ](e)\lambda (s,\hbox {d}e)\Big \}\hbox {d}s\\&\quad - \int _t^TM_s \hbox {d}W'_s-\int _t^T\int _EJ_s(e){\tilde{N}}'(\hbox {d}s,\hbox {d}e), \end{aligned} \end{aligned}$$

(21)

that is a BSDE with bounded terminal value $-\ln (1+F(c_T))$ and generator

$$\begin{aligned} L_s(g,m,j)= & {} -\beta ( g-\ln c)+\frac{m^2}{2}+\int _E [(\varphi _1(-j)\\&+\,\varphi _1(j))(1-\iota _s)(1+\theta _s) -\varphi _1(-j)](e)\lambda (s,\hbox {d}e). \end{aligned}$$

Since $\iota \in (0,1)$ and $\theta \in (-1+\epsilon ,0)$, $0<(1-\iota )(1+\theta )<1$, we can say that

$$\begin{aligned}&-\beta (|\ln c|+\kappa )-\beta g-\frac{m^2}{2}-[[-j]]_1\\&\quad \le L_t(g,m,j)\le \beta (|\ln c|+\kappa )+\beta g+\frac{m^2}{2}+[[j]]_1, \end{aligned}$$

with $\kappa$ such that $\beta (|\ln c|+\kappa )\ge \beta$. Moreover, given $p\ge \frac{2\mathrm e^{\beta T}}{\beta }$, we have that

$$\begin{aligned} \mathbb {E}\Big [\int _0^T\mathrm e^{p(|\ln c_{t-}|+\kappa )}\hbox {d}t\Big ]\le \mathrm e^{p\kappa }\Big (\mathbb {E}\Big [\int _0^Tc^{-p}_{t-}\hbox {d}t\Big ]+\mathbb {E}\Big [\int _0^Tc^p_{t-}\hbox {d}t\Big ]\Big )<\infty , \end{aligned}$$

but c is a stochastic exponential and the integrability of $c^p$ and $c^{-p}$ is ensured by hypothesis (17).

Moreover $L_t(g,m,j)$ is Lipschitz in g, increasing in $m^2$ and, by virtue of the boundedness hypotheses on $\iota$ and $\theta$ and the mean value theorem applied to $\varphi _1$, it verifies

$$\begin{aligned} L_s(g,m,j)-L_s(g,m,j^{\prime })\le \int _E\tau ^{g,m,j,j^{\prime },\iota ,\theta }_s(e)(j(e)-j^{\prime }(e))\lambda (s,\hbox {d}e) \end{aligned}$$

for $j,j^{\prime }\in L^2(\gamma )$, where $\tau ^{g,m,j,j^{\prime },\iota ,\theta }_s$ is such that $D_1\le \tau ^{g,m,j,j^{\prime },\iota ,\theta }_s(e)\le D_2$ with $-1< D_1\le 0$ and $D_2\ge 0$ for all $s,g,m,j,j^{\prime },\iota ,\theta ,e$. Thus we can apply the above result to say that there exists a solution for the BSDE (21).

Finally noting that

$$\begin{aligned} L_s(g,m,j)-L_s(g,m^{\prime },j)\le \frac{1}{2}(|m|+|m^{\prime }|)|m-m^{\prime }|, \end{aligned}$$

we obtain also the uniqueness of the solution.

Consequently there exists a unique solution also for Eq. (19). $\square$