arXiv subjects

Abhishek Tilva

Publications and source records attributed to Abhishek Tilva.

3 recordsLinked to original sources

arXiv2023

Arbitrage theory in a market of stochastic dimension

This paper studies an equity market of stochastic dimension, where the number of assets fluctuates over time. In such a market, we develop the fundamental theorem of asset pricing, which provides the equivalence of the following statements: (i) there exists a supermartingale numéraire portfolio; (ii) each dissected market, which is of a fixed dimension between dimensional jumps, has locally finite growth; (iii) there is no arbitrage of the first kind; (iv) there exists a local martingale deflator; (v) the market is viable. We also present the optional decomposition theorem, which characterizes a given nonnegative process as the wealth process of some investment-consumption strategy. Furthermore, similar results still hold in an open market embedded in the entire market of stochastic dimension, where investors can only invest in a fixed number of large capitalization stocks. These results are developed in an equity market model where the price process is given by a piecewise continuous semimartingale of stochastic dimension. Without the continuity assumption on the price process, we present similar results but without explicit characterization of the numéraire portfolio.

q-fin.MF↗

arXiv2023

Quantifying dimensional change in stochastic portfolio theory

In this paper, we develop the theory of functional generation of portfolios in an equity market with changing dimension. By introducing dimensional jumps in the market, as well as jumps in stock capitalization between the dimensional jumps, we construct different types of self-financing stock portfolios (additive, multiplicative, and rank-based) in a very general setting. Our study explains how a dimensional change caused by a listing or delisting event of a stock, and unexpected shocks in the market, affect portfolio return. We also provide empirical analyses of some classical portfolios, quantifying the impact of dimensional change in portfolio performance relative to the market.

q-fin.MF↗

arXiv2019

Continuous Breuer-Major theorem for vector valued fields

Let $ξ: Ω\times \mathbb{R}^n \to \mathbb{R}$ be zero mean, mean-square continuous, stationary, Gaussian random field with covariance function $r(x) = \mathbb{E}[ξ(0)ξ(x)]$ and let $G : \mathbb{R} \to \mathbb{R}$ such that $G$ is square integrable with respect to the standard Gaussian measure and is of Hermite rank $d$. The Breuer-Major theorem in it's continuous setting gives that, if $r \in L^d(\mathbb{R}^n)$, then the finite dimensional distributions of $Z_s(t) = \frac{1}{(2s)^{n/2}} \int_{[-st^{1/n},st^{1/n}]^n} \Big[G(ξ(x)) - \mathbb{E}[G(ξ(x))]\Big]dx$ converge to that of a scaled Brownian motion as $s \to \infty$. Here we give a proof for the case when $ξ: Ω\times \mathbb{R}^n \to \mathbb{R}^m$ is a random vector field. We also give a proof for the functional convergence in $C([0,\infty))$ of $Z_s$ to hold under the condition that for some $p>2$, $G\in L^p(\mathbb{R}^m, γ_m)$ where $γ_m$ denotes the standard Gaussian measure on $\mathbb{R}^m$ and we derive expressions for the asymptotic variance of the second chaos component in the Wiener chaos expansion of $Z_s(1)$.

math.PR↗