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Optimal investment and consumption under partial information

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Abstract

We present a unified approach for partial information optimal investment and consumption problems in a non-Markovian Itô process market. The stochastic local mean rate of return and the Wiener process cannot be observed by the agent, whereas the path-dependent volatility, the path-dependent interest rate and the asset prices can be observed. The main assumption is that the asset price volatility is a nonanticipative functional of the asset price trajectory. The utility functions are general and satisfy standard conditions. First, we show that the corresponding full information market is complete and in this setting we solve the problem using standard methods. Second, we transform the original partial information problem into a corresponding full information problem using filtering theory, and show that it follows that the market is observationally complete in the sense that any contingent claim adapted to the observable filtration is replicable. Using the solutions of the full information problem we then easily derive solutions to the original partial information problem.

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Notes

  1. The term complete with respect to \(\underline{\mathcal {F}}^S\) is also used, see e.g. Hahn et al. (2007).

  2. The proof of the first part of the proposition is found in Kallianpur (1980, Theorem 5.1.2) and in Liptser and Shiryayev (2001, Theorem 4.6). The proof of the second part is easiest found in Kallianpur (1980, Section 5.2) but the reasoning can also be found in the proof of Theorem 5.16 and Theorem 5.17 in Liptser and Shiryayev (2001). Regarding Definition 2.2: several different Lipschitz and growth conditions guaranteeing the existence of a unique strong solution are used in the literature (see e.g. Kallianpur 1980 or Liptser and Shiryayev 2001, or, for less general stochastic differential equations, Karatzas and Shreve 1991 or Øksendal 2003). Similar remarks apply to Definition 2.1.

  3. Here \('\) denotes transposition. However, \('\) will sometimes denote the derivative. What is meant should be clear from the context.

  4. Note that we in this proposition restrict the process \(\gamma \) to be \(\underline{{\mathcal {F}}}^\xi \)-progressively measurable, rather than just \(\underline{{\mathcal {F}}}\)-progressively measurable.

  5. See e.g. Fujisaki et al. (1972), Kallianpur (1980, Chapter 8) or Liptser and Shiryayev (2001, Chapter 8) for similar results and definitions, and also justifications of the definitions. Recall (for a given filtration) that a progressively measurable process is adapted and measurable, that a process which is measurable and adapted has a progressively measurable modification, and that an adapted process with every sample path right-continuous or left-continuous is progressively measurable, see Karatzas and Shreve (1991, Chapter 1).

  6. Karatzas and Shreve (1998) also have a slightly different setup in other respects. For example, they analyze the portfolio process \(\pi _t\) rather than our portfolio weights process \(u_t\), see Definition 3.2. The filtration of their market is moreover generated by the Wiener process driving the asset prices. The proofs of Karatzas and Shreve (1998) can easily be modified to our setup and are not included.

  7. \(\mathbf{1}\) is an n-dimensional vector with each element equal to 1.

  8. Thus \((1-u_t\mathbf{1})\) represents the weight in the bank account \(B_t\).

  9. As noted in Björk et al. (2010), it is not necessarily the case that \(\underline{{\mathcal {F}}}^S\) and \(\underline{{\mathcal {F}}}^W\) coincide, see Tsirelson’s counterexample in e.g. Rogers and Williams (1987).

  10. For further details see e.g. Karatzas and Shreve (1998).

  11. A slight modification is needed in the proof of Lemma 4.9 compared to the analogous result in Karatzas and Shreve (1998). Instead of referring to the standard martingale representation theorem one must refer to a more general result, such as Proposition 2.5.

  12. Corollary 2.11 is a more formal version of this reasoning.

  13. \(\int _0^T||\sigma _t^{-1}(S)\hat{\alpha }_t||^2{ dt} < \infty \) a.s. can be shown to follow from Assumption 5.1.

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Authors and Affiliations

  1. Department of Finance, Stockholm School of Economics, Box 6501, 113 83, Stockholm, Sweden

    Kristoffer Lindensjö

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  1. Kristoffer Lindensjö
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Correspondence to Kristoffer Lindensjö.

Additional information

Financial support from the Tom Hedelius and Jan Wallander Foundation and The Fund Carl Silfvén is gratefully acknowledged. The author is also grateful to Professor Tomas Björk for helpful comments on this paper.

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Lindensjö, K. Optimal investment and consumption under partial information. Math Meth Oper Res 83, 87–107 (2016). https://doi.org/10.1007/s00186-015-0521-1

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