Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes whe…
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48 results for “Functional Ito formula”
Extends Itô's formula for path-dependent functions in finance.
problem Modeling and hedging of path-dependent financial options.
method Functional extension of Itô's formula for C^{0,1}-functions of continuous weak Dirichlet processes.
result Validates the hedging or superhedging problems for path-dependent options.
Derives functional Itô formula for non-anticipative maps of rough paths.
problem Functional Itô formula for non-anticipative maps of càdlàg rough paths.
method Approximation properties of the signature and Marcus transformation.
result Functional Taylor expansion for sufficiently regular non-anticipative maps.
We use pathwise Itô calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of Föllmer's pathwise Itô calculus and works for portfolios generated from functions that may depend on the current states of the market port…
Developed a machine-checked Itô calculus for Brownian motion.
problem Formal verification of Itô calculus for Brownian motion.
method Machine-checked formalization in Lean over Mathlib.
result First machine-checked constructions of the Itô integral and Itô's formula.
Paper introduces cubature method for stochastic Volterra equations.
problem Solving stochastic Volterra integral equations efficiently.
method Derive stochastic Taylor expansion, introduce cubature measure.
result Cubature method is more efficient than Euler scheme under certain conditions.
A machine-checked Itô calculus for Brownian motion on
problem Developing an Itô calculus for Brownian motion
method Formalized in Lean 4 on top of Mathlib and the BrownianMotion package
result First machine-checked proof of Itô's formula and construction of Itô integral as martingale-valued process
A change of variable formula with applications to multi-dimensional optimal stopping problemsmath.PR
Derives a new formula for optimal stopping problems with exploding derivatives.
problem Optimal stopping problems with complex boundary conditions.
method Develops a change of variable formula for functions with exploding derivatives near a surface.
result Derives a formula similar to Itô's but with less restrictive conditions.
The paper develops a method for stochastic differential equations on manifolds using Schwartz morphisms and diffusion generators.
problem Representing stochastic differential equations on smooth manifolds.
method Using Schwartz morphisms and diffusion generators to construct SDEs on manifolds.
result An extended Ito formula for SDEs on manifolds.
On the Skew Stickiness Ratioq-fin.MF
Derives formula for skew stickiness ratio in asset price and volatility dynamics.
problem Capturing joint dynamics of asset price and volatility.
method Uses Itô-Wentzell and Clark-Ocone formulae to derive representation.
result Derives asymptotics of skew stickiness ratio under stochastic volatility models.
An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which a…
Dupire's functional Itô calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Itô calculus framework. Name…
The article constructs stochastic integration in Riemannian manifolds.
problem No specific problem stated; focuses on the construction of stochastic integration.
method Functional-analytic approach to stochastic integration in Riemannian manifolds.
result There are infinitely many stochastic integrals, and they are related by a simple formula.
We consider a standard optimal investment problem in a complete financial market driven by a Wiener process and derive an explicit formula for the optimal portfolio process in terms of the vertical derivative from functional It^o calculus. An advantage with this approach compared to the Malliavin calculus approach is t…
A new option pricing model uses a time-varying Hurst exponent for more accurate financial predictions.
problem Inaccurate modeling of financial time series due to constant memory parameter limitations.
method Modeling price fluctuations with multifractional Brownian motion and deriving option pricing formula.
result Empirical performance shows the multifractional model fits market quotes better than standard models.
Formula for option pricing in a stochastic volatility model with jumps.
problem Developing a formula for European option pricing in a complex stochastic volatility model.
method Fractional integral of a diffusion process, martingale representation, and Itô calculus for processes with jumps.
result A first-order approximation formula for option prices.
Counterexample shows Ito integrand needn't be locally square integrable.
problem Ito integrand's square integrability condition is not always met.
method Provided a counterexample to Ito's Lemma's integrability condition.
result Ito integrand needn't be locally square integrable.
Develops a new calculus for stochastic processes with occupation flows.
problem Analyzing the behavior of stochastic processes with occupation flows.
method Itô calculus for occupied processes, Feynman-Kac approach.
result Unified Markovian lifts for pricing financial derivatives.
This paper extends Markovian projections to semimartingales with jumps.
problem Extending Markovian projections to semimartingales with jumps.
method Using Markovian projections to match marginal laws of Itô semimartingales with jumps.
result Existence of Markovian projections for Itô semimartingales with jumps.
In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic diff…
Derives new equations for stochastic volatility models.
problem Modeling local-stochastic-volatility models and their derivatives.
method Conditional forward equation, Dupire stochastic PDE, rolling expiry vanilla option SPDE.
result New equations for LSV models and their derivatives.
The paper provides an efficient method to price path-dependent derivatives using multiscale stochastic volatility models.
problem Pricing path-dependent derivatives under multiscale stochastic volatility models.
method Derives a Malliavin representation for the first-order approximation of the price of path-dependent derivatives.
result An efficient Monte Carlo approximation for pricing path-dependent derivatives is derived.
The study examines insurance demand under rough volatility and path-dependent shocks.
problem Optimal insurance and investment strategies under rough volatility and path-dependent shocks.
method Rough volatility model and Hawkes process with power kernel, Functional Ito formula extension.
result Individuals demand more catastrophe insurance when path-dependent effects are considered.
Study on spin random fields using chaos decomposition for cosmic microwave background modeling.
problem Modeling polarization of Cosmic Microwave Background using spin random fields.
method Explicit Wiener-Itô chaos decomposition of area measures of level sets.
result Reveals a clear difference between high frequency regime and zero spin case.
G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index . This process has sta…
Extends Alòs' formula to Barndorff-Nielsen and Shephard model.
problem Modeling call option prices in a stochastic volatility model.
method Uses Alòs' decomposition formula and Ito's formula for an Ornstein-Uhlenbeck model with infinite jumps.
result First Alòs type decomposition formula for Barndorff-Nielsen and Shephard model.
New method decomposes profits and losses continuously, avoiding discrete reporting issues.
problem Analyzing profits and losses at discrete dates ignores detailed paths.
method Constructs a large class of continuous-time decompositions using extended Itô's formula.
result Identifies a preferred decomposition from exactness, symmetry, and normalization axioms.
Study pricing of American put options with stochastic interest rate and finite maturity.
problem Pricing American put options with stochastic interest rate and finite maturity.
method Applied stochastic calculus and Ito's lemma to derive the option value's formula and optimal exercise boundary.
result Existence and parametrisation of the optimal exercise boundary for the Vasicek model.
Derives new equations for volatility models and option pricing.
problem Modeling and pricing options in local-stochastic-volatility models.
method Develops conditional forward equations and Dupire stochastic PDEs.
result Derives new SPDE for vanilla options.
This paper derives a portfolio decomposition formula when the agent maximizes utility of her wealth at some finite planning horizon. The financial market is complete and consists of multiple risky assets (stocks) plus a risk free asset. The stocks are modelled as exponential Brownian motions with drift and volatility b…
Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times. The local time is constructed from discrete approximations, and it is shown that it is -Hölder …
We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model extending the decomposition obtained by E. Alòs in [2] for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used are non anticipative. …
Derives FPDE for equity-linked insurance pricing.
problem Calculating prices for insurance policies with complex payment histories.
method Variational techniques in functional Itô calculus.
result Derives a functional partial differential equation.
We link SVEs to SPDEs and derive Kolmogorov equations for singular kernels.
problem Solving stochastic Volterra equations with singular kernels.
method Establishing connections between SVEs and SPDEs, using stochastic calculus in Hilbert spaces.
result Solutions of SVEs can be expressed in terms of backward Kolmogorov equations.
This work focuses on the indifference pricing of American call option underlying a non-traded stock, which may be partially hedgeable by another traded stock. Under the exponential forward measure, the indifference price is formulated as a stochastic singular control problem. The value function is characterized as the …
American option of stochastic volatility model with negative Fichera function on degenerate boundaryq-fin.PR
In this paper we study a general framework of American put option with stochastic volatility whose value function is associated with a 2-dimensional parabolic variational inequality with degenerate boundaries. We apply PDE methods to analyze the existences of the strong solution and the properties of the 2-dimensional …
We examine in this article the pricing of target volatility options in the lognormal fractional SABR model. A decomposition formula by Ito's calculus yields a theoretical replicating strategy for the target volatility option, assuming the accessibilities of all variance swaps and swaptions. The same formula also sugges…
This paper first describes a class of uncertain stochastic control systems with Markovian switching, and derives an Itô-Liu formula for Markov-modulated processes. And we characterize an optimal control law, which satisfies the generalized Hamilton-Jacobi-Bellman (HJB) equation with Markovian switching. Then, by using …
Motivated by marginals-mimicking results for Itô processes via SDEs and by their applications to volatility modeling in finance, we discuss the weak convergence of the law of a hypoelliptic diffusions conditioned to belong to a target affine subspace at final time, namely if $X_{\cdot}=(Y_\cd…
New proof and formula linking fusion trees to quantum knot invariants.
problem Quantum knot invariants encoding in non-semisimple TQC.
method Connection between fusion trees and Lawrence representations, using graphical calculus.
result Explicit encoding of quantum knot invariants via fusion trees.
Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distri…
Novel approach to financial derivatives pricing using rough path theory.
problem No-arbitrage conditions in financial markets necessitating precise integration methods.
method Developed a polynomial-based approximation class for rough path functionals, extending to non-geometric rough paths.
result Motivated a hypothesis for payoff functionals in financial markets, facilitating analysis.
The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.
problem Approximating functionals of non-geometric rough paths.
method Extending rough paths with time and quadratic variation terms, proving uniform approximation.
result Linear functionals of extended signatures uniformly approximate continuous functionals.
Localizing Volatilitiesmath.PR
We propose two main applications of Gyöngy (1986)'s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous Itô processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based stochastic volatility mod…
Local signature-based expansionsq-fin.MF
Study local expansions of continuous-time processes using Ito signature properties.
problem Analyzing local expansions of continuous-time processes and their moments.
method Using the Ito signature, a basis of iterated integrals, to conduct expansions of the process' characteristic function.
result Explicit coefficients and stochastic representations for asymptotics as time shrinks or diverges.
This paper analyzes sampling from heavy-tailed distributions using discretized Itô diffusions.
problem Sampling from heavy-tailed distributions with finite variance.
method Mean-square analysis of discretized Itô diffusions with weighted Poincaré inequalities.
result Explicit iteration complexity for obtaining samples close to target distributions in Wasserstein-2 metric.