Sanjay P. Bhat

Modeling and analysis of mass-action kinetics

Mass-action kinetics are used in chemistry and chemical engineering to describe the dynamics of systems of chemical reactions, that is, reaction networks. These models are a special form of compartmental systems, which involve mass- and energy-balance relations. Aside from their role in chemical engineering applications, mass-action kinetics have numerous analytical properties that are of inherent interest from a dynamical systems perspective. Because of physical considerations, however, mass- action kinetics have special properties, such as nonnegative solutions, that are useful for analyzing their behavior. With this motivation in mind, this article has several objectives. First, a general construction of the kinetic equations based on the reaction laws is provided in a state-space form. Next, the nonnegativity of solutions to the kinetic equations is considered. The realizability problem, which is concerned with the inverse problem of constructing a reaction network having specified essentially nonnegative dynamics, is also considered. In particular, an explicit construction of a reaction network for essentially nonnegative polynomial dynamics involving a scalar state is provided. Next, the reducibility of the kinetic equations is considered as well as the stability of the equilibria of the kinetic equations. Lyapunov methods are applied to the kinetic equations, and semistability is guaranteed through the convergence to a Lyapunov- stable equilibrium that depends on the initial concentrations. Semistability is the appropriate notion of stability for compartmental systems in general, and reaction networks in particular, where the limiting concentration maybe nonzero and may depend on the initial concentrations. Finally, the zero deficiency result for mass-action kinetics in standard matrix terminology is presented and semistability is proven.

Arc-length-based Lyapunov tests for convergence and stability with applications to systems having a continuum of equilibria

In this paper, fundamental relationships are established between convergence of solutions, stability of equilibria, and arc length of orbits. More specifically, it is shown that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and semistable if the arc length is continuous in a neighborhood of the equilibrium. Next, arc-length-based Lyapunov tests are derived for convergence and stability. These tests do not require the Lyapunov function to be positive definite. Instead, these results involve an inequality relating the right-hand side of the differential equation and the Lyapunov function derivative. This inequality makes it possible to deduce properties of the arc length function and thus leads to sufficient conditions for convergence and stability. Finally, it is shown that the converses of all the main results hold under additional assumptions. Examples are included to illustrate how our results are particularly suited for analyzing stability of systems having a continuum of equilibria.

Discrete-time optimal hedging for multi-asset path-dependent European contingent claims

In this paper, we consider the problem of discretetime optimal hedging for a European contingent claim (ECC) written on multiple assets where the underlying assets are assumed to follow a vector Ito differential equation. Specifically, since the underlying asset is assumed to be a continuous-time process all discrete-time hedging strategies are non-replicable and lead to hedging errors. First, we present a framework for finding hedging strategies that minimize the variance of hedging errors due to discrete-time hedging. The general framework is valid for all ECCs whose underlying assets are martingales and the minimum variance hedging strategies are in terms of conditional covariance matrices. Next, we specialize the conditional covariance matrix formulas to the case of geometric Brownian motion. These results extend the existing formula for single asset European call and put options to simple and pathdependent ECCs written on multiple assets.

Discrete-time, minimum-variance hedging of European contingent claims

This paper addresses minimum-variance hedging of European contingent claims (ECC) in the case where trading dates are discrete and fixed. A simple derivation of the minimum-variance hedging strategy is first given in a general setting. The strategy is then applied to a general class of European contingent claims written on an underlying asset whose price process is a martingale modeled by a geometric Brownian motion. A Wiener space setting is used to show that the minimum-variance strategy requires the asset holding to equal the ratio of conditional expectations of the changes in the ECC payoff and the underlying asset price that occur when sample paths of the Wiener process are modified in a certain manner. In the case of specific claims, the minimum-variance hedging strategy can be further expressed in terms of pricing functions. Unlike previous work, the results of this paper apply equally well to simple as well as path-dependent claims.

Closed Rotation Sequences

A finite sequence of rotations is closed if a sequential application of all the rotations from the sequence results in no net orientation change. A complete characterization of closed rotation sequences involving a given set of rotation axes is presented, and the set of such sequences is shown to be a smooth manifold under a nondegeneracy condition on the rotation axes. The characterization is used to derive several examples of closed rotation sequences, some of which are then shown to specialize to classical examples of such sequences provided by the Rodrigues–Hamilton theorem and the Donkin’s theorem. Discrete versions of the Goodman–Robinson and Ishlinskii theorems are also derived and illustrated using the so-called Codman’s paradox.

Optimal Static Hedging of Uncertain Future Foreign Currency Cash Flows Using FX Forwards

An exporter is invariably exposed to currency risk due to unpredictable fluctuations in the exchange rates, and it is of paramount importance to minimize risk emanating from these forex exposures. The currency risk problem is further compounded if the future foreign currency receivables are uncertain in quantity. In this paper, we address the currency risk hedging problem for an uncertain foreign currency receivables. We consider a static hedging strategy defined by the positions in a FX forward contract, and present a minimum-risk static hedging problem to determine an optimal hedging strategy that minimizes conditional value at risk (CVaR) due to the loss. This risk minimization problem can be solved numerically by using Monte-Carlo simulations which, however, turn out to be computationally intensive, because of the presence of two sources of randomness. We consider bounding functions that bound the CVaR due to the loss above and below, and numerically obtain an outer estimate of the set of solutions of the minimum-risk static hedging problem. We present several bounding functions obtained under different assumptions on the risk factors.

Discrete-Time Quadratic-Optimal Hedging Strategies for European Contingent Claims

We revisit the problem of optimally hedging a European contingent claim (ECC) using a hedging portfolio consisting of a risky asset that can be traded at pre-specified discrete times. The objective function to be minimized is either the second-moment or the variance of the hedging error calculated in the market probability measure. The main outcome of our work is to show that unique solutions exist in a larger class of admissible strategies under integrability and non-degeneracy conditions on the hedging asset price process that are weaker than popular descriptions provided previously. Specifically, we do not require the hedging asset price process to be square-integrable, and do not use the bounded mean-variance trade off assumption. Our criterion for admissible strategies only requires the cumulative trading gain, and not the incremental trading gains, to be square integrable. We derive explicit expressions for the second-moment and the variance of the hedging error to arrive at the respective optimal hedging strategies. Further, we explain the connections between our work and those of the previous formulations.

Explicit formulas for optimal hedging stratergies for European contingent claims

In this paper, we consider the problem of discrete-time optimal hedging for a portfolio of (illiquid) European contingent claims (ECCs) written on multiple underlying assets. First, we present a framework to find discrete-time hedging strategies that minimize the variance of terminal wealth using a hedging portfolio of liquid assets, also assumed to ECCs written on the same underlying assets. Next, we specialize the framework to the case of illiquid portfolio consisting of a simple ECC written on a single underlying asset and a hedging portfolio consisting of the underlying asset and another simple ECC written on the same underlying asset. For this special case, we provide a (computable) formula for the minimum variance hedging strategy. Finally, we show that the minimum variance hedging strategy converges to the Δ-Γ-neutral hedging strategy as the interspacing between the hedging times converge to zero.