Functional Itô calculus and stochastic integral representation of martingales

Cont, Rama; Fournié, David-Antoine

doi:10.1214/11-aop721

Cited by 247 publications

(381 citation statements)

References 30 publications

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“…[14], [4], [1], [17], [21]. See also [7], [11], [18] and references therein for a related approach based on functional Itô calculus. Suppose the risk-neutral dynamics of the underlying asset of a European option is driven by a stochastic differential equation (for short SDE) of the form dX x t = b(X x t )dt + σ(X x t )dB t , X x 0 = x ∈ R , where b : R → R and σ : R → R are some given drift and volatility coefficients, respectively.…”

Section: Introductionmentioning

confidence: 99%

Computing deltas without derivatives

Baños

Meyer‐Brandis²,

Proske

et al. 2017

Finance Stoch

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A well-known application of Malliavin calculus in Mathematical Finance is the probabilistic representation of option price sensitivities, the socalled Greeks, as expectation functionals that do not involve the derivative of the pay-off function. This allows for numerically tractable computation of the Greeks even for discontinuous pay-off functions. However, while the pay-off function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example excludes already simple regime switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to Itô diffusions with irregular drift coefficients, whereat we here focus on the computation of the Delta, which is the option price sensitivity with respect to the initial value of the underlying. To this purpose we first show existence, Malliavin differentiability, and (Sobolev) differentiability in the initial condition of strong solutions of Itô diffusions with drift coefficients that can be decomposed into the sum of a bounded but merely measurable and a Lipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin and Sobolev derivatives in terms of the local time of the diffusion, respectively. We then turn to the main objective of this article and analyse the existence and probabilistic representation of the corresponding Deltas for European and path-dependent options. We conclude with a small simulation study of several regime-switching examples.

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Section: Introductionmentioning

confidence: 99%

Computing deltas without derivatives

Baños

Meyer‐Brandis²,

Proske

et al. 2017

Finance Stoch

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show abstract

“…The question of existence and uniqueness of a solution to a path-dependent martingale problem is addressed in [3] in a general setting of diffusions with a path-dependent jump term. In the case where there is no jump term and under Lipschitz conditions on the coefficients, the existence and uniqueness of a solution has been already established in [8] from the stochastic differential equation point of view.…”

Section: For Example D X φ(U ω) Is Defined As D X φ(U ω) := D X φ(mentioning

confidence: 99%

Dynamic Risk Measures and Path-Dependent Second Order PDEs

Bion-Nadal

2015

Stochastics of Environmental and Financial Economics

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We propose new notions of regular solutions and viscosity solutions for path-dependent second order partial differential equations. Making use of the martingale problem approach to path-dependent diffusion processes, we explicitly construct families of time-consistent dynamic risk measures on the set of càdlàg paths IR n valued endowed with the Skorokhod topology. These risk measures are shown to have regularity properties. We prove then that these time-consistent dynamic risk measures provide viscosity supersolutions and viscosity subsolutions for path-dependent semi-linear second order partial differential equations.

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“…In contrast to Markovian optimal control problems, for control problems where the state systems are described by SDDEs, the backward equation of the value function, obtained by using the Bellman principle in the context of dynamic programming, depends on the initial path of the state process, and so it is generally infinite-dimensional. Note that, although recently developed functional Itô calculus (see [5] and [6]) may be applied to the delayed trajectory, the classical Itô formula cannot be applied to such a trajectory, Hence, it is generally difficult to obtain a corresponding finite-dimensional Hamilton-Jacobi-Bellman equation to solve the problem, except for some special cases. See for example [10] and [11].…”

Section: Introductionmentioning

confidence: 99%

Conjugate duality in stochastic controls with delay

Wang

Hodge

2017

Adv. Appl. Probab.

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This paper uses the method of conjugate duality to investigate a class of stochastic optimal control problems where state systems are described by stochastic differential equations with delay. For this, we first analyse a stochastic convex problem with delay and derive the expression for the corresponding dual problem. This enables us to obtain the relationship between the optimalities for the two problems. Then, by linking stochastic optimal control problems with delay with a particular type of stochastic convex problem, the result for the latter leads to sufficient maximum principles for the former.

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Functional Itô calculus and stochastic integral representation of martingales

Cited by 247 publications

References 30 publications

Computing deltas without derivatives

Computing deltas without derivatives

Dynamic Risk Measures and Path-Dependent Second Order PDEs

Conjugate duality in stochastic controls with delay

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