Optimal portfolios for different anticipating integrals under insider information

We consider the non-adapted version of a simple problem of portfolio optimization in a financial market that results from the presence of insider information. We analyze it via anticipating stochastic calculus and compare the results obtained by means of the Russo-Vallois forward, the Ayed-Kuo, and the Hitsuda-Skorokhod integrals. We compute the optimal portfolio for each of these cases. Our results give a partial indication that, while the forward integral yields a portfolio that is financially meaningful, the Ayed-Kuo and the Hitsuda-Skorokhod integrals do not provide an appropriate investment strategy for this problem.


Introduction
Many mathematical models in the applied sciences are expressed in terms of stochastic differential equations such as dx dt = a(x, t) + b(x, t) ξ(t), (1.1) where ξ(t) is a "white noise". Of course, this equation cannot be understood in the sense of the classical differential calculus of Leibniz and Newton. Instead, the use of stochastic calculus provides a precise meaning to these models. However, the choice of a particular notion of stochastic integration has generated a debate that has expanded along decades [24,27]. This debate, which has been particularly intense in the physics literature, has been focused mainly on the choice between the Itô integral [19,20], which leads to the notation dx = a(x, t) dt + b(x, t) dB(t), and the Stratonovich integral [29] usually denoted as where B(t) is a Brownian motion. Also, quite often in the physics literature, one finds different meanings associated to equation (1.1) [24]. The use of different notions of stochastic integration leads, in general, to different dynamics and, when they exist, to different stationary probability distributions [17]; but it may also lead to different numbers of solutions [7,14].
Since the preeminent place for this debate on the noise interpretation has been the physics literature, the focus has been put on diffusion processes, perhaps due to the influence of the seminal works by Einstein [12] and Langevin [22]. Herein we move out of even the Markovian setting and, following the steps of [3,13], we concentrate on stochastic differential equations with non-adapted terms. Nonadaptedness arises in financial markets concomitantly to the presence of insider traders. Let us exemplify this with a model that is composed by two assets, the first of which is free of risk such as a bank account and the second one is risky such a stock This model is characterized by the set of positive parameters {M 0 , M 1 , ρ, µ, σ}, each one of which has an economic significance: • M 0 is the initial wealth to be invested in the bank account, • M 1 is the initial wealth to be invested in the stock, • ρ is the interest rate of the bank account, • µ is the appreciation rate of the stock, • σ is the volatility of the stock. Therefore the total initial wealth is M = M 0 + M 1 . Moreover we assume the inequality µ > ρ that imposes the higher expected return of the risky investment. We allow only buy-and-hold strategies in which the trader divides the fixed initial amount M between the two assets; that is, long only strategies are allowed. The total wealth at time t is This optimization problem is simple enough to allow for an analytical approach to the extension we will consider herein. In particular, we will permit random and non-adapted initial conditions that will model the knowledge of an insider trader. Under these conditions, we will derive the optimal portfolio for different notions of stochastic integration. The precise problem in presented in the next section.

Insider trading
We consider a financial market in which an insider trader, who possesses information on the future price of a stock, is present. Precisely we assume the trader knows the value S 1 (T ) at time t = 0, what we will implement mathematically as if she knew the value B(T ). Moreover we only consider buy-and-hold strategies for which shorting is not allowed. Mathematically this translates into her control of the anticipating initial condition f (B(T )), where f ∈ L ∞ (R) and 0 ≤ f ≤ 1, for the stock. Therefore, in our two assets market, we find the following two equations that model the insider wealth: and This system of equations, as written in (2.2a) and (2.2b), is ill-posed. While problem (2.1a) and (2.1b) could still be considered a random differential equation, the non-adaptedness of initial condition (2.2b) implies that equation (2.2a) is ill-posed as an Itô stochastic differential equation. There is a way out of this pitfall that consists in changing the notion of stochastic integration from the Itô integral to one of its generalizations that admit non-adapted integrands. The following sections analyze three different possibilities: the Russo-Vallois forward, the Ayed-Kuo and the Hitsuda-Skorokhod stochastic integrals. We compute the optimal investment strategy provided by each of these integrals, and subsequently we compare them. We will show that, while any of these anticipating stochastic integrals guarantees the well-posedness of the problem at hand, the financial consequences of each choice might be very different. The aim of this work is to clarify the suitability of the use of each of these stochastic integrals in this particular portfolio optimization. We anticipate that the forward integral is the only one among these that provides an optimal investment strategy that takes advantage of the anticipating condition in the financial sense. Therefore our analysis further supports the use of this integral in a financial context as employed, for instance, in [4,8,9,10,11,23,25].

The Russo-Vallois integral
The Russo-Vallois forward integral was introduced by F. Russo and P. Vallois in 1993 in [26]. This stochastic integral generalizes the Itô one, in the sense that it allows to integrate anticipating processes, but it produces the same results as the latter when the integrand is adapted [8].
in probability. In this case, I(t) is the forward integral of ϕ(t) with respect to B(t) on [a, b] and we denote When the choice is the Russo-Vallois integral, we face the initial value problem where d − denotes the Russo-Vallois forward stochastic differential, and f was defined in the previous section. It follows directly from the results in [8] that this problem possesses a unique solution. In the next statement, we establish the optimal investment strategy for the insider trader under Russo-Vallois forward integration.
The optimal investment strategy under Russo-Vallois integration is Proof. The Russo-Vallois integral preserves Itô calculus [8] so using this classical stochastic calculus it is possible to solve problem (3.1a)-(3.1b) explicitly to find Our goal is to find a strategy f such that where we have used that B(T ) ∼ N (0, T ). Now consider Hence, we get The sign of this integrand is determined by the value of x, and it possesses a unique root x c at The inequality x > x c corresponds to and to the positivity of the integrand. Clearly, in this interval we should take f as large as possible in order to maximize E M RV (T ) . On the other hand, if x < x c then the integrand is negative and we should take f as small as possible for the same reason. This, together with the assumption 0 ≤ f ≤ 1, renders the optimal investment strategy Remark 3.3. The function f from Theorem 3.2 implies that the trader should invest the whole amount M in the bank account or the stock according to the value of B(T ), in particular, in the asset which actual value is larger at the maturity time t = T (something that is known by the insider). Hence, this investment strategy not only maximizes the expected value E M RV (T ) , but it does also take advantage of the anticipating condition in an intuitive way. Thus, the Russo-Vallois integral works as one would expect from the financial point of view, at least for this formulation of the insider trading problem.
Remark 3.4. The expected wealth of the Russo-Vallois insider under this optimal strategy was computed in [13] and reads is the cumulative distribution function of the standard normal distribution. In the same reference it was proven that a fact that matches well with what one expects from the financial viewpoint.

The Ayed-Kuo integral
The Ayed-Kuo integral was introduced in [1,2]. As the previous theory, it generalizes the Itô integral to anticipating integrands. Let us now consider a Brownian motion {B(t), t ≥ 0} and a filtration {F t , t ≥ 0} such that: For the Ayed-Kuo integral, the integrand is assumed to be a product of an adapted stochastic process with respect to the filtration F t and an instantly independent stochastic process. Next, we recall its definition as given in [1].  For the Ayed-Kuo integral we denote the initial value problem as where d * denotes the Ayed-Kuo stochastic differential, and f is as before. This anticipating stochastic differential equation, as the previous case, has a unique and explicitly computable solution at least when f ∈ C(R) [18]. Now, we establish the optimal investment strategy for the insider under Ayed-Kuo integration. For this we assume as before the no-shorting condition 0 ≤ f ≤ 1 along with the regularity constraint f ∈ C(R) for technical reasons. Proof. Using the Itô formula for the Ayed-Kuo integral [18] it is possible to solve problem (4.1a)-(4.1b) explicitly to find Our aim is to find the strategy f such that E [M (T )] is maximized. Hence, we have where we have used that B(T ) ∼ N (0, T ). By changing variables where the first change is implemented in the second integral only, we find Therefore, we may summarize this as By assumption, f is a function of B(T ) such that 0 ≤ f ≤ 1, and µ > ρ; so we have a convex linear combination and, since the exponential function is strictly monotone, we Equivalently we have and consequently f (B(T )) = 1. in this case as well. Let us argue why. We start considering {f n } ∞ n=1 , a sequence of functions such that f n ∈ C(R) and 0 ≤ f n ≤ 1 for all n ∈ N. Then, we have By Lusin Theorem [21], there exists such a family {f n } n∈N so that as n → ∞.
In consequence this means that we can approximate any strategy f ∈ L ∞ (R) by a sequence of strategies f n ∈ C(R). This in turn suggests that the strategy f found in Theorem 4.3 does not change if we look for the optimal investment allocation in the bigger space L ∞ (R) C(R). A complete proof would require the solution of (4.1a)-(4.1b) under the more general condition f ∈ L ∞ (R).
Remark 4.5. The function f from Theorem 4.3, and also from Remark 4.4, suggests that the formalization of the problem based on the Ayed-Kuo integral does not take advantage of the anticipating initial condition, as this optimal investment strategy is the same as the one of the honest trader. Indeed, it implies to invest the whole amount M in the stock, in such a way that Thus, the result of the use of the Ayed-Kuo integral seems to be counterintuitive in the financial sense, at least for this formulation of the insider trading problem.

The Hitsuda-Skorokhod integral
The Hitsuda-Skorokhod integral is an anticipating stochastic integral that was introduced by Hitsuda [15] and Skorokhod [28] by means of different methods. The following definition makes use of the Wiener-Itô chaos expansion; background on this topic can be found for instance in [8,16].
× Ω) be a square integrable stochastic process. By the Wiener-Itô chaos expansion, X can be decomposed into an orthogonal series in L 2 (Ω), where f n,t ∈ L 2 ([0, T ] n ) are symmetric functions for all non-negative integers n. Thus, we write f n,t (t 1 , . . . , t n ) = f n (t 1 , . . . , t n , t), which is a function defined on [0, T ] n+1 and symmetric with respect to the first n variables. The symmetrization of f n (t 1 , . . . , t n , t n+1 ) is given bŷ because we only need to take into account the permutations which exchange the last variable with any other one. Then, the Hitsuda-Skorokhod integral of X is defined by provided that the series converges in L 2 (Ω).
For the Hitsuda-Skorokhod integral we arrive at the initial value problem for a Hitsuda-Skorokhod stochastic differential equation, where δ denotes the Hitsuda-Skorokhod stochastic differential. The existence and uniqueness theory for linear stochastic differential equations of Hitsuda-Skorokhod type, which covers the present case, can be found in [6]; as before, f is a function of B(T ), such that f ∈ L ∞ (R) and 0 ≤ f ≤ 1. In this case we have the following result.
Theorem 5.2. The unique solution of (5.1a) and (5.1b) is Proof. The statement is a particular case of the existence and uniqueness result in [5], which in turn states that the solution to stochastic differential equations of the form × Ω), and η ∈ L p (Ω), p > 2 is given by and, for 0 ≤ s ≤ t ≤ T , the Girsanov transformation is Now taking σ s = σ and µ s = µ as two constant processes, and η = M f (B(T )), the different factors in (5.2) become the result follows. Proof. The statement is a direct consequence of Theorem 5.2 and the proof of Theorem 4.3.
Remark 5.4. Note that this result is the same as the one found in the previous section for the Ayed-Kuo integral but in the original general case f ∈ L ∞ (R) (so in particular Remark 4.4 is not necessary here). Therefore the same financial conclusions hold in this case as well.

Conclusions
In this work we have considered a version of insider trading that allows us to compare the optimal investment strategies provided by three different anticipating stochastic integrals: the Russo-Vallois forward, the Ayed-Kuo and the Hitsuda-Skorokhod integrals.
Specifically, we have considered the following formulation of insider trading for the bank account f (B(T ))) , and for the stockd where the anticipating initial condition f is a function of B(T ), such that f ∈ L ∞ (R) and 0 ≤ f ≤ 1, andd ∈ {d − , d * , δ} is a stochastic differential of one of the types under consideration. That is, we have only considered buy-and-hold strategies for which shorting is not allowed.
Our main task has been to establish the optimal investment strategy for each of the anticipating integrals. When the choice is the Russo-Vallois integral, the investment strategy that maximizes the expected wealth is to invest the whole amount M in the asset which actual value is larger at maturity time. Thus, For the Ayed-Kuo (in this case under the stronger condition f ∈ C(R), but see Remark 4.4) and the Hitsuda-Skorokhod integrals, the optimal investment strategy is the same as the one in the Itô case (that is, the uninformed case), which implies to invest the whole amount M in the stock. Hence, f (B(T )) = 1.
These results suggest that the Russo-Vallois integral works as one would expect from a financial perspective, while the Ayed-Kuo and Hitsuda-Skorokhod integrals provide a solution that seems to be counterintuitive in the financial sense. Indeed, from our present results along with those in [13] it follows that where we have chosen the optimal investment allocation in each case. If we selected the optimal Russo-Vallois strategy for all the three anticipating cases the situation becomes even worse, see [13]; and it could be even more critical if we allowed different strategies, see [3]. All in all, our results suggest that while the Russo-Vallois forward integral allows the insider to make full use of the privileged information, both the Ayed-Kuo and Hitsuda-Skorokhod integrals effectively transform the insider into an uninformed trader.