Utility maximization in a multidimensional semimartingale model with nonlinear wealth dynamics

Abstract

We explore martingale and convex duality techniques to maximize expected risk-averse utility from consumption in a general multi-dimensional (non-Markovian) semimartingale market model with jumps and non-linear wealth dynamics. The model allows to incorporate additional cash flows via non-linear margin payment functions in the drift term that depend on the allocation proportion. These can be used to cast frictions such as the impact of the portfolio choices of a ‘large’ investor on the expected assets’ returns, funding costs arising from differential borrowing and lending rates, and the cash inflow of a firm in a neoclassical economy with constant return-to-scale Cobb–Douglas technology subject to exogenous aggregate shocks. We provide a general verification theorem for random utility fields satisfying the usual Inada conditions, find conditions under which jumps in our model lead to precautionary saving, and present an explicit characterization for CRRA. We report two-fund separation-type results which assert that optimal allocations move along one-dimensional segments, and illustrate our results numerically for various margin payment functions and bounded variation tempered $\alpha$-stable jumps.

Appendix

Proof of Theorem 1

We prove first part (b). Since $Y_0^{x,\hat{\varphi }}/H_0^{\hat{\varphi }}=\mathcal {X}^{\varphi }(\mathcal {Y}^{\varphi }(x))=x,$ it suffices to show that $Y^{x,\hat{\varphi }}/H^{\hat{\varphi }}$ satisfies the wealth equation (9) for the pair $(\hat{\pi },\hat{\gamma }).$

Using Itô’s formula for semi-martingales (see e.g. Theorem 4.57 in Jacod and Shiryaev [20, Ch. I]) the differential of $1/H^{\hat{\varphi }}$ satisfies

$$\begin{aligned} d\Bigl (\frac{1}{H_t}\Bigr )&=\frac{-1}{H_{t-}^2}\,dH_t+\frac{1}{2}\frac{2}{H_{t-}^3}\,d\left\langle H^\mathsf {c}\right\rangle _t\\&\quad +\frac{1}{H_{t-}}{\int _{\mathbb {R}^d}\Bigl [\frac{1}{\hat{\varphi }^2(t,y)}-1+(\hat{\varphi }^2(t,y)-1)\Bigr ]\,\mu (dy,dt)}\\&=\frac{1}{H_{t-}}\left\{ [r_t+\tilde{g}_K(t,\hat{\zeta }_t)+{\varphi ^1_t\cdot a_t\varphi ^1_t}]\,dA_t-\varphi ^1_t\cdot dX_t^{\mathsf {c}}-\int _{\mathbb {R}^d}(\varphi ^2(t,y)-1)\,\tilde{\mu }(dy,dt)\right. \\&\quad +\left. \int _{\mathbb {R}^d}\Bigl [\frac{1}{\hat{\varphi }^2(t,y)}-1+(\hat{\varphi }^2(t,y)-1)\Bigr ]\,\mu (dy,dt)\right\} \\&=\frac{1}{H_{t-}}\left\{ \Bigl [r_t+\tilde{g}_K(t,\hat{\zeta }_t)+\varphi ^1_t\cdot a_t\varphi ^1_t+\int _{\mathbb {R}^d}(\varphi ^2(t,y)-1)\,\nu _t(dy)\Bigr ]\,dA_t-\varphi ^1_t\cdot dX_t^{\mathsf {c}}\right. \\&\quad +\left. {\int _{\mathbb {R}^d}\Bigl [\frac{1}{\hat{\varphi }^2(t,y)}-1\Bigr ]\,\mu (dy,dt)}\right\} \end{aligned}$$

From (19), the differential of $Y^{x,\hat{\varphi }}$ is given by

$$\begin{aligned} dY_t^{x,\hat{\varphi }}=-H_t^{\hat{\varphi }}\gamma _t^{x,\hat{\varphi }}\,d{\kappa _t}+\alpha _t^{x,\varphi }\cdot dX_t^{\mathsf {c}}+\int _{\mathbb {R}^d}\beta ^{x,\hat{\varphi }}(t,y)\,\tilde{\mu }(dy,dt) \end{aligned}$$

Using the product rule for semimartingales, we have

$$\begin{aligned}&d\biggl (\frac{Y_t^{x,\hat{\varphi }}}{H_t^{\hat{\varphi }}}\biggr )=Y_{t-}^{x,\hat{\varphi }}\,d\Bigl (\frac{1}{H_t^{\hat{\varphi }}}\Bigr ) +\frac{1}{H_{t-}^{\hat{\varphi }}}\,dY_t^{x,\hat{\varphi }}+d\Bigl \langle Y^{x,\hat{\varphi },\mathsf {c}},\frac{1}{H^{\hat{\varphi },\mathsf {c}}}\Bigr \rangle _t\\&\qquad +\frac{1}{H_{t-}^{\hat{\varphi }}}\int _{\mathbb {R}^d}\beta ^{x,\hat{\varphi }}(t,y)\Bigl [\frac{1}{\hat{\varphi }^2(t,y)}-1\Bigr ]\,\mu (dy,dt)\\\\&\quad =\frac{Y_{t-}^{x,\hat{\varphi }}}{H_{t-}^{\hat{\varphi }}}\left\{ \Bigl [r_t+\tilde{g}_K(t,\hat{\zeta }_t)+\varphi ^1_t\cdot a_t\varphi ^1_t+\int _{\mathbb {R}^d}(\varphi ^2(t,y)-1)\,\nu _t(dy)\Bigr ]\,dA_t-\varphi ^1_t\cdot dX_t^{\mathsf {c}}\right. \\&\qquad +\left. \int _{\mathbb {R}^d}\Bigl [\frac{1}{\hat{\varphi }^2(t,y)}-1\Bigr ]\,\mu (dy,dt)\right\} -\gamma _t^{x,\hat{\varphi }}\,d{\kappa _t}+\frac{1}{H_{t-}^{\hat{\varphi }}}\Bigl \{\alpha _t^{x,\varphi }\cdot dX_t^{\mathsf {c}}\Bigr .\\&\qquad +\int _{\mathbb {R}^d}\beta ^{x,\hat{\varphi }}(t,y)\,\tilde{\mu }(dy,dt)-\alpha _t^{x,\hat{\varphi }}\cdot a_t\varphi _t^1\,dA_t\\&\qquad +\Bigl .\int _{\mathbb {R}^d}\beta ^{x,\hat{\varphi }}(t,y)\Bigl [\frac{1}{\hat{\varphi }^2(t,y)}-1\Bigr ]\,\mu (dy,dt)\Bigr \} \end{aligned}$$

Multiplying and dividing by $Y_{t-}^{x,\hat{\varphi }}$ in the last bracket and using conditions (20) and (21), we obtain

$$\begin{aligned}&d\biggl (\frac{Y_t^{x,\hat{\varphi }}}{H_t^{\hat{\varphi }}}\biggr ) =\frac{Y_{t-}^{x,\hat{\varphi }}}{H_{t-}^{\hat{\varphi }}}\left\{ \Bigl [r_t+\tilde{g}_K(t,\zeta ^{\hat{\varphi }}_t)+\Bigl [\varphi ^1_t-\frac{1}{Y_{t-}^{x,\hat{\varphi }}}\alpha ^{x,\hat{\varphi }}_t\Bigr ]\cdot a_t\varphi ^1_t\Bigr .\right. \\&\qquad +\Bigl .\int _{\mathbb {R}^d}\Bigl (\varphi ^2(t,y)-1-\frac{\beta ^{x,\hat{\varphi }}(t,y)}{Y_{t-}^{x,\hat{\varphi }}}\Bigr )\,\nu _t(dy)\Bigr ]\,dA_t +\Bigl [\frac{\alpha ^{x,\hat{\varphi }}_t}{Y_{t-}^{x,\hat{\varphi }}}-\varphi _t^1\Bigr ]\cdot dX_t^{\mathsf {c}}\\&\qquad \left. +\int _{\mathbb {R}^d}\Bigl (\frac{1}{\hat{\varphi }^2(t,y)}-1+\frac{\beta ^{x,\hat{\varphi }}(t,y)}{\hat{\varphi }^2(t,y)Y_{t-}^{x,\hat{\varphi }}}\Bigr )\,\mu (dy,dt)\right\} -\gamma _t^{x,\hat{\varphi }}\,d\kappa _t\\&\quad =\frac{Y_{t-}^{x,\hat{\varphi }}}{H_{t-}^{\hat{\varphi }}}\biggl \{\Bigl [r_t+\tilde{g}_K(t,\zeta ^{\hat{\varphi }}_t)-\hat{\pi }_t\cdot a_t\varphi ^1_t-\hat{\pi }_t\int _{\mathbb {R}^d}(e^y-\underline{1})\varphi ^2(t,y)\nu _t(dy)\Bigr ]\,dA_t\biggr .\\&\qquad +\hat{\pi }_t\cdot dX_t^{\mathsf {c}}+\biggl .\hat{\pi }_t\cdot \int _{\mathbb {R}^d}(e^y-\underline{1})\,\mu (dy,dt)\biggr \} -\gamma _t^{x,\hat{\varphi }}\,d\kappa _t \end{aligned}$$

Finally, we add and subtract $\hat{\pi }_t\cdot \left[ r_t\underline{1}-b_t-\frac{1}{2}\mathrm {diag}[a_t]-\int _{\mathbb {R}^d}h(y)\,\nu _t(dy)\right]$ in the $dA_t$ term, and use the definition (12) of $\hat{\zeta }=\zeta ^{\hat{\varphi }}$ and condition (22) to get

$$\begin{aligned} d\biggl (\frac{Y_t^{x,\hat{\varphi }}}{H_t^{\hat{\varphi }}}\biggr )&=\frac{Y_{t-}^{x,\hat{\varphi }}}{H_{t-}^{\hat{\varphi }}}\left\{ \Bigl [r_t+\tilde{g}_K(t,\hat{\zeta }_t)+\hat{\pi }_t\cdot \hat{\zeta }_t\Bigr .\right. \\&\quad -\left. \Bigl .\hat{\pi }_t\cdot \left( r_t\underline{1}-b_t-\frac{1}{2}\mathrm {diag}[a_t]-\int _{\mathbb {R}^d}h(y)\,\nu _t(dy)\right) \Bigr ]\,dA_t\right. \\&\quad +\biggl .\,\hat{\pi }_t\cdot dX_t^{\mathsf {c}}+\hat{\pi }_t\cdot \int _{\mathbb {R}^d}(e^y-\underline{1})\,\mu (dy,dt)\biggr \}-\gamma _t^{x,\hat{\varphi }}\,d\kappa _t\\&=\frac{Y_{t-}^{x,\hat{\varphi }}}{H_{t-}^{\hat{\varphi }}}\biggl \{\Bigl [r_t+g(t,\hat{\pi }_t)\Bigr .\biggr .\\&\quad \left. \Bigl .+\pi _t\cdot \Bigl (b_t+\frac{1}{2}\,\mathrm {diag}[a_t]-r_t\underline{1}+\int _{\left| y\right| \le 1}(e^y-y-\underline{1})\,\nu _t(dy)\Bigr )\Bigr ]\,dA_t\right. \\&\quad +\biggl .\pi _t\cdot \Bigl [dX_t^{\mathsf {c}}+\int _{\left| y\right| \le 1}(e^{y}-\underline{1})\,\tilde{\mu }(dy,dt)+\int _{\left| y\right| > 1}(e^{y}-\underline{1})\,\mu (dy,dt)\Bigr ]\biggr \}-\gamma _t^{x,\hat{\varphi }}\,d\kappa _t \end{aligned}$$

Hence, $Y^{x,\hat{\varphi }}/{H^{\hat{\varphi }}}$ solves the wealth equation (9). Part (b) follows by uniqueness of solution to Eq. (9). This in turn implies optimality of $(\hat{\pi },\hat{\gamma }).$ The proof of $(\hat{\pi },\hat{\gamma })\in \tilde{\mathcal {A}}(x)$ is the same as the proof of part (ii) of Lemma 1 in Michelbrik and Le [33]. Part (c) follows easily since $\vartheta (x)=\tilde{\vartheta }(x)=J(\hat{\gamma }).$ $\square$

Lemma 3

Let $x,z\ge 0$ be fixed. Suppose $U'''>0$ (resp. $<0$) and $z>1$ (resp. $z<1$). Then

$$\begin{aligned} I(U'(x)z)-x+\frac{1}{\mathrm {AR}(x)}(z-1)>0. \end{aligned}$$

Proof

Define $f(\lambda ):=I(U'(x)\lambda ).$ Suppose $U'''>0$ and $z>1.$ By the the mean value theorem, there exists $z^*\in (1,z)$ such that

$$\begin{aligned} \frac{f(z)-f(1)}{z-1}=f'(z^*)=I'(U'(x)z^*)U'(x)=\frac{U'(x)}{U''(I(U'(x)z^*))}. \end{aligned}$$

Since I is decreasing and $U''$ is increasing, we have $I(U'(x)z^*)<I(U'(x))=x$ and

$$\begin{aligned} U''(I(U'(x)z^*))<U''(x) \end{aligned}$$

and the desired result follows. The same argument can be used if $z\in (0,1)$ and $U''$ is decreasing. $\square$