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Register a new userA Differentiation Theory for It^os Calculus
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Authors
Hassan Allouba
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A peculiar feature of It^os calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to It^os integral calculus? From It^os definition of his integral, such a derivative must be based on the quadratic covariation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolles theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications some of which we address in an upcoming article.
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This paper establishes It^os formula along a flow of probability measures associated with gene-ral semimartingales. This generalizes existing results for flow of measures on It^o processes. Our approach is to first prove It^os formula for cylindrical polynomials and then use function approximation and localization techniques for the general case. This general form of It^os formula enables derivation of dynamic programming equations and verification theorems for McKean-Vlasov controls with jump diffusions and for McKean-Vlasov mixed regular-singular control problems. It also allows for generalizing the classical relation between the maximum principle and the dynamic programming principle to the McKean-Vlasov singular control setting, where the adjoint process is expressed in term of the derivative of the value function with respect to probability measures.
In a 2006 article (cite{A1}), Allouba gave his quadratic covariation differentiation theory for It^os 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 differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to It^os integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory to obtain variants of the Clark-Ocone and Stroock formulas, with and without change of measure. We prove our variants of the Clark-Ocone formula under $L^{2}$-type conditions; with no Malliavin calculus, without the use of weak distributional or Radon-Nikodym type derivatives, and without the significant machinery of the Hida-Malliavin calculus. Unlike Malliavin or Hida-Malliavin calculi, the form of our variant of the Clark-Ocone formula under change of measure is as simple as it is under no change of measure, and without requiring any further differentiability conditions on the Girsanov transform integrand beyond Novikovs condition. This is due to the invariance under change of measure of the first authors derivative in cite{A1}. The formulations and proofs are natural applications of the differentiation theory in cite{A1} and standard It^o integral calculus. Iterating our Clark-Ocone formula, we obtain variants of Stroocks formula. We illustrate the applicability of these formulas by easily, and without Hida-Malliavin methods, obtaining the representation of the Brownian indicator $F=mathbb{I}_{[K,infty)}(W_{T})$, which is not standard Malliavin differentiable, and by applying them to digital options in finance. We then identify the chaos expansion of the Brownian indicator.
Given strong uniqueness for an It^os stochastic equation, we prove that its solution can beconstructed on any probability space by using, for example, Eulers polygonal approximations. Stochastic equations in $mathbb{R}^{d}$ and in domains in $mathbb{R}^{d}$ are considered. This is almost a copy of an old article in which we correct errors in the original proof of Lemma 4.1 found by Martin Dieckmann in 2013. We present also a new result on the convergence of tamed Euler approximations for SDEs with locally unbounded drifts, which we achieve by proving an estimate for appropriate exponential moments.
We propose to study a new type of Backward stochastic differential equations driven by a family of It^os processes. We prove existence and uniqueness of the solution, and investigate stability and comparison theorem.
Using Dupires notion of vertical derivative, we provide a functional (path-dependent) extension of the It^os formula of Gozzi and Russo (2006) that applies to C^{0,1}-functions of continuous weak Dirichlet processes. It is motivated and illustrated by its applications to the hedging or superhedging problems of path-dependent options in mathematical finance, in particular in the case of model uncertainty
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