Related papers: Bubbles in discrete time models
Considering a simple economy, we derive a new Hamilton-Jacobi equation which is satisfied by the value of a ''bubble'' asset. We then show, by providing a rigorous mathematical analysis of this equation, that a unique non-zero stable…
We introduce two new exactly solvable (stochastic) interacting particle systems which are discrete time versions of q-TASEP. We call these geometric and Bernoulli discrete time q-TASEP. We obtain concise formulas for expectations of a large…
Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long-range…
This paper shows how the theory of dynamic risk measures provides viscosity solutions to a family of second-order parabolic partial differential equations, even in the degenerate case. First, motivated by the martingale problem approach of…
A dynamical model is introduced for the formation of a bullish or bearish trends driving an asset price in a given market. Initially, each agent decides to buy or sell according to its personal opinion, which results from the combination of…
We prove the existence of weak solutions for distribution-dependent stochastic Volterra equations under linear growth and continuity conditions on the coefficients and mild regularity assumptions on the kernels, including singular kernels.…
It was demonstrated previously that the stochastic volatility emerges as the gauge field necessary for restoring the local symmetry under changes of the prices of the stocks inside the Black-Scholes (BS) equation. When this occurs, then a…
We study a market model in which the volatility of the stock may jump at a random time from a fixed value to another fixed value. This model was already described in the literature. We present a new approach to the problem, based on partial…
Given a stock price process, we analyse the potential of arbitrage by insiders in a context of short-selling prohibitions. We introduce the notion of minimal supermartingale measure, and we analyse its properties in connection to the…
We consider a stochastic volatility model with jumps where the underlying asset price is driven by the process sum of a 2-dimensional Brownian motion and a 2-dimensional compensated Poisson process. The market is incomplete, resulting in…
In the present paper, we give some examples of stochastic differential equations which have delicateness in the Markov and strong Markov properties, the uniqueness locally in time and globally in time, and initial conditions. Moreover, we…
We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. More precisely, one is given access to a discrete time observation sequence which at any time $k$ depends only on the state of an…
We derive a general multivariate theory for realised characteristics of `model-free discretisation-invariant swaps', so-called because the standard no-arbitrage assumption of martingale forward prices is sufficient to derive fair-value swap…
We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in…
Recently, D. Williams \cite{williams} gave an explicit example of a random time $\rho $ associated with Brownian motion such that $\rho $ is not a stopping time but $\mathbb{E}M_{\rho}=\mathbb{E}M_{0}$ for every bounded martingale $M$. The…
We study the optimal timing of derivative purchases in incomplete markets. In our model, an investor attempts to maximize the spread between her model price and the offered market price through optimally timing her purchase. Both the…
In this article integro-differential Volterra equations whose convolution kernel depends on the vector variable are considered and a connection of these equations with a class of semi-Markov processes is established. The variable order…
This paper introduces a new approach for bubble detection based on mixed causal and noncausal autoregressive processes and their tail process representation during an explosive episode. Departing from traditional definitions of bubbles as…
We develop a Bayesian inference method for diffusions observed discretely and with noise, which is free of discretisation bias. Unlike existing unbiased inference methods, our method does not rely on exact simulation techniques. Instead,…
This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to…