Stochastic Calculus & Continuous-Time Models

We study nearly unstable bivariate cumulative heavy-tailed INAR($\infty$) processes and show that, under a one-factor parameterization and a suitable scaling, they converge to the rough Heston model. This yields a discrete-time microstructural route to the joint price-variance dynamics and gives exp

In this work, we propose a model to quantify the impact of the climate transition on a property in housing market. We begin by noting that property is an asset in an economy. That economy is organized in sectors, driven by its productivity which is a multidimensional Ornstein-Uhlenbeck process, whil

This paper addresses the problem of robust option pricing within the framework of Vectorial Martingale Optimal Transport (VMOT). We investigate the geometry of VMOT solutions for $N$-period market models and demonstrate that, when the number of underlying assets is $d=2$ and the payoff is sub- or su

The increasing availability of granular and big data on various objects of interest has made it necessary to develop methods for condensing this information into a representative and intelligible map. Financial regulation is a field that exemplifies this need, as regulators require diverse and often

We reconcile rough volatility models and jump models using a class of reversionary Heston models with fast mean reversions and large vol-of-vols. Starting from hyper-rough Heston models with a Hurst index $H \in (-1/2,1/2)$, we derive a Markovian approximating class of one dimensional reversionary H

In this paper, we study the Euler–Maruyama scheme for a particle method to approximate the McKean–Vlasov dynamics of calibrated local-stochastic volatility (LSV) models. Given the open question of well-posedness of the original problem, we work with regularised coefficients and prove that under ce

Motivated by recent results on the dual formulation of optimal stopping problems, we investigate in this short paper how the knowledge of an approximating dual martingale can improve the efficiency of primal methods. In particular, we show on numerical examples that accurate approximations of a dual

Financial markets exhibit temporal organization that is not fully captured by volatility measures or linear correlation structure. We study a null validated topological approach for quantifying market complexity and apply it to Bitcoin daily log returns. The analysis uses the $L^1$ norm of persisten

This paper studies the continuous-time reinforcement learning (RL) for optimal switching problems across multiple regimes. We consider a type of exploratory formulation under entropy regularization where the agent randomizes both the timing of switches and the selection of regimes through the genera

Parallel physical information neural networks (P-PINNs) have been widely used to solve systems with multiple coupled physical fields, such as the coupled Stokes-Darcy equations with Beavers-Joseph-Saffman (BJS) interface conditions. However, excessively high or low physical constants in partial diff

The lifted Heston model is a stochastic volatility model emerging as a Markovian lift of the rough Heston model and the class of rough volatility processes. The model encodes the path dependency of volatility on a set of N square-root state processes driven by a common stochastic factor. While the s

Myopic optimization (MO) outperforms reinforcement learning (RL) in portfolio management: RL yields lower or negative returns, higher variance, larger costs, heavier CVaR, lower profitability, and greater model risk. We model execution/liquidation frictions with mark-to-market accounting. Using Mall

We present a Markovian market model driven by a hidden Brownian efficient price. In particular, we extend the queue-reactive model, making its dynamics dependent on the efficient price. Our study focuses on two sub-models: a signal-driven price model where the mid-price jump rates depend on the effi

This paper provides evidence that stock returns, after truncation, might be modeled by a special type of continuous mixtures or normals, so-called $q$-Gaussians. Negative binomial distributions might model the counts for extreme returns. A generalized jump-diffusion model is proposed, and an explici

We present a Pontryagin-Guided Direct Policy Optimization (PG-DPO) framework for Merton’s portfolio problem, unifying modern neural-network-based policy parameterization with the adjoint viewpoint from Pontryagin’s maximum principle (PMP). Instead of approximating the value function (as done in deep

This article presents a generic hybrid numerical method to price a wide range of options on one or several assets, as well as assets with stochastic drift or volatility. In particular for equity and interest rate hybrid with local volatility.

The strong relative arbitrage problem in Stochastic Portfolio Theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative cont

We present weak approximations schemes of any order for the Heston model that are obtained by using the method developed by Alfonsi and Bally (2021). This method consists in combining approximation schemes calculated on different random grids to increase the order of convergence. We apply this metho

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based upon a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure

Predicting the conditional evolution of Volterra processes with stochastic volatility is a crucial challenge in mathematical finance. While deep neural network models offer promise in approximating the conditional law of such processes, their effectiveness is hindered by the curse of dimensionality

This paper studies the optimal liquidation of stocks in the presence of temporary and permanent price impacts, and we focus in the case of cryptocurrencies. We start by presenting analytical solutions to the problem with linear temporary impact, and linear and quadratic permanent impact. Then, using

This paper extends the existing drawdown modulation control policy to include a novel restart mechanism for trading. It is known that the drawdown modulation policy guarantees the maximum percentage drawdown no larger than a prespecified drawdown limit for all time with probability one. However, whe

Local Stochastic Volatility (LSV) models have been used for pricing and hedging derivatives positions for over twenty years. An enormous body of literature covers analytical and numerical techniques for calibrating the model to market data. However, the literature misses a potent approach commonly u

Signature stochastic differential equations (SDEs) constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are linear maps of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence

We propose a novel asset allocation model using a Markov process of states defined by clustered efficient frontier coefficients. While most research in Markov models of the market characterize regimes using return and volatility, we instead propose characterizing these states using efficient frontie

This paper investigates the valuation of variable annuity contracts with an early surrender option under non-Markovian models. Moreover, policyholders are provided with guaranteed minimum maturity and death benefits to protect against the downside risk. Unlike the existing literature, our variable a

We present a strictly monotone, provably convergent two-dimensional (2D) integration method for multi-period mean-conditional value-at-risk (mean-CVaR) reward-risk stochastic control in models whose one-step increment law is specified via a closed-form characteristic function (CF). When the transiti

We consider the pricing of derivatives written on accumulated marks, such as weather derivatives or aggregate loss claims, using a self-exciting marked point process. The jump intensity mean-reverts between events and increases at jump times by an amount proportional to the mark. The resulting state

We develop a neural-network framework for multi-period risk–reward stochastic control problems with constrained two-step feedback policies that may be discontinuous in the state. We allow a broad class of objectives built on a finite-dimensional performance vector, including terminal and path-depen

In life insurance, life tables are used to estimate the survival distribution of individuals from a given population. However, these tables only provide survival probabilities at integer ages but no information about the distribution of deaths between two consecutive integer values. This incompleten

We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of the resulting rough paths approximate continuous functionals

We present a study of the short-maturity asymptotics for VIX and European option prices in local-stochastic volatility models with compound Poisson jumps. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. The leading-order asymptotics are obtained in closed-form. We appl

In this paper we use Malliavin Calculus techniques in order to obtain expressions for the short-time behavior of the at-the-money implied volatility (ATM-IV) level and skew for a jump-diffusion stock price. The diffusion part is assumed to be the stochastic volatility Bachelier model and the jumps a

Empirical studies with publicly available life tables identify long-range dependence (LRD) in national mortality data. Although the longevity market is supposed to benchmark against the national force of mortality, insurers are more concerned about the forces of mortality associated with their own p

This paper introduces an analytical formula for the fractional-order conditional moments of nonlinear drift constant elasticity of variance (NLD-CEV) processes under regime switching, governed by continuous-time finite-state irreducible Markov chains. By employing a hybrid system approach, we derive

We define a new model using a Hawkes process as a subordinator in a standard Brownian motion. We demonstrate that this Hawkes subordinated Brownian motion or more succinctly, variance-Hawkes process can be fit to 2018 and 2019 natural gas and crude oil front-month futures log returns. This variance-

The significance of mortality modeling extends across multiple research areas, ranging from life insurance valuation to optimal lifetime decision-making. Existing approaches, such as mortality laws and factor-based models, often fall short in capturing the complexity of individual mortality, hinderi

In this paper we provide a quantum Monte Carlo algorithm to solve multidimensional Black-Scholes PDEs with correlation for option pricing. The payoff function of the option is of general form and is only required to be continuous and piecewise affine, which covers most of the relevant payoff functio

Given the promising results on joint modeling of SPX/VIX smiles of the recently introduced quadratic rough Heston model, we consider a multi-asset market making problem on SPX and its derivatives, e.g. VIX futures, SPX and VIX options. The market maker tries to maximize its profit from spread captur

This article studies a retirement planning problem from a new perspective in which a retiree delegates an initial lump sum to a professional fund manager. The fund is managed dynamically to deliver lifelong benefits while satisfying a guarantee that, at all times, wealth remains above a prescribed s

We study finite-horizon portfolio optimization with proportional transaction costs and trading opportunities arriving at the jump times of a Cox process. Borrowing and short-selling are prohibited, while utility functions need not be concave, increasing, or differentiable. The admissible class inclu

We study a finite-horizon reversible investment problem in which a risk-neutral firm adjusts capacity at a proportional purchase cost and a lower salvage value under multi-factor geometric Brownian motion. Via the singular control–optimal switching correspondence, the marginal value of capacity sol

Variance swaps are a convenient instrument for trading vega and convexity, and a listed contract now trades on Cboe. We ask when a trader should put such a position on and when she should take it off, and for a perpetual, continuously settled contract we answer both in closed form: each threshold is

We study continuous-time portfolio choice in diffusion markets with parameter $θ\in Θ$ and uncertainty law $q(dθ)$. Nature draws latent $θ\sim q$ at time 0; the investor cannot observe it and must deploy a single $θ$-blind feedback policy maximizing an ex-ante CRRA objective averaged over diffusion

As the FX markets continue to evolve, many institutions have started offering passive access to their internal liquidity pools. Market makers act as principal and have the opportunity to fill those orders as part of their risk management, or they may choose to adjust pricing to their external OTC fr

We consider an investor who wants to hedge a path-dependent option with maturity $T$ using a static hedging portfolio using cash, the underlying, and vanilla put/call options on the same underlying with maturity $ t_1$, where $0 < t_1 < T$. We propose a model-free approach to construct such a portfo

Local stochastic volatility refers to a popular model class in applied mathematical finance that allows for “calibration-on-the-fly”, typically via a particle method, derived from a formal McKean-Vlasov equation. Well-posedness of this limit is a well-known problem in the field; the general case is

We consider an equity market subject to risk from both unhedgeable shocks and default. The novelty of our work is that to partially offset default risk, investors may dynamically trade in a credit default swap (CDS) market. Assuming investment opportunities are driven by functions of an underlying d

We introduce the deep multi-FBSDE method for robust approximation of coupled forward-backward stochastic differential equations (FBSDEs), focusing on cases where the deep BSDE method of Han, Jentzen, and E (2018) fails to converge. To overcome the convergence issues, we consider a family of FBSDEs t

This paper investigates an infinite horizon, discounted, consumption-portfolio problem in a market with one bond, one liquid risky asset, and one illiquid risky asset with proportional transaction costs. We consider an agent with liquidity preference, modeled by a Cobb-Douglas utility function that

We explore the interplay between sovereign debt default/renegotiation and environmental factors (e.g., pollution from land use, natural resource exploitation). Pollution contributes to the likelihood of natural disasters and influences economic growth rates. The country can default on its debt at an

In this paper, we study the computation of sensitivities with respect to spot of path dependent financial derivatives by means of path weighting. We propose explicit path weighting formula and variance reduction adjustment in order to address the large variance happening when the first simulation ti

The paper presents a Bayesian framework for the calibration of financial models using neural stochastic differential equations (neural SDEs), for which we also formulate a global universal approximation theorem based on Barron-type estimates. The method is based on the specification of a prior distr

This paper presents a novel approach to pricing American options using piecewise diffusion Markov processes (PDifMPs), a type of generalised stochastic hybrid system that integrates continuous dynamics with discrete jump processes. Standard models often rely on constant drift and volatility assumpti

In this paper, we propose the Continuous Time Fractional Topic Model (cFTM), a new method for dynamic topic modeling. This approach incorporates fractional Brownian motion~(fBm) to effectively identify positive or negative correlations in topic and word distribution over time, revealing long-term de

Merton portfolio management problem is studied in this paper within a stochastic volatility, non constant time discount rate, and power utility framework. This problem is time inconsistent and the way out of this predicament is to consider the subgame perfect strategies. The later are characterized

This paper analyzes the robust long-term growth rate of expected utility and expected return from holding a leveraged exchange-traded fund (LETF). When the Markovian model parameters in the reference asset are uncertain, the robust long-term growth rate is derived by analyzing the worst-case paramet

In this paper, we propose a machine learning algorithm for time-inconsistent portfolio optimization. The proposed algorithm builds upon neural network based trading schemes, in which the asset allocation at each time point is determined by a a neural network. The loss function is given by an empiric

Introduced in the late 90s, the passport option gives its holder the right to trade in a market and receive any positive gain in the resulting traded account at maturity. Pricing the option amounts to solving a stochastic control problem that for $d>1$ risky assets remains an open problem. Even in a

We propose a virtual bidding strategy by modeling the price differences between the day-ahead market and the real-time market as Brownian motion with drift, where the drift rate and volatility are functions of meteorological variables. We then transform the virtual bidding problem into a mean-varian

This research addresses accurate option pricing by employing models beyond the traditional Black-Scholes framework. While Black-Scholes provides a closed-form solution, it is limited by assumptions of constant volatility, no dividends, and continuous price movements. To overcome these limitations, w

We propose the Compound BSDE method, a fully forward, deep-learning-based approach for solving a broad class of problems in financial mathematics, including optimal stopping. The method is based on a reformulation of option pricing problems in terms of a system of backward stochastic differential eq

Wiesel and Zhang [2023] established that two probability measures $μ,ν$ on $\mathbb{R}^d$ with finite second moments are in convex order (i.e. $μ\preceq_c ν$) if and only if $W_2(ν,ρ)^2-W_2(μ,ρ)^2 \leq \int |y|^2ν(dy) - \int |x|^2μ(dx).$ Let us call a measure $ρ$ maximizing $W_2(ν,ρ)^2-W_2(μ,ρ)^2$ t

We study an optimal execution strategy for purchasing a large block of shares over a fixed time horizon. The execution problem is subject to a general price impact that gradually dissipates due to market resilience. This resilience is modeled through a potentially arbitrary limit-order book shape. T

This paper explores continuous-time and state-space optimal stopping problems from a reinforcement learning perspective. We begin by formulating the stopping problem using randomized stopping times, where the decision maker’s control is represented by the probability of stopping within a given time-

A framework to learn a multi-modal distribution is proposed, denoted as the Conditional Quantum Generative Adversarial Network (C-qGAN). The neural network structure is strictly within a quantum circuit and, as a consequence, is shown to represent a more efficient state preparation procedure than cu

This article explores the optimisation of trading strategies in Constant Function Market Makers (CFMMs) and centralised exchanges. We develop a model that accounts for the interaction between these two markets, estimating the conditional dependence between variables using the concept of conditional

We consider two data-driven approaches to hedging, Reinforcement Learning and Deep Trajectory-based Stochastic Optimal Control, under a stepwise mean-variance objective. We compare their performance for a European call option in the presence of transaction costs under discrete trading schedules. We

The debate between active and passive investment strategies has been ongoing for many years and is far from being over. In this paper, we show that the choice of an optimal portfolio management strategy depends on an investment climate, which we measure via the parameters of the Heston model calibra

Filtering problems with jumps in both the signal and the observation have been extensively studied, typically under the assumption that jump times are totally inaccessible. In many applications, however, jump times are known in advance (i.e., predictable), such as scheduled clinical visits, dividend

This paper investigates optimal investment and pension policies in a Pay-As-You-Go (PAYG) system supplemented by a buffer fund used as an intergenerational risk-sharing mechanism. The social planner’s preference criterion is represented by non-zero volatility forward Constant Relative Risk Aversion

It is well-known that, in the Bachelier model, when asset prices and volatilities are uncorrelated, the implied volatility coincides with the fair value of the volatility swap. In this paper, via classical Itô calculus and Taylor expansions, we write the price for out-of-the-money (OTM) and in-the-m

This paper introduces a transformative framework for managing path-dependent financial risk by shifting from traditional distribution-centric models to a geometry-based approach. We propose the SigSwap as a new regulatory instrument that allows market participants to decompose complex risk into term

Banks must optimize risky investments, dividend payouts, and capital structure under tight Basel III solvency and liquidity constraints, while costly equity issuance serves as a distress-recovery tool. We formulate this as a stochastic control problem that reduces the high-dimensional balance-sheet

This study examines the effects of Trump-era tariffs on financial market efficiency by applying multifractal detrended fluctuation analysis to the return and absolute return time series of six major financial assets: the S&P 500, SSEC, VIX, BTC/USD, EUR/USD, and Gold. Using the Hurst exponent $h(2)

We introduce a Geometry Informed Model for financial forecasting by embedding high dimensional market data onto constant curvature 2manifolds. Guided by the uniformization theorem, we model market dynamics as Brownian motion on spherical S2, Euclidean R2, and hyperbolic H2 geometries. We further inc

We examine a delayed stochastic interest rate model with super-linearly growing coefficients and develop several new mathematical tools to establish the properties of its true and truncated EM solutions. Moreover, we show that the true solution converges to the truncated EM solutions in probability

We provide series expansions for the tempered stable densities and for the price of European-style contracts in the exponential Lévy model driven by the tempered stable process. These formulas recover several popular option pricing models, and become particularly simple in some specific cases such a

This paper investigates the continuous-time limit of score-driven models with long memory. By extending score-driven models to incorporate infinite-lag structures with coefficients exhibiting heavy-tailed decay, we establish their weak convergence, under appropriate scaling, to fractional Ornstein-U

This paper studies the pricing problem in which the underlying asset follows a non-Markovian stochastic volatility model. Classical partial differential equation methods face significant challenges in this context, as the option prices depend not only on the current state, but also on the entire his

The study of probabilistic models for the analysis of complex networks represents a flourishing research field. Among the former, Exponential Random Graphs (ERGs) have gained increasing attention over the years. So far, only linear ERGs have been extensively employed to gain insight into the structu

In this paper, we propose an analytical method to compute the collateral liquidation probability in decentralized finance (DeFi) stablecoin single-collateral lending. Our approach models the collateral exchange rate as a zero-drift geometric Brownian motion, and derives the probability of it crossin

High-frequency trading (HFT) is an investing strategy that continuously monitors market states and places bid and ask orders at millisecond speeds. Traditional HFT approaches fit models with historical data and assume that future market states follow similar patterns. This limits the effectiveness o

A new deep-learning neural network architecture based on high-order weak approximation algorithms for stochastic differential equations (SDEs) is proposed. The architecture enables the efficient learning of martingales by deep learning models. The behaviour of deep neural networks based on this arch

This Ph.D. thesis explores approximations and regularity for the Heston stochastic volatility model through three interconnected works. The first work focuses on developing high-order weak approximations for the Cox-Ingersoll-Ross (CIR) process, essential for financial modelling but challenging due

This study examines the impact of the coronavirus disease 2019 (COVID-19) pandemic on market efficiency by analyzing three time series – price returns, absolute returns, and volatility increments – in stock (Deutscher Aktienindex, Nikkei 225, Shanghai Stock Exchange (SSE), and Volatility Index) and

Fourier-based methods are central to option pricing and hedging when the Fourier-Laplace transform of the log-price and integrated variance is available semi-explicitly. This is the case for the Volterra Stein-Stein stochastic volatility model, where the characteristic function is known analytically

In financial market microstructure, there are two enigmatic empirical laws: (i) the market-order flow has predictable persistence due to metaorder splitters by institutional investors, well formulated as the Lillo-Mike-Farmer model. However, this phenomenon seems paradoxical given the diffusive and

Multilevel Monte Carlo (MLMC) reduces the total computational cost of financial option pricing by combining SDE approximations with multiple resolutions. This paper explores a further avenue for reducing cost and improving power efficiency through the use of low precision calculations on configurabl

We study operator learning in the context of linear propagator models for optimal order execution problems with transient price impact à la Bouchaud et al. (2004) and Gatheral (2010). Transient price impact persists and decays over time according to some propagator kernel. Specifically, we propose t

We characterise the solutions to a continuous-time optimal liquidity provision problem in a market populated by informed and uninformed traders. In our model, the asset price exhibits fads – these are short-term deviations from the fundamental value of the asset. Conditional on the value of the fad,

In a reinforcement learning (RL) framework, we study the exploratory version of the continuous time expected utility (EU) maximization problem with a portfolio constraint that includes widely-used financial regulations such as short-selling constraints and borrowing prohibition. The optimal feedback

We introduce a new identification strategy for uncertainty shocks to explain macroeconomic volatility in financial markets. The Chicago Board Options Exchange Volatility Index (VIX) measures market expectations of future volatility, but traditional methods based on second-moment shocks and time-vary

This paper addresses a novel \emph{“cost-sensitive”} distributionally robust log-optimal portfolio problem, where the investor faces \emph{“ambiguous”} return distributions, and a general convex transaction cost model is incorporated. The uncertainty in the return distribution is quantified using th

We develop a deep reinforcement learning (RL) framework for an optimal market-making (MM) trading problem, specifically focusing on price processes with semi-Markov and Hawkes Jump-Diffusion dynamics. We begin by discussing the basics of RL and the deep RL framework used, where we deployed the state

We investigate the optimal investment-reinsurance problem for insurance company with partial information on the market price of the risk. Through the use of filtering techniques we convert the original optimization problem involving different filtrations, into an equivalent stochastic control proble

We consider an insurance company which faces financial risk in the form of insurance claims and market-dependent surplus fluctuations. The company aims to simultaneously control its terminal wealth (e.g. at the end of an accounting period) and the ruin probability in a finite time interval by purcha

This paper studies the equity holders’ mean-variance optimal portfolio choice problem for (non-)protected participating life insurance contracts. We derive explicit formulas for the optimal terminal wealth and the optimal strategy in the multi-dimensional Black-Scholes model, showing the existence o

This paper extends the possibility to examine the underlying curvature of data through the lens of topology by using the Betti curves, tools of Persistent Homology, as key topological descriptors, building on the clique topology approach. It was previously shown that Betti curves distinguish random

Introducing an algebraic framework for modeling limit order books (LOBs) with tools from physics and stochastic processes, our proposed framework captures the creation and annihilation of orders, order matching, and the time evolution of the LOB state. It also enables compositional settings, accommo