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

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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.

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References

  1. Ancona, A.: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst. Fourier 28, 169–213 (1978)

    Article  MathSciNet  Google Scholar 

  2. Berrier, F.P.Y.S., Rogers, L.C.G., Tehranchi, M.R.: A characterization of forward utility functions. Working paper (2009). http://www.statslab.cam.ac.uk/~mike/papers/forward-utilities.pdf

  3. Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (2012)

    MATH  Google Scholar 

  4. Bogachev, V.I.: Measure Theory, Vol. II. Springer, Berlin (2007)

    Book  Google Scholar 

  5. Duffie, D.: Dynamic Asset Pricing Theory, 3rd edn. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

  6. El Karoui, N., Mrad, M.: An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE. SIAM J. Financ. Math. 4, 697–736 (2013)

    Article  MathSciNet  Google Scholar 

  7. El Karoui, N., Mrad, M.: Stochastic utilities with a given optimal portfolio: approach by stochastic flows. Working paper (2013). arXiv:1004.5192

  8. Ferus, D.: Analysis III (2008). http://page.math.tu-berlin.de/~ferus/ANA/Ana3.pdf

    Google Scholar 

  9. Filipović, D., Mayerhofer, E.: Affine diffusion processes: theory and applications. In: Albrecher, H., et al. (eds.) Advanced Financial Modelling. Radon Series Comp. Appl. Math., vol. 8, pp. 125–164. Walter de Gruyter, Berlin (2009)

    Google Scholar 

  10. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  11. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin/New York (1977)

    Book  Google Scholar 

  12. Henderson, V., Hobson, D.: Horizon-unbiased utility functions. Stoch. Process. Appl. 117, 1621–1641 (2007)

    Article  MathSciNet  Google Scholar 

  13. Källblad, S., Obłój, J., Zariphopoulou, T.: Dynamically consistent investment under model uncertainty: the robust forward criteria. Finance Stoch. 22, 879–918 (2018)

    Article  MathSciNet  Google Scholar 

  14. Karatzas, I.: Lectures on the Mathematics of Finance. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  15. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)

    MATH  Google Scholar 

  16. Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, New York (1998)

    Book  Google Scholar 

  17. Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1968)

    Book  Google Scholar 

  18. Liang, G., Zariphopoulou, T.: Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE. SIAM J. Financ. Math. 8, 344–372 (2017)

    Article  MathSciNet  Google Scholar 

  19. Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)

    Article  Google Scholar 

  20. Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373–413 (1971)

    Article  MathSciNet  Google Scholar 

  21. Murata, M.: Structure of positive solutions to \((-\delta + v) u = 0\) in \(\mathbb{R}^{n}\). Duke Math. J. 53, 869–943 (1986)

    Article  MathSciNet  Google Scholar 

  22. Musiela, M., Zariphopoulou, T.: Investments and forward utilities. Preprint (2006). ma.utexas.edu/users/zariphop/pdfs/TZ-TechnicalReport-4.pdf

  23. Musiela, M., Zariphopoulou, T.: Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model. In: Fu, M.C., et al. (eds.) Advances in Mathematical Finance. Applied and Numerical Harmonic Analysis, pp. 303–334. Birkhäuser, Boston (2007)

    Chapter  Google Scholar 

  24. Musiela, M., Zariphopoulou, T.: Portfolio choice under dynamic investment performance criteria. Quant. Finance 9, 161–170 (2010)

    Article  MathSciNet  Google Scholar 

  25. Musiela, M., Zariphopoulou, T.: Portfolio choice under space-time monotone performance criteria. SIAM J. Financ. Math. 1, 326–365 (2010)

    Article  MathSciNet  Google Scholar 

  26. Musiela, M., Zariphopoulou, T.: Stochastic partial differential equations and portfolio choice. In: Chiarella, C., Novikov, A. (eds.) Contemporary Quantitative Finance, Essays in Honour of Eckhard Platen, pp. 195–216. Springer, Berlin (2010)

    Chapter  Google Scholar 

  27. Nadtochiy, S., Tehranchi, M.: Optimal investment for all time horizons and Martin boundary of space-time diffusions. Math. Finance 27, 438–470 (2015)

    Article  MathSciNet  Google Scholar 

  28. Nadtochiy, S., Zariphopoulou, T.: A class of homothetic forward investment performance processes with non-zero volatility. In: Kabanov, Y., et al. (eds.) Inspired by Finance. The Musiela Festschrift, pp. 475–504. Springer, Cham (2014)

    Chapter  Google Scholar 

  29. Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  30. Sheu, S-J.: Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann. Probab. 19, 538–561 (1991)

    Article  MathSciNet  Google Scholar 

  31. Shkolnikov, M., Sircar, R., Zariphopoulou, T.: Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations. SIAM J. Financ. Math. 7, 588–618 (2016)

    Article  MathSciNet  Google Scholar 

  32. Widder, D.V.: The role of the Appell transformation in the theory of heat conduction. Trans. Am. Math. Soc. 109, 121–134 (1963)

    Article  MathSciNet  Google Scholar 

  33. Zariphopoulou, T.: A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 61–82 (2001)

    Article  MathSciNet  Google Scholar 

  34. Zitkovič, G.: A dual characterization of self-generation and exponential forward performances. Ann. Appl. Probab. 19, 2176–2210 (2009)

    Article  MathSciNet  Google Scholar 

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Correspondence to Mykhaylo Shkolnikov.

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M. Shkolnikov was partially supported by the NSF grant DMS-1506290. L. Avanesyan was partially supported by a Gordon Y.S. Wu Fellowship in Engineering.

Appendix A

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. □

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Avanesyan, L., Shkolnikov, M. & Sircar, R. Construction of a class of forward performance processes in stochastic factor models, and an extension of Widder’s theorem. Finance Stoch 24, 981–1011 (2020). https://doi.org/10.1007/s00780-020-00436-1

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