scispace - formally typeset
Search or ask a question
Author

Siu Ah Ng

Bio: Siu Ah Ng is an academic researcher from University of KwaZulu-Natal. The author has contributed to research in topics: Simple (abstract algebra) & Infinitesimal. The author has an hindex of 1, co-authored 1 publications receiving 9 citations.

Papers
More filters
Journal ArticleDOI
TL;DR: In this paper, a simple derivation of the Levy-Khintchine formula via an explicit construction of certain laws of the infinitesimal increments is given, where any arbitrary Levy process is representable as the standard part of a hyperfinite sum of infiniteimal increments.
Abstract: We use methods from nonstandard analysis to obtain a short and simple derivation of the Levy-Khintchine formula via an explicit construction of certain laws of the infinitesimal increments. Consequently, any arbitrary Levy process is representable as the standard part of a hyperfinite sum of infinitesimal increments.

10 citations


Cited by
More filters
Journal ArticleDOI
TL;DR: In this article, the authors prove the existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete, i.e., the number of securities is at least one more than the independent sources of uncertainty, and they prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model.
Abstract: We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and from a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information. We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy.

108 citations

Journal ArticleDOI
TL;DR: The paper develops a notion of nonlinear stochastic integrals for hyperfinite Lévy processes and uses it to find exact formulas for expressions which are intuitively of the form $$sum_{s=0}^t\phi(\omega,dl_{s},s)$$ and $$prod_{s =0}^{t\psi(\omegas,dl?,s),s).
Abstract: I develop a notion of nonlinear stochastic integrals for hyperfinite Levy processes and use it to find exact formulas for expressions which are intuitively of the form $$\sum_{s=0}^t\phi(\omega,dl_{s},s)$$ and $$\prod_{s=0}^t\psi(\omega,dl_{s},s)$$ , where l is a Levy process. These formulas are then applied to geometric Levy processes, infinitesimal transformations of hyperfinite Levy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic calculus for processes with jumps on manifolds, and the paper may be regarded as a reworking of his ideas in a different setting and with totally different techniques.

10 citations

Journal ArticleDOI
TL;DR: In this article, for a continuous-time financial market with a single agent, the authors establish equilibrium pricing formulae under the assumption that the dividends follow an exponential Levy process, and the resulting equilibrium asset price processes depend on the agent's risk aversion (through the felicity functions).
Abstract: For a continuous-time financial market with a single agent, we establish equilibrium pricing formulae under the assumption that the dividends follow an exponential Levy process. The agent is allowed to consume a lump at the terminal date; before that, only flow consumption is allowed. The agent’s utility function is assumed to be additive, defined via strictly increasing, strictly concave smooth felicity functions which are bounded below (thus, many CRRA and CARA utility functions are included). For technical reasons we require for our equilibrium existence result that only pathwise continuous trading strategies are permitted in the demand set. The resulting equilibrium asset price processes depend on the agent’s risk aversion (through the felicity functions). Even in our simple, straightforward economy, the equilibrium asset price processes will essentially only be (stochastic) exponential Levy processes when they are already geometric Brownian motions. Our equilibrium asset pricing formulae can also be modified to obtain explicit equilibrium derivative pricing formulae.

5 citations

Journal ArticleDOI
TL;DR: In this article, the hyper-finite theory of stochastic integration with respect to certain hyperfinite Levy processes with finite-variation jump part was linked with the elementary theory of path-wise stochastically integration for pure-jump Levy processes.
Abstract: This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Levy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Levy processes with finite-variation jump part. Since the hyperfinite Ito integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Levy jump-diffusions with finite-variation jump part. As an application, we provide a short and direct nonstandard proof of the generalized Ito formula for stochastic differentials of smooth functions of Levy jump-diffusions whose jumps are bounded from below in norm.

5 citations

Posted Content
TL;DR: In this article, for a continuous-time financial market with a single agent, the authors established equilibrium pricing formulae under the assumption that the dividends follow an exponential Levy process, and the resulting equilibrium prices depend on the agent's risk-aversion through the felicity functions.
Abstract: For a continuous-time financial market with a single agent, we establish equilibrium pricing formulae under the assumption that the dividends follow an exponential Levy process. The agent is allowed to consume a lump at the terminal date; before, only flow consumption is allowed. The agent's utility function is assumed to be additive, defined via strictly increasing, strictly concave smooth felicity functions which are bounded below (thus, many CRRA and CARA utility functions are included). For technical reasons we require that only pathwise continuous trading strategies are permitted in the demand set. The resulting equilibrium prices depend on the agent's risk-aversion through the felicity functions. It turns out that these prices will be the (stochastic) exponential of a Levy process essentially only if this process is geometric Brownian motion.

2 citations