The paper studies martingales and super-martingales under a convex set of measures.
problem Understanding martingales and super-martingales in a convex set of equivalent measures.
method Introduced local regular super-martingales and proved necessary and sufficient conditions for their regularity.
result Generalized Doob's decomposition theorem for super-martingales under a convex set of measures.
The paper studies incomplete financial markets and risk assets.
problem Incomplete financial markets and risk assets.
method Study of martingales and super-martingales, introduction of local regular super-martingales, and presentation of all local regular super-martingales.
result A new formula for the fair price of super-hedge is founded for the discrete geometric Brownian motion.
We consider the fundamental theorem of asset pricing (FTAP) and hedging prices of options under non-dominated model uncertainty and portfolio constrains in discrete time. We first show that no arbitrage holds if and only if there exists some family of probability measures such that any admissible portfolio value proces…
A well known result in stochastic analysis reads as follows: for an R-valued super-martingale X=(Xt)0≤t≤T such that the terminal value XT is non-negative, we have that the entire process X is non-negative. An analogous result holds true in the no arbitrage theory of mathematical finance…
We propose a new definition for tameness within the model of security prices as Itô processes that is risk-aware. We give a new definition for arbitrage and characterize it. We then prove a theorem that can be seen as an extension of the second fundamental theorem of asset pricing, and a theorem for valuation of contin…
Researchers develop a pricing method for contingent claims under partial information and short selling constraints.
problem Pricing contingent claims with partial information and short selling restrictions.
method Derive a dual problem using conjugate duality theory and conditions for strong duality.
result Characterization of contingent claim prices involving martingale and super-martingale conditions.
We propose an optimal portfolio problem in the incomplete market where the underlying assets depend on economic factors with delayed effects, such models can describe the short term forecasting and the interaction with time lag among different financial markets. The delay phenomenon can be recognized as the integral ty…
Study dynamic risk measures and performance indices using distortion functions.
problem Investigate time consistency of dynamic risk measures and performance indices generated by distortion functions.
method Analyze dynamic coherent risk measures (DCRMs) and dynamic weighted value at risk measures, proving their equivalence. Establish properties of families of DCRMs generated by distortion functions and define corresponding dynamic coherent acceptability indices (DCAIs). Examine time consistency of DCRMs and DCAIs.
result DCRM generated by distortion functions are sub-martingale time consistent but not super-martingale time consistent and not weakly acceptance time consistent.
The article constructs a forward utility for markets with multiple default risks.
problem Characterizing forward performance processes in a market with multiple default risks.
method Using Jacod-Pham decomposition and recursive BSDEs, the article constructs a forward utility and proves its existence and uniqueness.
result The article identifies the risk-sensitive long-run growth rate of the optimal wealth process in a stochastic factor model with ergodic dynamics.
Neural networks improve pricing and hedging of complex financial claims.
problem Pricing and hedging of high-dimensional, path-dependent contingent claims.
method Regress later Monte Carlo approach using neural networks for interpretability.
result Any contingent claim can be semi-statically hedged using a portfolio of short maturity options.
In our recent paper, we showed that in exponential family, contrastive divergence (CD) with fixed learning rate will give asymptotically consistent estimates \cite{wu2016convergence}. In this paper, we establish consistency and convergence rate of CD with annealed learning rate ηt. Specifically, suppose CD-m gener…
New method for non-arbitrage pricing in risky assets.
problem Non-arbitrage pricing in markets with non-negative risky assets.
method Constructing martingale measures and proving optional decomposition theorem.
result Deriving fair prices for European option contracts.
Paper introduces online tensor inference for real-time data analysis.
problem Real-time processing of high-dimensional tensor data.
method Stochastic Gradient Descent (SGD) for efficient online inference.
result Establishes non-asymptotic convergence and optimal estimation error rate.