Construction of Forward Performance Processes in Stochastic Factor Models and an Extension of Widder’s Theorem
Levon Avanesyan\@footnotetext2010 Mathematics Subject Classification. Primary: 35K55, 91G10; secondary: 35J15, 60H10.\@footnotetextKey words and phrases. Factor models, forward performance processes, generalized Widder’s theorem, Hamilton-Jacobi-Bellman equations, ill-posed partial differential equations, incomplete markets, Merton problem, optimal portfolio selection, positive eigenfunctions, time-consistency. Mykhaylo Shkolnikov111M. Shkolnikov was partially supported by the NSF grant DMS-1506290. Ronnie Sircar
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 with power-utility initial data, as well as the corresponding optimal portfolios. This is done by solving the associated non-linear parabolic partial differential equations (PDEs) posed in the “wrong” time direction, for stock-factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. Along the way we establish on domains an explicit form of the generalized Widder’s theorem of Nadtochiy and Tehranchi [nadtochiy2015optimal, Theorem 3.12] and rely hereby on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the “right” time direction.
In this paper we study the optimal portfolio selection problem under forward investment criteria in incomplete markets, specifically stochastic factor models. Our setup is that of a continuous-time market model with multiple stocks whose growth rates and volatilities are functions of multiple observable stochastic factors following jointly a diffusion process. The incompleteness arises hereby from the imperfect correlation between the Brownian motions driving the stock prices and the factors. The factors themselves can model various market inputs, including stochastic interest rates, stochastic volatility and major macroeconomic indicators, such as inflation, GDP growth or the unemployment rate.
The optimal portfolio problem in continuous time was originally considered by Merton in his pioneering work [merton1969lifetime], [merton1971optimum], and is commonly referred to as the Merton problem. In this framework an investor looks to maximize her expected terminal utility from wealth acquired in the investment process within a geometric Brownian motion market model. Good compilations of classical results can be found in the books [duffie2010dynamic], [karatzas1998methods]. As fundamental as this setup is, it has two important drawbacks. First, the investor must decide on her terminal utility function before entering the market, and thereby cannot adapt it to changes in market conditions. Second, before settling on an investment strategy, the investor must firmly set her time horizon. That is, the portfolio derived in this framework is optimal only for one specific utility function over one time horizon.
External factors such as the economic cycle, natural disasters, and the political climate can lead to dynamic changes in one’s level of risk aversion. This would change the terminal utility function, thereby affecting the optimal portfolio allocation. Even if the terminal utility function stays the same, the investor might decide to exit the market at an earlier or a later time than originally planned. For two investment horizons there is no natural relation between the two respective optimal portfolios. Thus, if the investor initially decided to stay in the market until time , but later on decided to continue the investment activities until time , she would have to either incur significant transaction costs to rebalance her portfolio, or continue investing at a suboptimal level of expected utility from terminal wealth. In both cases she would regret her past decisions, thereby making the classical approach terminal time inconsistent. We call performance criteria terminal time consistent if the optimal dynamic portfolio on the time interval restricted to the interval yields the optimal dynamic portfolio on the time interval . Finding such criteria is essential in solving portfolio optimization problems with an uncertain investment horizon. For this purpose forward investment performance criteria were introduced and developed in [musiela2006investments] and [musiela2007investment], as well as in [HENDERSON20071621].
Instead of looking to optimize the expectation of a deterministic utility function at a single terminal point in time, this approach looks to maximize the expectation of a stochastic utility function at every single point in time. Forward performance processes (FPPs) capture the time evolutions of such stochastic utility functions. They are increasing and strictly concave in the wealth argument, intrinsically incorporate the randomness stemming from the market, and most importantly yield terminal time-consistent investment strategies. Other than completely specifying the market and the factors that affect it, the only piece of information a portfolio manager needs is the investor’s initial utility function. The portfolio manager can infer the shape of this function (or, equivalently, the level of risk aversion) by observing the return targets and the error bounds around them set by the investor.
A comprehensive description of all FPPs remains a challenging open problem. Much work towards this goal has been carried out throughout the last ten years, see [berrier2009characterization], [nicole2013exact], [nicole2013stochastic], [HENDERSON20071621], [musiela2010stochastic], and [zitkovic2009] for some important results. In [musiela2010stochastic], Musiela and Zariphopoulou proposed a construction of FPPs by means of solutions to a stochastic partial differential equation (SPDE). The SPDE can be thought of as the forward stochastic analogue of the Hamilton-Jacobi-Bellman (HJB) equation that arises in the optimization of the expected utility from terminal wealth. Every classical solution of this SPDE which is increasing and strictly concave in the wealth argument is a local FPP, but no existence theory for such SPDEs is available, and additional conditions (to be checked on a case-by-case basis) are needed to ensure that the local FPP is a true FPP. The key novelty and difficulty in dealing with this SPDE is the introduction of the forward volatility process. It reflects the investor’s uncertainty about her preferences in the future and is subject to her choice. To find all the FPPs characterized by the SPDE, one would have to find all forward volatility processes, along with initial utility functions, for which the SPDE has a classical solution. The case of zero forward volatility yields time-monotone FPPs, and was extensively discussed in [musiela2010portfolio2] and [musiela2010portfolio]. In [nicole2013exact] and [nicole2013stochastic], El Karoui and M’rad find a functional representation of the forward volatility for which, given an initial utility function and a wealth process satisfying certain regularity conditions, the SPDE has a classical solution. Moreover, if the solution is a true FPP, it renders the chosen wealth process optimal. This is an important result, as it helps to infer investors’ performance criteria from the portfolios they pick in a given market. Here, we are concerned with the complementary problem of constructing an FPP and an associated optimal portfolio for an investor entering a new market equipped with her initial utility function.
We consider factor-driven market models and FPPs into which the randomness enters only through the underlying stochastic factors. Assuming such a form, with a compatible forward volatility process, the SPDE mentioned above reduces to an HJB equation set in the “wrong” time direction. We will call its classical solutions factor-form local FPPs if they are increasing and strictly concave in the wealth argument. In a complete market one can use the Fenchel-Legendre transform to linearize the HJB equation, and arrive at a linear second-order parabolic PDE set in the “wrong” time direction (see [nadtochiy2015optimal]). In an incomplete market no such linearizing transformation is available in general. To the best of our knowledge, the only exception is the special case of power utility in a one-factor market model, where a linearization is possible through a distortion transformation, as discovered in [zariphopoulou2001solution] for the Merton problem, and used for the construction of FPPs in [nadtochiy2015optimal], [nadtochiy2014class], and [shkolnikov2015asymptotic]. We show that for a multiple factor market model with a special stock-factor correlation matrix structure (see Assumption 2.6 below) the distortion transformation still simplifies the HJB equation to a linear second-order parabolic equation set in the “wrong” time direction.
Motivated by such a simplification in one-factor market models, Nadtochiy and Tehranchi [nadtochiy2015optimal, Theorem 3.12] exhibited a characterization of all positive solutions to such linear parabolic equations. Their theorem constitutes a generalization of the celebrated Widder’s theorem (see [widder1963Appel]), which describes all positive solutions of the heat equation set in the “wrong” time direction. The generalized Widder’s theorem reveals that positive solutions of a linear second-order parabolic equation set in the “wrong” time direction must be linear combinations of exponentially scaled positive eigenfunctions for the corresponding elliptic operator according to a positive finite Borel measure. Moreover, each solution is uniquely identified with a pairing of the eigenfunctions and the measure.
In our first main theorem (Theorem 2.11) we give a new version of [nadtochiy2015optimal, Theorem 3.12] on domains in the multiple stocks multiple factor setup with an initial utility function of power type to describe a new class of FPPs. Note that generalized Widder’s theorems do not provide a way to construct the pairings of the eigenfunctions and the measure. Our second set of results (see Theorem 2.14 and Remark 2.15) addresses this issue: in Theorem 2.14 we give the Laplace transform of the measure in terms of the solution to a linear parabolic equation set in the “right” time direction, and we provide a method (see Remark 2.15) of finding the only possible corresponding eigenfunctions as well. Thus, we indeed obtain a large explicit class of FPPs.
The rest of the paper is structured as follows. In Section 2 we state our main results, postponing their proofs to later sections. In Section 3 we introduce relevant facts about FPPs and subsequently prove Theorem 2.11. In Section 4 we show Theorem 2.14, summarize some results from the theory of linear elliptic operators, and use them to establish Propositions 2.18, 2.23 and 2.24. In Section 5 we discuss the Merton problem within the framework of our market model. Lastly, in Section 6 we discuss the meaning of the main assumption in Theorem 2.11 (Assumption 2.6).
2 Main results
Consider an investor with initial capital aiming to invest in a market with stocks, the prices of which follow a process , and a riskless bank account with zero interest rate. The stock prices depend on an observable -dimensional stochastic factor process taking values in , and are driven by a -dimensional standard Brownian motion . The factor process is itself driven by a -dimensional standard Brownian motion , whose correlation with is given by a matrix with singular values in . Without loss of generality we assume that (see [karatzaslectures, Remark 0.2.6]). The investor’s filtration is generated by a pair of processes satisfying
where the superscript denotes transposition and is a -dimensional standard Brownian motion independent of . We write for and for throughout.
For the convenience of the reader we summarize the dimensions of all the quantities we have introduced thus far:
Note that there is no loss of generality in using the representation (2.3) for the standard Brownian motion , since we can let be the square root of the positive semidefinite matrix (recall that the singular values of belong to ), and .
The functions , are continuous, the stochastic differential equation (SDE) (2.2) possesses a unique weak solution, and the columns of belong to the range of left-multiplication by for all .
Under Assumption 2.1 it holds for all , where is the Moore-Penrose pseudoinverse of . Indeed, , so that the columns of (and consequently the vectors in their span, that is, the range of the left-multiplication by ) are invariant under the left-multiplication by . This is true, in particular, if has rank for all .
Our investor dynamically allocates her wealth in the market using a self-financing trading strategy that at any time yields a portfolio allocation among the stocks with the associated wealth process
where is the Sharpe ratio. Apart from the self-financeability, we impose additional conditions on the trading strategies to ensure that their wealth processes are well-defined by (2.4).
An -progressively measurable self-financing trading strategy is called admissible if its portfolio allocation among the stocks fulfills
with probability one. In this case, we write .
Next, we define (local) forward performance processes, which capture how the utility functions of an investor evolve over time as she continues to invest in the financial market above. Part of the definition is an optimality criterion for portfolio allocations that reflects the dynamic programming principle time-consistent optimal portfolio allocations must satisfy.
An -progressively measurable is referred to as a (local) forward performance process (FPP) if
with probability one, all functions , are strictly concave and increasing,
for each , the process , is an (local) supermartingale,
there exists an optimal for which , is an (local) martingale.
2.2 Separable power factor form FPPs in EVE models
We consider (local) FPPs of factor-form into which the randomness enters only through the stochastic factor process, that is,
for a deterministic function . To be able to construct functions such that the corresponding is a (local) FPP in the generality of the setup (2.1), (2.2) we focus on the situation when the initial utility function is of product form and a power function in the wealth variable:
The crucial simplification arising from the structure in (2.7) lies in its propagation to positive times. In fact, we will construct (local) FPPs of the form
where is continuously differentiable in (its first argument) and twice continuously differentiable in (the second argument). We propose to call them separable power factor form (local) FPPs.
We are able to characterize all separable power factor form local FPPs under the next assumption on the correlation matrix .
For some ,
For any orthonormal matrix , we may replace by and by in (2.2) without changing the dynamics of the pair . Since is a -dimensional standard Brownian motion and is diagonal for an appropriate choice of , we could have assumed without loss of generality from the very beginning that is diagonal. Thus, the only true restriction imposed by Assumption 2.6 lies in the equality of the eigenvalues of . We refer to market models that satisfy the condition (2.9) as eigenvalue equality (EVE) models. Note that for EVE models, since is a -matrix, at least one of the following two has to hold true:
2.3 Characterizing the FPPs
In order to describe our construction of separable power factor form FPPs, we need to introduce some quantities related to linear elliptic operators of the second order. Consider on such an operator
under the following assumption.
The operator is locally uniformly elliptic with locally -Hölder continuous and globally bounded coefficients. That is, with and , there exists an such that, for any bounded subdomain of satisfying ,
, where ,
We define the Hölder space as the subspace consisting of functions whose second-order partial derivatives are locally -Hölder continuous (in the same sense as in condition (ii) of Assumption 2.8). Next, we introduce the sets of positive eigenfunctions for the operator , which correspond to eigenvalues , and are normalized at some fixed :
Moreover, we let be the spectrum of associated with positive eigenfunctions:
Finally, we call a functional such that for all , a selection of positive eigenfunctions, and recall the definition of Bochner integrability in this setting.
Given a positive finite Borel measure on , we refer to a selection of positive eigenfunctions as -Bochner integrable if, for all compact , , where .
We are now ready to state our first main result.
For any positive finite Borel measure on and a -Bochner integrable selection of positive eigenfunctions the unique separable power factor form local FPP with the initial condition
is given by
Moreover, any that solves
is an associated optimal portfolio.
Given a function , there exists a local FPP of separable power factor form with the initial condition
if and only if there exists a positive finite Borel measure on and a -Bochner integrable selection of positive eigenfunctions such that
We note that the equation (2.16) for optimal portfolios does not involve the initial wealth . This is a consequence of the local FPP being of separable power factor form. In the setting of the Merton problem, the same statement is true (and well-known) for terminal utility functions of power form.
Part (ii) of Theorem 2.11 shows that, once a portfolio manager has an estimate for an investor’s level of risk-aversion and the functional dependence (encoded by ) of her current utility function on the value of the factor process, he can extrapolate the future values of her utility function according to (2.15) and acquire a portfolio fulfilling (2.16) (e.g. the portfolio in (2.19)) on her behalf, provided is of the form (2.18). It is therefore crucial to understand which functions admit the representation (2.18) and to be able to determine the pairings for such.
2.4 Finding selections of positive eigenfunctions and measures
The next set of results addresses the problem of solving the equation (2.18) for the pairing , when it exists. The equation (2.18) stems from a further generalization of the generalized Widder’s theorem of Nadtochiy and Tehranchi [nadtochiy2015optimal, Theorem 3.12] (see Theorem 3.4 below) and, thus, our results can be viewed as yielding explicit versions of such theorems. The following theorem is also of independent interest, as it relates the pairing arising in the positive solution of a linear second-order parabolic PDE posed in the “wrong” time direction to the solution of the same PDE posed in the “right” time direction.
Let satisfy Assumption 2.8 and let be a positive function such that
is locally bounded on for the weak solution of the SDE associated with and , where is the first exit time of from . Then, there exists a classical solution to
Moreover, for a positive finite Borel measure on and a -Bochner integrable selection of positive eigenfunctions the function can be expressed as if and only if, for every , the function on is the Laplace transform of the measure , that is,
In this case, it holds, in particular,
Theorem 2.14 reveals that, whenever a pairing exists, it can be inferred by finding the measure through a one-dimensional Laplace inversion of (recall that the values of the Laplace transform on a non-trivial interval determine the underlying positive finite Borel measure, see [billingsley2012probability, Section 30]) and then the functions , from , through additional one-dimensional Laplace inversions.
As a by-product we obtain the following uniqueness result for linear second-order parabolic PDEs posed in the “wrong” time direction by combining the generalized Widder’s theorem on domains (Theorem 3.4 below) with Theorem 2.14 and the uniqueness of the Laplace transform (see [billingsley2012probability, Section 30]).
We stress that Corollary 2.16 is not an immediate consequence of the generalized Widder’s theorem on domains (Theorem 3.4) by itself. The latter does ensure that every pairing corresponds to exactly one positive solution of (2.24). However, it is not clear a priori whether the representation is unique for all functions with the property (2.20). Theorem 2.14 and the uniqueness of the Laplace transform (see [billingsley2012probability, Section 30]) show that this representation is, indeed, unique.
For arbitrary operators relatively little is known about the sets of positive eigenfunctions . Nevertheless, in certain situations additional information on the sets is available and can be exploited to find the selection of positive eigenfunctions for a given function by a finite number of Laplace inversions.
Let satisfy Assumption 2.8, then
If, in addition, the potential is constant and is such that the corresponding solution of the generalized martingale problem on (see [pinsky1995positive, Section 1.13]) is recurrent, then and .
The quantity of (2.25) is commonly referred to as the critical eigenvalue of the operator on .
The structure of the eigenspaces can differ widely depending on the choice of the dimension , the restrictions on the operator , and the domain . The case corresponds to having a single factor and leads to eigenspaces of dimension at most .
Suppose satisfies Assumption 2.8 on a domain . Then, the number of extreme points of the convex set is for all and belongs to for .
Proposition 2.20 reveals that, in the setting of Theorem 2.14 with , one can determine the pairing via a three-step procedure: first, one recovers by a one-dimensional Laplace inversion of ; second, one finds by a one-dimensional Laplace inversion of for an arbitrary ; third, for all , one solves the second-order linear ordinary differential equation for with the obtained boundary conditions at and to end up with the selection .
When , the variability in the dimensionality of the eigenspaces is illustrated by the following two scenarios, in which the eigenspaces have dimensions and , respectively.
A potential on is called principally radially symmetric if
where the functions and are locally integrable to power for some , with being radially symmetric ( for some ), and vanishing outside of a compact set.
Consider a positive with bounded and , as well as an operator on with a locally -Hölder continuous bounded principally symmetric potential . Then, has the property for any such that
where is the unique solution of
In the situation of Proposition 2.23, we must pick as the unique element of . On the other hand, in the case of a multidimensional factor process on a bounded domain with a Lipschitz boundary, the eigenspaces are infinite-dimensional.
Let , be a bounded domain with a Lipschitz boundary and satisfy (i)-(iii) in Assumption 2.8 with instead of and . Then, the convex set has infinitely many extreme points for all .
Thus, one cannot assert that the number of extreme points of is finite in the generality of Assumption 2.8. Therefore, the procedure of Remark 2.15 cannot always be reduced to a finite number of Laplace inversions. In such cases, we propose to determine the selection on a finite number of grid points .
3 Proof of Theorem 2.11 and a new Widder’s theorem
The goal of this section is to prove Theorem 2.11. Recall that we are interested in separable power factor form local FPPs defined in Remark 2.5. We start by focusing on the function and give a sufficient condition for to be a local FPP.
Under Assumption 2.1 let be continuously differentiable in (its first argument) and twice continuously differentiable in and (the second and third arguments). Suppose further that is strictly concave and increasing in and a classical solution of the HJB equation
where is the generator of the factor process . Then, is a local FPP. Moreover, the corresponding optimal portfolio allocations among the stocks are of a feedback form and characterized by
For the former statement, one only needs to repeat the derivation of [shkolnikov2015asymptotic, equation (1.6)] mutatis mutandis and to use (see Remark 2.2). For the latter statement, we apply Itô’s formula to and substitute for to conclude that the drift coefficient of is the negative of
The process of Proposition 3.1 is a true FPP if is a true supermartingale for every and a true martingale for every optimal portfolio allocation of (3.2). In view of Fatou’s lemma, the supermartingale property is fulfilled if is integrable for all and . The martingale property is valid if the diffusion coefficients , of are -square integrable on each .
The HJB equation (3.1) is a fully non-linear PDE and one does not expect to find explicit formulas for its solutions in general. However, for initial conditions of separable power type and under the Assumption 2.6, the HJB equation (3.1) can be linearized.
Let Assumption 2.6 be satisfied, , and . Then, the HJB equation (3.1) with an initial condition , where , has a classical solution in separable power form, , with if and only if there exists a positive solution to the linear PDE problem
posed in the “wrong” time direction. Hereby, is the linear elliptic operator of the second order with the coefficients of (2.13). In that case, the two solutions are related through
Since we are looking for solutions of the HJB equation (3.1) in separable power form, we plug in the ansatz to arrive at
Next, we employ the distortion transformation and get the PDE
equipped with the initial condition . Moreover, the assumed positivity of translates to , so that we can divide both sides of (3.7) by . In addition, we insert the identity of Assumption 2.6 to end up with
The crucial observation is now that the non-linear term in the PDE (3.8) drops out thanks to . Hence, is a positive solution of (3.4). The converse follows by carrying out the transformations we have used in the reverse order. ∎
Proposition 3.3 reduces the task of finding solutions of the HJB equation (3.1) in separable power form to solving the linear PDE problem (3.4) set in the “wrong” time direction. The latter has been studied in [widder1963Appel] with being the Laplace operator on and in [nadtochiy2015optimal] for more general linear second-order elliptic operators on . We establish subsequently a further generalization of [nadtochiy2015optimal, Theorem 3.12] that allows for linear second-order elliptic operators on arbitrary domains .
Under Assumption 2.8 a function is a classical solution of with if and only if it admits the representation
where is a Borel probability measure on and is a -Bochner integrable selection of positive eigenfunctions. In this case, the pairing is uniquely determined by the function .
We can adapt the proof of [nadtochiy2015optimal, Theorem 3.12] to the situation at hand. Consider any subdomain satisfying and . We endow the space of continuous functions on with the topology of uniform convergence on the compact subsets of the sets
Next, we repeat the proofs of [nadtochiy2015optimal, Theorem 3.6, Lemmas 3.7, 3.9, 3.10, and Theorem 3.11] and the necessity part of the proof of [nadtochiy2015optimal, Theorem 3.12], just replacing their by our and the Harnack’s inequality employed therein by the one in [Lieberman, Chapter VII, Corollary 7.42], to deduce that every function as in the statement of the theorem can be expressed as
with a Borel probability measure on and , . This conclusion for a sequence of the described subdomains increasing to and the uniqueness of the Laplace transform (see [billingsley2012probability, Section 30]) imply that (3.11) applies with the same and , for all in the sequence, so that (3.9) and the uniqueness of the pairing readily follow. Conversely, proceeding as in the sufficiency part of the proof of [nadtochiy2015optimal, Theorem 3.12] we find that, for every subdomain as above, the right-hand side of (3.9) is a classical solution of on with . Picking a sequence of subdomains increasing to as before we obtain the sufficiency part of Theorem 3.4. ∎
We now have all the ingredients needed to prove Theorem 2.11.
Proof of Theorem 2.11.
(i). Take a pairing as specified in point (i) of the theorem. By Proposition 3.1 it is enough to provide a classical solution of the HJB equation (3.1) with the properties as in that proposition and satisfying the initial condition
In view of Proposition 3.3, such a function can be constructed by solving
Conversely, for a separable power factor form local FPP and a portfolio allocation , we apply Itô’s formula to and infer from the conditions (ii) and (iii) in Definition 2.4 that the resulting drift coefficient must be non-positive for all and equal to for any maximizer . Equating the maximum of the drift coefficient over all to we end up with the PDE in (3.6) for . Moreover, the proof of Proposition 3.3 reveals that the function associated with via solves the problem (3.4) with . At this point, the identity (2.15) follows from Theorem 3.4. Finally, the characterization (2.16) of the optimal portfolios is a direct consequence of (3.2) and (2.15).
(ii). Arguing as in the second half of the proof of part (i) we deduce that, for any separable power factor form local FPP with the initial condition of (2.17), the function is a classical solution of the problem (3.6). The substitution and Theorem 3.4 show the necessity and sufficiency of the representation (2.18). We conclude as in the second half of the proof of part (i). ∎
4 Proof of Theorem 2.14 and further ramifications
4.1 Proof of Theorem 2.14
We start our analysis of the pairing by establishing Theorem 2.14.
Proof of Theorem 2.14.
Let be a bounded subdomain with a boundary and be a thrice continuously differentiable function with compact support in . Then, by [Lady, Chapter IV, Theorem 5.2] the problem
(posed in the “right” time direction) has a unique classical solution with -Hölder continuous , in the variable, -Hölder continuous , in the variable, and -Hölder continuous in the variable. In particular, obeys the Feynman-Kac formula