Quadratic expansions in optimal investment with respect to perturbations of the semimartingale model

Abstract

We study the response of the optimal investment problem to small changes of the stock price dynamics. Starting with a multidimensional semimartingale setting of an incomplete market, we suppose that the perturbation process is also a general semimartingale. We obtain second-order expansions of the value functions, first-order corrections to the optimisers, and provide the adjustments to the optimal control that match the objective function up to the second order. We also give a characterisation in terms of the risk-tolerance wealth process, if it exists, by reducing the problem to the Kunita–Watanabe decomposition under a change of measure and numéraire. Finally, we illustrate the results by examples of base models that allow closed-form solutions, but where this structure is lost under perturbations of the model where our results allow an approximate solution.

Appendix:  The structure of \(\mathcal{M}^{\infty}\) and \(\mathcal{N}^{\infty}\)

We recall that Mostovyi [36, Lemma 4.1] shows that every element of $\mathcal{M}^{\infty}$ can be represented as a stochastic integral with respect to $R^{\pi}$. The following lemma establishes the opposite direction.

Lemma A.1

Fix $x>0$ and set $y = u_{x}(x,0)$. Suppose $M\in \mathcal{H}^{2}_{ \mathrm{loc}}(\mathbb{P})$, that (2.2) and Assumptions 2.1and 2.2hold and that $R^{\pi}$ is sigma-bounded. Then we have

$$ \mathcal{M}^{\infty }= \{\textit{$\alpha \cdot R^{\pi}: \alpha $ is predictable, $R^{\pi}$-integrable and such that $\alpha \cdot R^{\pi}$ is bounded}\}. $$

Remark A.2

The proof goes through without the sigma-boundedness assumption. The latter is imposed to ensure that $\mathcal{M}^{\infty}$ is non-degenerate and that the closure of $\mathcal{M}^{\infty}$ in $\mathcal{H}^{2}_{0}(\mathbb{R})$ is equal to $\mathcal{M}^{2}$. Also, the proof goes through with NUPBR or, equivalently, (2.10), instead of $M\in \mathcal{H}^{2}_{\mathrm{loc}}(\mathbb{P})$ and (2.2), and with the Inada conditions instead of Assumption 2.1; all we need is that together with Assumption 2.2, the standard assertions of utility maximisation theory hold.

Proof of Lemma A.1

Let $\alpha$ be predictable and $R^{\pi}$-integrable and such that $\alpha \cdot R^{\pi}$ is bounded. Then there exists a constant $C>0$ such that $C + \alpha \cdot R^{\pi}$ is strictly positive. By Jacod and Shiryaev [19, Theorem II.8.3], there exists a predictable $R^{\pi}$-integrable process $\widetilde{\alpha}$ such that

$$ C+ \alpha \cdot R^{\pi }= C\mathcal{E}( \widetilde{\alpha}\cdot R^{ \pi })= C \frac{\mathcal{E}\left ((\pi + \widetilde{\alpha})\cdot R\right )}{\mathcal{E}\left (\pi \cdot R\right )}, $$

where the second equality uses (3.2). We deduce that the bounded process $\alpha \cdot R^{\pi}$ admits the representation

$$ \alpha \cdot R^{\pi }= C \frac{\mathcal{E}\left ((\pi + \widetilde{\alpha})\cdot R\right ) - \mathcal{E}\left (\pi \cdot R\right )}{\mathcal{E}\left (\pi \cdot R\right )} $$

which is an element of $\mathcal{M}^{\infty}$ by the definition of $\mathcal{M}^{\infty}$. As $\alpha$ was arbitrary, the proof is complete. □

Lemma A.3

Fix $x>0$ and set $y = u_{x}(x,0)$. Impose the assumptions of Lemma A.1and that both $M$ and $H$ are in $\mathcal{H}^{2}_{\mathrm{loc}}(\mathbb{P})$. Recall that H is defined in Assumption 2.3and is such that $Y = y\mathcal{E}(H)$. Then we have

$$ \begin{aligned}\mathcal{N}^{\infty }= \{N^{H}:\,& \textit{$N^{H}$ is bounded, $N\in \mathcal{H}^{2}_{\mathrm{loc}}(\mathbb{P})$} \\ &\textit{and $N$ is orthogonal to each component of $M$}\}. \end{aligned} $$

Proof

Take $N\in \mathcal{H}^{2}_{\mathrm{loc}}(\mathbb{P})$ such that $N^{H}$ is bounded and fix $\widetilde{M}\in \mathcal{M}^{\infty}$. By [36, Lemma 4.1], we have $\widetilde{M} = \alpha \cdot R^{\pi}$ for some predictable $R^{\pi}$-integrable process $\alpha$. Let us approximate $\alpha$ by the $M$-integrable processes

$$ \alpha ^{n} :=( -n \vee \alpha \wedge n )1_{[\!\![0,\tau _{n} ]\!\!]},\qquad n\in \mathbb{N}, $$

where every component of $\alpha$ is truncated from above by $n$ and below by $-n$ and where $\tau _{n}$, $n\in \mathbb{N}$, is a localising sequence for both $M$ and $N$. Then for a fixed $n\in \mathbb{N}$ and every stopping time $\tau$, similarly to Lemma 5.13, we get

$$ \begin{aligned}\mathbb{E}_{\mathbb{R}} [ (\alpha ^{n}\cdot R^{\pi }_{ \tau }) N^{H}_{\tau }] = \mathbb{E}_{\mathbb{R}}[ \langle \alpha ^{n} \cdot M, N \rangle _{\tau _{n}\wedge \tau }] = 0. \end{aligned} $$

(A.1)

As $N^{H}$ is bounded and $\alpha ^{n}\cdot R^{\pi}$, $n\in \mathbb{N}$, converges to $\alpha \cdot R^{\pi}$ in $\mathcal{H}^{2}(\mathbb{R})$, we deduce from (A.1) that $N^{H}$ is orthogonal to $\alpha \cdot R^{\pi}$. Now from Lemma A.1, we deduce that $N^{H}$ is orthogonal to $\mathcal{M}^{\infty}$. Since additionally, the closure of $\mathcal{M}^{\infty}$ in $\mathcal{H}^{2}_{0}(\mathbb{R})$ is equal to $\mathcal{M}^{2}$ by Kramkov and Sîrbu [30, Lemma 6], we get

$$ \begin{aligned} \mathcal{N}^{\infty }\supseteq \{N^{H}:\,& \text{$N^{H}$ is bounded, $N\in \mathcal{H}^{2}_{\mathrm{loc}}(\mathbb{P})$} \\ &\text{and $N$ is orthogonal to each component of $M$}\}. \end{aligned} $$

To show the opposite inclusion, we proceed as follows. We fix $K\in \mathcal{N}^{\infty}$ and set $N := K+ [K, H] = K^{-H}$. Then $N$ is locally square-integrable under ℙ because $K$ is bounded and $H$ is locally ℙ-square-integrable. We suppose that $\mathcal{E}(K)>0$, as otherwise we may multiply $K$ by a sufficiently small constant $\varepsilon$ and conduct the proof for $\varepsilon K$.

For $\alpha = -\pi$, as $\mathcal{E}(\alpha \cdot R^{\pi}) = \frac {1}{\mathcal{E}(\pi \cdot R)}>0$, using the sigma-boundedness of $R^{\pi}$ and the Ansel–Stricker theorem (see [8, Corollary 7.3.8]), we deduce that $\mathcal{E}(\alpha \cdot R^{\pi})$ is a local martingale under ℝ and hence in $\mathcal{H}^{1}_{\mathrm{loc}}(\mathbb{R})$. Let $\sigma _{n}$, $n\in \mathbb{N}$, be a localising sequence for $\mathcal{E}(\alpha \cdot R^{\pi})$ such that on $[\!\![0,\sigma _{n}]\!\!]$, $\mathcal{E}(\alpha \cdot R^{\pi})$ is in $\mathcal{H}^{1}(\mathbb{R})$, where by boundedness of $K$, we also suppose that $\mathcal{E}(K)$ is bounded on $[\!\![0,\sigma _{n}]\!\!]$. Further, as $R^{\pi}$ is sigma-bounded, we can use [30, Theorem 4] to approximate $\mathcal{E}(\alpha \cdot R^{\pi})$ on $[\!\![0,\sigma _{n}]\!\!]$ in $\mathcal{H}^{1}(\mathbb{R})$ by some bounded stochastic integrals with respect to $R^{\pi}$. These integrals are in $\mathcal{M}^{\infty}$ by Lemma A.1. By convergence in $\mathcal{H}^{1}(\mathbb{R})$ and boundedness of $\mathcal{E}(K)$ on $[\!\![0,\sigma _{n} ]\!\!]$, we obtain

$$\begin{aligned} \mathbb{E}_{\mathbb{R}} [\mathcal{E}(\alpha \cdot R^{\pi})_{t\wedge \sigma _{n}} \mathcal{E}(K)_{t\wedge \sigma _{n}}|\mathcal{F}_{s \wedge \sigma _{n}}] &= \mathcal{E}(\alpha \cdot R^{\pi})_{s\wedge \sigma _{n}} \mathcal{E}\left (K\right )_{s\wedge \sigma _{n}} \\ &= \frac{\mathcal{E}\left (N+H\right )_{s\wedge \sigma _{n}}}{\mathcal{E}(H)_{s\wedge \sigma _{n}}\mathcal{E}(\pi \cdot R)_{s\wedge \sigma _{n}}} , \end{aligned}$$

(A.2)

and by a change of measure, we have

$$\begin{aligned} &\mathbb{E}_{\mathbb{R}}[\mathcal{E}(\alpha \cdot R^{\pi})_{t\wedge \sigma _{n}} \mathcal{E}\left (K\right )_{t\wedge \sigma _{n}}| \mathcal{F}_{s\wedge \sigma _{n}}] \\ &= \mathbb{E}\left [ \frac{\mathcal{E}(H)_{t\wedge \sigma _{n}}\mathcal{E}(\pi \cdot R)_{t\wedge \sigma _{n}}}{\mathcal{E}(H)_{s\wedge \sigma _{n}}\mathcal{E}(\pi \cdot R)_{s\wedge \sigma _{n}}} \mathcal{E}(\alpha \cdot R^{\pi})_{t\wedge \sigma _{n}} \mathcal{E}(N^{H})_{t \wedge \sigma _{n}}\bigg|\mathcal{F}_{s\wedge \sigma _{n}}\right ] \\ &= \mathbb{E}\left [ \frac{\mathcal{E}(H)_{t\wedge \sigma _{n}} }{\mathcal{E}(H)_{s\wedge \sigma _{n}}\mathcal{E}(\pi \cdot R)_{s\wedge \sigma _{n}}} \frac{\mathcal{E}\left (N+H\right )_{t\wedge \sigma _{n}}}{\mathcal{E}(H)_{t\wedge \sigma _{n}} } \bigg|\mathcal{F}_{s\wedge \sigma _{n}}\right ] \\ &= \frac{1}{\mathcal{E}(H)_{s\wedge \sigma _{n}}\mathcal{E}(\pi \cdot R)_{s\wedge \sigma _{n}}} \mathbb{E}[ {\mathcal{E}\left (N+H\right )_{t\wedge \sigma _{n}}} | \mathcal{F}_{s\wedge \sigma _{n}}]. \end{aligned}$$

(A.3)

Comparing (A.2) and (A.3), we deduce that $\mathcal{E}\left (N+H\right )$ is a local martingale under ℙ. As $\mathcal{E}\left (N+H\right )$ and $\mathcal{E}\left (N+H\right )_{-}$ are both non-vanishing by construction because $\mathcal{E}\left (N+H\right ) = \mathcal{E}(K)\mathcal{E}(H)$, the stochastic logarithm of $\mathcal{E}\left (N+H\right )$ is well defined by [19, Theorem II.8.3] and is equal to $N+H$ by [19, Corollary II.8.7]. Further, from [19, Theorem II.8.3] and Protter [40, Theorem III.29], we conclude that $N+H$ is a local martingale under ℙ. Moreover, $N$ is locally ℙ-square-integrable as seen in the preceding paragraph and $H\in \mathcal{H}^{2}_{\mathrm{loc}}(\mathbb{P})$ by assumption, and so we conclude that $N\in \mathcal{H}^{2}_{\mathrm{loc}}(\mathbb{P})$.

Second, we show that $N$ is orthogonal to $M$. For this, take an element of $\mathcal{M}^{\infty}$ of the form $\alpha \cdot R^{\pi}$, where by sigma-boundedness, we suppose that each component of $\alpha$ takes values in $(0,1]$. Choose a stopping time $\sigma$ such that $\mathbb{E}[\int _{0}^{\sigma }\alpha ^{\top}_{s} d\left \langle M \right \rangle _{s} \alpha _{s} ]<\infty$ and $\mathbb{E}[\left \langle N \right \rangle _{\sigma}]<\infty$. Then as $K = N^{H}\in \mathcal{N}^{\infty}$, similarly to the proof of Lemma 5.13, we have

$$ 0 =\mathbb{E}_{\mathbb{R}}\big[ [\alpha \cdot R^{\pi}, N^{H}]_{\sigma} \big]=\mathbb{E}_{\mathbb{R}}[ \left \langle \alpha \cdot M, N \right \rangle _{\sigma}]. $$

(A.4)

From (A.4), we deduce that $\left \langle \alpha \cdot M, N \right \rangle$ is an ℝ-martingale on $[\!\![0, \sigma ]\!\!]$. Further, since $\left \langle \alpha \cdot M, N \right \rangle$ is predictable and of finite variation, we deduce that $\left \langle \alpha \cdot M, N \right \rangle \equiv 0$ on $[\!\![0, \sigma ]\!\!]$. As $M$ and $N$ are locally square-integrable and each component of $\alpha$ in the previous paragraph was $(0,1]$-valued, thus non-vanishing, we can deduce by localisation that each component of $M$ is orthogonal to $N$ on $[0,T]$. □