Construction of a class of forward performance processes in stochastic factor models, and an extension of Widder’s theorem

Abstract

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes, as well as the corresponding optimal portfolios, with power-utility initial data and for stock–factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. This is done by solving the associated nonlinear parabolic partial differential equations (PDEs) posed in the “wrong” time direction. Along the way, we establish on domains an explicit form of the generalised Widder theorem of Nadtochiy and Tehranchi (Math. Finance 27:438–470, 2015, Theorem 3.12) and rely for that on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the “right” time direction.

Appendix A

Lemma A.1

Let all the conditions of Theorem 2.14hold and let $Y$be a stochastic process on $D$with dynamics as in (2.2) under a probability measure ℙ. Then for any $t>0$and any set $A \subseteq D$of positive Lebesgue measure, we have $\mathbb{P}[Y_{t} \in A] > 0$.

Proof

Let us argue by contradiction. Suppose there exist $t>0$ and a set $A \subseteq D$ of positive Lebesgue measure such that $\mathbb{P}[Y_{t} \in A] = 0$. Consider the measure $\tilde{\mathbb{P}}$ defined by the Radon–Nikodým derivative

$$ \frac{\mathrm {d}\tilde{\mathbb{P}}}{\mathrm {d}\mathbb{P}} = \mathcal{E}\bigg( \int \Gamma \rho ^{\top }\lambda (Y_{s}) \mathrm {d}B_{s}\bigg)_{\!t}\, , $$

where ℰ denotes the stochastic exponential. From Assumption 2.10, it follows that $\lambda$ is bounded; hence Novikov’s condition yields $\tilde{\mathbb{P}} \approx \mathbb{P}$. Thus $\mathbb{P}[Y_{t} \in A] = 0$ holds if and only if $\tilde{\mathbb{P}}[Y_{t} \in A] = 0$. Note that under $\tilde{\mathbb{P}}$, the process $Y$ has the dynamics

$$\begin{aligned} \mathrm {d}Y_{t} = \big(\alpha (Y_{t}) + \Gamma \kappa (Y_{t}) \smash{^{\top }}\rho \smash{^{\top }}\lambda (Y_{t}) \big) \mathrm {d}t + \kappa (Y_{t})^{\top }\mathrm {d}\tilde{B}_{t}. \end{aligned}$$

Let the set $C \subseteq \mathbb{R}^{k}$ be the image of the set $A \subseteq D$ under the diffeomorphism $\Xi : D \to \mathbb{R}^{k}$ and denote by $Z$ the image $\Xi (Y)$ of the process $Y$. Then $\mathbb{P}[Y_{t} \in A] = 0$ is equivalent to $\tilde{\mathbb{P}}[Z_{t} \in C]=0$. Since $\Xi$ is a diffeomorphism, it follows that $C$ has positive Lebesgue measure. The process $Z$ is a diffusion on $\mathbb{R}^{k}$ with the generator

$$ \mathcal{L}_{z} = \frac{1}{2}\sum _{i,j=1}^{k} \overline{a}_{ij}(z) \partial _{z_{i} z_{j}} + \sum _{i=1}^{k} \overline{b}_{i}(z) \partial _{z_{i}}, $$

where $\overline{a}(\cdot )$, $\overline{b}(\cdot )$ are as in (2.9) and $a(\cdot )$, $b(\cdot )$ are as in (2.11). Since $a(\cdot )$, $b(\cdot )$ satisfy Assumption 2.10 and $C$ has positive Lebesgue measure, it follows from [30, Theorem A] that $\tilde{\mathbb{P}}[Z_{t} \in C]>0$, which is the desired contradiction. □