We study a mean-variance investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. We construct a mean-variance efficient portfolio through the solutions of backward stochastic differential equations. We also give sufficient conditions under which an explicit analytic expression is available for the mean-variance optimal wealth of the investor. problem using the Hamilton-Jacobi-Bellman equation of dynamic programming. About 15 years later, the equivalent martingale measure concept developed by Harrison and Pliska [7] paved the way for solving several utility maximization problems in a market model with stochastic uniformly bounded parameters (see, for example, Karatzas [8]). Unfortunately one cannot adequately address the continuous-time mean-variance problem using one of the above techniques. This may explains in part why significant results were obtained only recently. Alternate methods had to be considered. For example, Duffie and Richardson [9] solved a deterministic market model mean-variance problem via orthogonal projection techniques. In the early 1990s Pardoux and Peng [10] introduced the notion of backward stochastic differential equations (BSDE) which provided the ideal tool for Lim and Zhou [11] to adequately solve the continuous-time mean-variance allocation problem in a Black-Scholes model with stochastic uniformly bounded market coefficients. It is worth mentioning that BSDE theory was also proved to be valuable in utility portfolio selection problems as shown in [12].Unfortunately, in these papers, the uniform boundedness hypothesis assumed for the interest rate process precludes the use of interest models such as Vasicek, Hull-White and Cox-Ingersoll-Ross (CIR) models which are highly valued by practitioners. In order to pass this limit a number of researchers drew on a more general market where in addition to the usual bank account and stocks an individual is allowed to invest part of his wealth in bonds or interest derivatives. Bajeux-Besnainou and Portait [13] incorporated a Vasicek interest rate model and zero-coupons bonds in their model and solved a mean-variance-type problem. But the Vasicek as well as the Hull-White model have a major drawback which is that there is a positive probability that one encounters negative values. Obviously, this is a highly unrealistic and undesirable feature in a real-world financial market. A few years later, Deelstra et al. [14] modified Bajeux-Besnainou and Portait's model by rather opting for a CIR interest rate process which is known to be almost surely positive under mild conditions. Applying essentially the martingale approach they obtained an optimal strategy for a power-utility maximization problem. Now the classical CIR process rely on three constant parameters, thus, from a practical point of view statistical fitting to real financial data may be greatly improved by considering a more flexible time-varying parameter model as shown in Maghsoodi's [15] study of 25 years of U.S. Treasury bill data....

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In this paper, we establish closed-form formulas for key probabilistic properties of the cone-constrained optimal mean-variance strategy, in a continuous market model driven by a multidimensional Brownian motion and deterministic coefficients. In particular, we compute the probability to obtain to a point, during the investment horizon, where the accumulated wealth is large enough to be fully reinvested in the money market, and safely grow there to meet the investor's financial goal at terminal time. We conclude that the result of Li and Zhou [Ann. Appl. Prob., v.16, pp.1751-1763, (2006] in the unconstrained case carries over when conic constraints are present: the former probability is lower bounded by 80% no matter the market coefficients, trading constraints, and investment goal. We also compute the expected terminal wealth given that the investor's goal is underachieved, for both the mean-variance strategy and the aforementioned hybrid strategy where transfer to the money market occurs if it allows to safely achieve the goal. The former probabilities and expectations are also provided in the case where all risky assets held are liquidated if financial distress is encountered. These results provide investors with novel practical tools to support portfolio decision-making and analysis.Appl. Stochastic Models Bus. Ind. 2014, 30 544-572 C. LABBÉ AND F. WATIER Li and Zhou [7] compute this probability and establish the following astonishing result that they call the 80% rule: the chances of achieving the goal when using the switch-when-safe strategy are at least 80% (whereas they might be as low as 50% with the optimal mean-variance strategy). The 80% lower bound is universal, in the sense that it holds no matter the market coefficients, the length of the investment horizon, or the target.A very natural question is whether the remarkable 80% rule carries over, for example, when there are trading constraints and/or within the frame of an incomplete market. Scott and Watier [8] provide an affirmative answer when short-selling of stocks is prohibited (in a complete market). In this paper, we establish that it actually extends to more general portfolio constraints that encompass short-selling prohibition and incomplete markets. Precisely, the vector whose components are the amounts invested in each of the n stocks must remain within a given closed convex conic subset of R n . As it is of interest for an investor to know what to expect if the goal were unachieved, we take one further step and study the expected value of terminal wealth in that event.Another drawback of the mean-variance strategy (in its classical formulation) is that it does not preclude the wealth from becoming negative, or even arbitrarily small. Therefore, unless an investor is willing (or can afford) to go deep into debt, then the strategy, with or without the switching feature, may be impossible to implement in practice. In order to account for this kind of risk, let us say that the investor falls in financial distress if at some point his wealth ...

<p class="MsoNormal"> <span lang="EN-US">We establish, through solving semi-infinite programming problems, bounds on the probability of safely reaching a de</span><span lang="EN-US">sired level of wealth on a finite horizon, when an investor starts with an optimal mean-variance financial investment strategy under a non-negative wealth restriction.</span> </p>