On Pathwise Formulae in Stochastic Calculus

Rajeeva Karandikar, Director, Chennai Mathematical Institute

Abstract

The standard treatment of stochastic calculus emphasizes that the integral is defined as a limit in probability and thus it does not make sense to talk of the integral path-by-path. However, in uses of stochastic calculus, in say stochastic control, we are required to obtain the optimal control as a functional of the observed path. Likewise, in statistical inference involving continuous time stochastic models, an estimate of a parameter should be a functional of the observed path. Both these requirements may not be met when the optimal control or the estimator involve stochastic integral. We are thus lead to a question: Can we identify a class of integrands for which we can express the path of stochastic integral as a functional of the path of the integrand and the path of integrator?

 

Lecture 1. Thursday June 1, 14:00-16:00, LT 340

 
We begin by obtaining by constructing a mapping F: D-> D on the space D of cadlag functions which coincides such that if M is a locally square integrable martingale then F(M)=[M] is the usual quadratic variation process.

The proof does not use Doob Meyer decomposition or any other result from stochastic calculus and only uses Doob’s maximal inequality.  This can be used as a starting point to develop the theory of stochastic integration (instead of Doob Meyer decomposition Theorem) which we will outline. 

 

Lecture 2. Tuesday June 6, 14:00-16:00, LT 311

Using the theory of stochastic integration developed in Lecture 1, we go onto construct a mapping I: D x D -> D such that if M =(M_t) is a locally square integrable martingale and H is an adapted r.c.l.l. process then I(H,M) is (a version of ) the stochastic integral of H- with respect to M. The mapping I is universal and does not depend upon the underlying probability space, probability measure, filtration or the distribution of the integrand or the integrator. The proof of this result only depends upon the treatment discussed in first lecture.

We will also show that if X is a semimartingale, then I(H,X) is (a version of ) the stochastic integral of H- with respect to X.

 

Lecture 3. Wednesday June 7, 14:00-16:00, LT 308

In this lecture, we consider a stochastic differential equation dZ= b(t,Z)dX where X is a semimartingale and b(t,Z) is a non-anticipating function of Z. Assuming that b satisfies appropriate Lipschitz condition (which ensures that the SDE admits a unique strong solution), we construct a pathwise formula for the solution Z in terms of the driving semimartingale X.

We achieve this by showing that a modified Euler-Peano approximation to the solution converges almost surely. The proof is based on the Metivier-Pellumail inequality, using which one can get a consolidated estimate on the growth of the stochastic integral without decomposing the semimartingale into a local martingale and a process with finite variation paths.

References:

[1] Bichteler, K.; Stochastic integration and L^p-theory of semi martingales, Annals of Probability, 9, (1981), 49-89.

[2] Karandikar, R.L.: Pathwise stochastic calculus of continuous semimartingales, Ph.D. Thesis, Indian Statistical Institute, 1981.

[3] Karandikar, R.L.: Pathwise solution of stochastic differential equations. Sankhya A, 43, (1981), 121-132.

[4] Karandikar, R.L.: On quadratic variation process of a continuous martingale, Illinois journal of Mathematics, 27, (1983), 178-181.

[5] Karandikar, R.L.: A. S. approximation results for multiplicative stochastic integrals, Seminaire de Probabilites XVI, Lecture notes in Mathematics, 920, (1982), 384-391, Springer-Verlag.

[6] Karandikar, R.L.: On Metivier-Pellaumail inequality, Emery topology and Pathwise formuale in Stochastic calculus. Sankhya A, 51, (1989), 121-143.

[7] Karandikar, R.L.: On a. s. convergence of modified Euler-Peano approximations to the solution of a stochastic differential equation, Seminaire de Probabilites XVII, Lecture notes in Mathematics, 1485, (1991), 113-120, Springer-Verlag.

[8] Karandikar, R.L.: On pathwise stochastic integration, Stochastic processes and their applications, 57, (1995), 11-18.

[9] Karandikar, R. L. and Rao, B. V. : On Quadratic Variation of Martingales. Proc. Indian Academy of Sciences, 124, (2014), 457-469.