This is the mathematical core of derivatives pricing and dynamic optimization: diffusions and jump processes for prices, stochastic volatility (Heston, SABR, rough Bergomi) for option surfaces, backward SDEs and PDEs for valuation under frictions, and HJB equations for consumption, liquidation, and hedging problems. Papers here score high on math complexity almost by construction; the question is what they do on the empirical axis.
What to check when reading. Many papers in this family are theoretical by design and should be read as such: the contribution is a closed form, an existence result, or a numerical scheme. When a paper does claim market relevance, look for calibration to real option surfaces or realized paths with reported fit errors, out-of-sample pricing or hedging error rather than in-sample calibration, and an honest statement of what the model cannot reproduce (the ATM skew term structure is the usual casualty).
This paper considers the pricing of long-term options on assets such as housing, where either government intervention or the economic nature of the asset is assumed to limit large falls in prices. The observed asset price is modelled by a geometric Brownian motion (the ’notional price’) reflected at
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
We study a dynamic model of a non-life insurance portfolio. The foundation of the model is a compound Poisson process that represents the claims side of the insurer. To introduce clusters of claims appearing, e.g. with catastrophic events, this process is time-changed by a Lévy subordinator. The sub
We study the Merton portfolio management problem within a complete market, non constant time discount rate and general utility framework. The non constant discount rate introduces time inconsistency which can be solved by introducing sub game perfect strategies. Under some asymptotic assumptions on
This work focuses on financial risks from a probabilistic point of view. The value of a firm is described as a geometric Brownian motion and default emerges as a first passage time event. On the technical side, the critical threshold that the value process has to cross to trigger the default is assu
We consider a stochastic volatility model where the price evolution depend on the exponential of the Ornstein–Uhlenbeck process. After a brief revision of the related theory the entropy-minimal equivalent martingale measure. is calculated.
We study an optimal execution problem in the infinite horizon setup. Our financial market is given by the Black-Scholes model with a linear price impact. The main novelty of the current note is that we study the constrained case where the number of shares and the selling rate are non-negative proces
In several recent works on infinite-dimensional systems of ODEs \cite{cao_derivation_2021,cao_explicit_2021,cao_iterative_2024,cao_sticky_2024}, which arise from the mean-field limit of agent-based models in economics and social sciences and model the evolution of probability distributions (on the s
The paper provides an overview of the theory and applications of risk-sensitive Markov decision processes. The term ‘risk-sensitive’ refers here to the use of the Optimized Certainty Equivalent as a means to measure expectation and risk. This comprises the well-known entropic risk measure and Condit
In Chakraborti’s yard-sale model of an economy, identical agents engage in trades that result in wealth exchanges, but conserve the combined wealth of all agents and each agent’s expected wealth. In this model, wealth condensation, that is, convergence to a state in which one agent owns everything a
This paper delves into the dynamics of asset pricing within Bachelier market model, elucidating the representation of risky asset price dynamics and the definition of riskless assets.
In this paper we formulate and solve an optimal problem for Stochastic process with a regime absorbing state. The solution for this problem is obtained through a system of partial differential equations. The method is applied to obtain an explicit solution for the Merton portfolio problem when an as
We introduce a discrete binary tree for pricing contingent claims with the underlying security prices exhibiting history dependence characteristic of that induced by market microstructure phenomena. Example dependencies considered include moving average or autoregressive behavior. Our model is marke
We synthesize and discuss some new developments in econophysics. In doing so, we focus on option pricing. We relax the assumptions of constant volatility and interest rate. In doing so, we rely on the square root of the Brownian motion. We also provide simple, closed-form pricing formulas for the Am
We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set $\mathcal{D}\subseteq [0,T]\times\mathbb{R}^d$, a diffusion $X$ in $\mathbb{R}^d$ must be linearly controlled in order to keep the time-space process $(t,X_t)$ inside the set $\
We develop a practical framework for identifying and quantifying the hidden layers of risks and optionality embedded in American options by introducing stochasticity into one or more of their underlying determinants. The heuristic approach remedies the problems of conventional pricing systems, which
This paper develops a theoretical mesoscopic model of the limit order book driven by multivariate Hawkes processes, designed to capture temporal self-excitation and the spatial propagation of order flow across price levels. In contrast to classical zero-intelligence or Poisson based queueing models,
A theoretical model of systemic-risk propagation of financial market is analyzed for stability. The state equation is an unsteady diffusion equation with a nonlinear logistic growth term, where the diffusion process captures the spread of default stress between interconnected financial entities and
It is known that the price of call options in the Heston model is determined in a non-unique way. In this paper, this problem is analyzed from the point of view of the existing mathematical theory of uniqueness classes for degenerate parabolic equations. For the special case of degeneracy, a new exa
This paper considers the ruin problem with random premiums, whose densities have rational Laplace transforms, and investments in a risky asset whose price follows a geometric Brownian motion. The asymptotic behavior of the ruin probability for large initial capital values is investigated.
Although the valuation of life contingent assets has been thoroughly investigated under the framework of mathematical statistics, little financial economics research pays attention to the pricing of these assets in a non-arbitrage, complete market. In this paper, we first revisit the Fundamental The
We study optimal execution in markets with transient price impact in a competitive setting with $N$ traders. Motivated by prior negative results on the existence of pure Nash equilibria, we consider randomized strategies for the traders and whether allowing such strategies can restore the existence
It is known that the value of a call option in the case of constant elasticity processes (CEV) with the indicator $α$ exceeding the critical $α=1$ is determined in a non-unique way. We show how, based on an already existing mathematical theory concerning the correctness of boundary conditions for de
In a seminal paper, F. Delbaen and W. Schachermayer proved that the classical NA (“no arbitrage”) condition implies the existence of an “absolutely continuous local martingale measure” (ACLMM). It is known that in general the existence of an ACLMM alone is not sufficient for NA. In this paper we inv
In this work we study the continuous time exponential utility maximization problem in the framework of an investor who is informed about the price changes with a delay. This leads to a non-Markovian stochastic control problem. In the case where the risky asset is given by a Gaussian process (with so
We develop methods for the solution of inhomogeneous Robin type boundary value problems (BVPs) that arise for certain linear parabolic Partial Differential Equations (PDEs) on a half line, as well as a second order generalisation. We are able to obtain non-standard solutions to equations arising in
In this note, we complete the analysis of the Martingale Wasserstein Inequality started in arXiv:2011.11599 by checking that this inequality fails in dimension $d\ge 2$ when the integrability parameter $ρ$ belongs to $[1,2)$ while a stronger Maximal Martingale Wasserstein Inequality holds whatever t
Calculating true volatility is an essential task for option pricing and risk management. However, it is made difficult by market microstructure noise. Particle filtering has been proposed to solve this problem as it favorable statistical properties, but relies on assumptions about underlying market
In this paper we study a class of HJB equations which solve for equilibria for general time-inconsistent deterministic linear quadratic control problems within the intra-personal game theoretic framework, where the inconsistency arises from non-exponential discount functions. We characterize the sol
We revisit the classical Merton consumption–investment problem when risky-asset returns are modeled by stochastic differential equations interpreted through a general $α$-integral, interpolating between Itô, Stratonovich, and related conventions. Holding preferences and the investment opportunity s
In this paper we aim to study viability and completeness in finite markets. In order to do that, we characterize the set of equivalent martingale measures of two-period markets as convex combinations of a finite number of martingale measures. We provide an algorithm for finding such measures, that c
We present a variation of the well-known binomial model of asset prices. This variation incorporates a bound to short-selling, inspired by a model from Gunduz Caginalp[2]. We formalize this model and prove a formula for all the moments of the logarithmic returns. We also derive a formula for the cas
We discuss the role of information entropy on the behaviour of random processes, and how this might take effect in the dynamics of financial market prices. We then go on to show how the Open Quantum Systems approach can be used as a more flexible alternative to classical methods in terms of modellin
Financial markets are often modelled as if time were unique and continuous across assets and markets. Financial markets are however asynchronous, order flow is event-driven, and waiting times between events are often random. Many of the most influential formulations of financial market models presup
This paper implements an efficient numerical algorithm for the time-fractional Black-Scholes model governing European options. The proposed method comprises the Crank-Nicolson approach to discretize the time variable and exponential B-spline approximation for the space variable. The implemented meth
On April 22, 2020, the CME Group switched to Bachelier pricing for a group of oil futures options. The Bachelier model, or more generally the arithmetic Brownian motion (ABM), is not so widely used in finance, though. This paper provides the first comprehensive survey of options pricing under ABM. U
This paper diverges from previous literature by considering the utility maximization problem in the context of investors having the freedom to actively acquire additional information to mitigate estimation risk. We derive closed-form value functions using CARA and CRRA utility functions and establis
In this short paper we introduce a new class of performance measures based on certainty equivalents defined via scaled utility functions. We analyse their properties, show that the corresponding portfolio optimization problem is well-posed under generic conditions, and analyse the link between portf
Environmental, Social, and Governance (ESG) finance is a cornerstone of modern finance and investment, as it changes the classical return-risk view of investment by incorporating an additional dimension of investment performance: the ESG score of the investment. We define the ESG price process and i
Ito’s Lemma implies that if $W$ is a Wiener process and $f$ is a twice continuously differentiable function, then the process $f(W)$ is the sum of a time integral and an Ito integral. The Ito integrand is not necessarily locally square integrable. This note provides a counterexample.
The integration of cryptocurrencies into institutional portfolios necessitates the adoption of robust risk modeling frameworks. This study is a part of a series of subsequent works to fine-tune model risk analysis for cryptocurrencies. Through this first research work, we establish a foundational be
We propose martingale consumption as a natural, desirable consumption pattern for any given (proportional) investment strategy. The idea is to always adjust current consumption so as to achieve level expected future consumption under the arbitrarily chosen investment strategy. This approach avoids t
We reformulate the neutral wealth tax framework of Froeseth (2026; arXiv:2603.05264) in the language of stochastic dynamics and statistical physics. Individual wealth under geometric Brownian motion satisfies a Langevin equation with multiplicative noise; the probability density of wealth across a p
Three traits of decentralized finance are studied. First, the market impact function is derived for optimal-growth liquidity providers. For a standard random walk, the classic square-root impact is recovered. An extension is then derived to fit general fractional Ornstein-Uhlenbeck processes. These
We identify the measurable absorbing obstruction to uniqueness of invariant probability measures for a Markov kernel. Ordinary absorbing decompositions obstruct global irreducibility and recurrence, but not necessarily uniqueness: an absorbing component may have full mass for no invariant probabilit
In Chakraborti’s yard-sale model of an economy, identical agents engage in pairwise trades, resulting in wealth exchanges that conserve each agent’s expected wealth. Doob’s martingale convergence theorem immediately implies almost sure wealth condensation, i.e., convergence to a state in which a sin
This paper refutes the claim that the expected rate of return of the underlying asset plays no role in the Black-Scholes-Merton option pricing model.
Standard real options theory predicts delay in exercising the option to invest or deploy when extreme asset volatility or technological uncertainty are present. However, in the current race to develop artificial general intelligence (AGI), sovereign actors are exhibiting behaviors contrary to theore
We use classical tools from calculus of variations to formally derive necessary conditions for a Markov control to be optimal in a standard finite time horizon stochastic control problem. As an example, we solve the well-known Merton portfolio optimization problem.
Benoit Mandelbrot’s scientific legacy spans an extraordinary range of disciplines, from linguistics and fluid turbulence to cosmology and finance, suggesting the intellectual temperament of a “fox” in Isaiah Berlin’s famous dichotomy of thinkers. This essay argues, however, that Mandelbrot was, at h
Given the marginal distribution information of the underlying asset price at two future times $T_1$ and $T_2$, we consider the problem of determining a model-free upper bound on the price of a class of American options that must be exercised at either $T_1$ or $T_2$. The model uncertainty consistent
We establish a first and second-order approximation for an infinite dimensional limit order book model (LOB) in a single (‘‘critical’’) scaling regime where market and limit orders arrive at a common time scale. With our choice of scaling we obtain non-degenerate first-order and second-order approxi
We develop a fitted value iteration (FVI) method to compute bicausal optimal transport (OT) where couplings have an adapted structure. Based on the dynamic programming formulation, FVI adopts a function class to approximate the value functions in bicausal OT. Under the concentrability condition and
This work defines and studies one-dimensional convolution kernels that preserve nonnegativity. When the past dynamics of a process is integrated with a convolution kernel like in Stochastic Volterra Equations or in the jump intensity of Hawkes processes, this property allows to get the nonnegativity