This paper provides a new version of the condition of Di Nunno et al. (2003), Ankirchner and Imkeller (2005) and Biagini and \{O}ksendal (2005) ensuring the semimartingale property for a large class of continuous stochastic processes. Unlike our predecessors, we base our modeling framework on the concept of portfolio proportions which yields a short self-contained proof of the main theorem, as well as a counterexample, showing that analogues of our results do not hold in the discontinuous setting.
Deep Dive into On the semimartingale property via bounded logarithmic utility.
This paper provides a new version of the condition of Di Nunno et al. (2003), Ankirchner and Imkeller (2005) and Biagini and {O}ksendal (2005) ensuring the semimartingale property for a large class of continuous stochastic processes. Unlike our predecessors, we base our modeling framework on the concept of portfolio proportions which yields a short self-contained proof of the main theorem, as well as a counterexample, showing that analogues of our results do not hold in the discontinuous setting.
In [5] the connection between the concept of no-arbitrage and the assumption that the financial assets are driven by semimartingales is initiated. Here it is shown there that if the financial market satisfies the condition of no free lunch with vanishing risk for simple trading strategies, then the traded securities allow for a semimartingale decomposition. This and similar results depend heavily on the mathematical constructs -the theory of stochastic integration employed, and the class of the integrands used -to describe the economic concept of no-arbitrage. [2] illustrates possible pitfalls resulting from an attempt of economic interpretation of mathematical results based on an integration theory at odds with the financial intuition.
The main result of the present paper is inspired by [3] and [6], and states (loosely) that a continuous process with finite quadratic variation is a semimartingale if the expected utility of a logarithmic investor is uniformly bounded from above over a specific natural class of trading strategies. Unlike [3] and [6], we do not replace the Itô integration with the anticipative forward integration, and we do not assume the existence of a trading strategy that achieves the optimal expected logarithmic utility. In fact, the existence of such a strategy is one of the conclusions of our main theorem.
The recent and independent paper [1] develops an idea similar to ours and relates the semimartingality of the stock-price process to the boundedness of the expected utility. The authors base their approach on simple buy-and-hold strategies thereby circumventing the lack of a stochastic integration theory. Our approach is different -it hinges on the observation that a canonical integration theory can be based on simple portfolio proportions, without falling into the traps described in [2]. Indeed, while one of the main results of [1] is that bounded utility implies semimartingality, regardless of the continuity properties of the process under scrutiny, our epilogue is different. We construct an example of a discontinuous non-semimartingale S with the property that the expected logarithmic utility is uniformly bounded over all strategies in which portfolio proportions are simple processes. Moreover, there exists a shrinkage of the original filtration under which the process S is a semimartingale. The existence of such a counterexample poses the following question: Can we describe (and work with) a class of stochastic processes, strictly larger than the class of semimartingales, for which the logarithmic investors will not be able to achieve arbitrarily large expected utilities? While the non-semimartingales in this class will surely admit free lunch with vanishing risk, the possibiliy of their use in financial modeling is not ruled out. Indeed, the logarithmic investors will not demand unlimited quantities of such securities. We leave this question for further research.
In the continuous case, the flavor of our results agrees with [1], but our approach provides new insights in several respects. First, the proof of our main theorem is short and self-contained, and uses a simple Hilbert-space argument. As a consequence of this, we are able to explicitly derive the semimartingale decomposition of the stock price in terms of the Riesz representation of a suitably defined linear functional. The proof of the related result in [1] is based on the already mentioned result of [5], and provides only the abstract existence of the semimartingale decomposition. Second, a byproduct of our analysis is the existence of the optimal trading strategy for an investor with logarithmic utility -the growth-optimal portfolio.
The paper is structured as follows: In the first section we describe the framework and prove our main result. The second section provides a counterexample which illustrates the fact that, when jumps are present, bounded logarithmic utility on simple portfolio proportions is not sufficient to grant the semimartingality of the price process.
As all our stochastic processes are defined on the time horizon [0, 1], we will consistently use the shorthand S for the process (S t ) t∈[0,1] , throughout the paper.
We consider a continuous stochastic process S, defined on the unit time horizon [0, 1], and adapted to a complete and right-continuous filtration F (F t ) t∈[0,1] , on some probability space (Ω, F , P). We assume that S has finite quadratic variation on [0, 1], meaning for ω ∈ Ω the following limit exists
and is finite. In that case, the process S defined by
is finite valued, non-decreasing and continuous. Of course, the sequence {0, 1 n , . . . , n-1 n , 1} of partitions in ( 2) is chosen for simplicity; any other sequence with comparable properties would lead to the same conclusions.
Remark 1 Arguably the most natural way to ensure the existence of the quadratic variation, as defined in (1), is to assume the existence of a filtration F ′ (F ′ t ) t∈[0,1] smaller than F (i.e., F
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