Andrew E. B. Lim

Mean-Variance Portfolio Selection with Random Parameters in a Complete Market

This paper concerns the continuous-time, mean-variance portfolio selection problem in a complete market with random interest rate, appreciation rates, and volatility coef.cients. The problem is tackled using the results of stochastic linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), two theories that have been extensively studied and developed in recent years. Specifically, the mean-variance problem is formulated as a linearly constrained stochastic LQ control problem. Solvability of this LQ problem is reduced, in turn, to proving global solvability of a stochastic Riccati equation. The proof of existence and uniqueness of this Riccati equation, which is a fully nonlinear and singular BSDE with random coefficients, is interesting in its own right and relies heavily on the structural properties of the equation. Efficient investment strategies as well as the mean-variance efficient frontier are then analytically derived in terms of the solution of this equation. In particular, it is demonstrated that the efficient frontier in the mean-standard deviation diagram is still a straight line or, equivalently, risk-free investment is still possible, even when the interest rate is random. Finally, a version of the Mutual Fund Theorem is presented.

Steering flexible needles under Markov motion uncertainty

When inserted into soft tissues, flexible needles with bevel tips have been shown experimentally to follow a path of constant curvature in the direction of the bevel. By controlling 2 degrees of freedom at the needle base (bevel direction and insertion distance), these needles can be steered around obstacles to reach targets inaccessible to rigid needles. Motion planning for needle steering is a type of nonholonomic planning for a Dubins car with no reversal. We develop a motion planning algorithm based on dynamic programming where the path of the needle is uncertain due to uncertainty in tissue properties, needle mechanics, and interaction forces. The algorithm computes a discrete control sequence of insertions and direction changes so the needle reaches a target in an imaging plane while minimizing expected cost due to insertion distance, direction changes, and obstacle collisions. We efficiently sample the state space of needle tip positions and orientations and define bounds on the errors due to discretization. We formulate the motion planning problem as a Markov decision process (MDP) and use infinite horizon dynamic programming to compute an optimal control sequence. We first apply the method to the deterministic motion case where the needle precisely follows a path of constant curvature and then to the uncertain motion case where state transitions are defined by a probability distribution. Our implementation generates motion plans for bevel-tip needles that reach targets inaccessible to rigid needles and demonstrates that accounting for uncertainty can lead to significantly different motion plans.

Quadratic Hedging and Mean-Variance Portfolio Selection with Random Parameters in an Incomplete Market

This paper concerns the problems ofquadratic hedging andpricing, andmean-variance portfolio selection in anincomplete market setting with continuous trading, multiple assets, and Brownian information. In particular, we assume throughout that the parameters describing the market model may be random processes. We approach these problems from the perspective oflinear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs); that is, we focus on the so-calledstochastic Riccati equation (SRE) associated with the problem. Excepting certain special cases, solvability of the SRE remains an open question. Our primary theoretical contribution is a proof of existence and uniqueness of solutions of the SRE associated with the quadratic hedging and mean-variance problems. In addition, we derive closed-form expressions for the optimal portfolios and efficient frontier in terms of the solution of the SRE. A generalization of theMutual Fund Theorem is also obtained.

Linear-Quadratic Control of Backward Stochastic Differential Equations

This paper is concerned with optimal control of linear backward stochastic differential equations (BSDEs) with a quadratic cost criteria, or backward linear-quadratic (BLQ) control. The solution of this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique. Two alternative, though equivalent, expressions for the optimal control are obtained. The first of these involves a pair of Riccati-type equations, an uncontrolled BSDE, and an uncontrolled forward stochastic differential equation (SDE), while the second is in terms of a Hamiltonian system. Contrary to the deterministic or stochastic forward case, the optimal control is no longer a feedback of the current state; rather, it is a feedback of the entire history of the state. A key step in our derivation is a proof of global solvability of the aforementioned Riccati equations. Although of independent interest, this issue has particular relevance to the BLQ problem since these Riccati equations play a central role in our solution. Last but not least, it is demonstrated that the optimal control obtained coincides with the solution of a certain forward linear-quadratic (LQ) problem. This, in turn, reveals the origin of the Riccati equations introduced.

Conditional value-at-risk in portfolio optimization: Coherent but fragile

We evaluate conditional value-at-risk (CVaR) as a risk measure in data-driven portfolio optimization. We show that portfolios obtained by solving mean-CVaR and global minimum CVaR problems are unreliable due to estimation errors of CVaR and/or the mean, which are magnified by optimization. This problem is exacerbated when the tail of the return distribution is made heavier. We conclude that CVaR, a coherent risk measure, is fragile in portfolio optimization due to estimation errors.

Mean-Variance Hedging When There Are Jumps

In this paper, we consider the problem of mean-variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by Brownian motion and doubly stochastic Poisson processes. This problem is formulated as a stochastic control problem, and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and the theory of backward stochastic differential equations. The results we have obtained show how backward stochastic differential equations can be used to obtain solutions to optimal investment and hedging problems when discontinuities in the underlying price processes are modeled by the arrivals of Poisson processes with stochastic intensities. Applications to the problem of hedging default risk are also discussed.

Discrete time LQG controls with control dependent noise

This paper presents some studies on partially observed linear quadratic Gaussian (LQG) models where the stochastic disturbances depend on both the states and the controls, and the measurements are bilinear in the noise and the states/controls. While the Separation Theorem of standard LQG design does not apply, suboptimal linear state estimate feedback controllers are derived based on certain linearizations. The controllers are useful for nonlinear stochastic systems where the linearized models include terms bilinear in the noise and states/controls and are significantly more accurate than if the bilinear terms are set to zero. The controllers are calculated by solving a generalized discrete time Riccati equation, which in turn has properties relating to well posedness of the associated LQG problem.

Pricing American-Style Derivatives with European Call Options

We present a new approach to pricing American-style derivatives that is applicable to any Markovian setting (i.e., not limited to geometric Brownian motion) for which European call-option prices are readily available. By approximating the value function with an appropriately chosen interpolation function, the pricing of an American-style derivative with arbitrary payoff function is converted to the pricing of a portfolio of European call options, leading to analytical expressions for those cases where analytical European call prices are available (e.g., the Merton jump-diffusion process). Furthermore, in many settings, the approach yields upper and lower analytical bounds that provably converge to the true option price. We provide computational results to illustrate the convergence and accuracy of the resulting estimators.

Robust Asset Allocation with Benchmarked Objectives

In this paper, we introduce a new approach for finding robust portfolios when there is model uncertainty. It differs from the usual worst�?case approach in that a (dynamic) portfolio is evaluated not only by its performance when there is an adversarial opponent (“nature�?), but also by its performance relative to a stochastic benchmark. The benchmark corresponds to the wealth of a fictitious benchmark investor  who invests optimally given knowledge of the model chosen by nature, so in this regard, our objective has the flavor of min–max regret. This relative performance  approach has several important properties: (i) optimal portfolios seek to perform well over the entire range of models and not just the worst case, and hence are less pessimistic than those obtained from the usual worst�?case approach; (ii) the dynamic problem reduces to a convex static optimization problem under reasonable choices of the benchmark portfolio for important classes of models including ambiguous jump�?diffusions; and (iii) this static problem is dual to a Bayesian  version of a single period asset allocation problem where the prior on the unknown parameters (for the dual problem) correspond to the Lagrange multipliers in this duality relationship. This dual static problem can be interpreted as a less pessimistic alternative to the single period worst�?case Markowitz problem. More generally, this duality suggests that learning and robustness are closely related when benchmarked objectives are used.

A new risk-sensitive maximum principle

In this paper, a new maximum principle for the risk-sensitive control problem is established. One important feature of this result is that it applies to systems in which the diffusion term may depend on the control. Such control dependence gives rise to interesting phenomena not observed in the usual setting where control independence of the diffusion term is assumed. In particular, there is an additional second order adjoint equation and additional terms in the maximum condition that involve this second order process as well as the risk-sensitive parameter. Moreover, contrary to a conventional maximum principle, the first-order adjoint equation involved in our maximum principle is a nonlinear equation. An advantage of considering this new type of adjoint equation is that the risk-sensitive maximum principle derived is similar in form to its risk-neutral counterpart. The approach is based on the logarithmic transformation and the relationship between the adjoint variables and the value function. As an example, a linear-quadratic risk-sensitive problem is solved using the maximum principle derived.