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# Stochastic Differential Equations: An Introduction With Applications (Universitext)l

Stochastic differential equations (SDEs) model quantities that evolve under the influence of noise and random perturbations. They have found many applications in diverse disciplines such as biology, physics, chemistry and the management of risk. Classic well-posedness theory for ordinary differential equations does not apply to SDEs. However, stochastic integration allows to develop a new calculus for such equations (Ito calculus). This leads to many new and interesting insights about quantities that evolve under randomness, that have found many real-world applications. This course is a introduction to stochastic differential equations.

## Stochastic Differential Equations: An Introduction With Applications (Universitext)l

A stochastic differential equation model is naturally associated with a probability measure μ. Our non-i.i.d. sampling strategy can be thought of as the assignment of a set of probability measures μ1, μ2,... to . Each unique sample σi is associated with an implied probability measure μi and is generated from under μi in an i.i.d. manner. Our proofs require that all the implied probability measures are equivalent. That is, an event is possible (resp. impossible) under a probability measure if and only if it is possible (resp. impossible) under the original probability measure μ.

In what follows, will refer to a system of stochastic differential equations. We note that a system of stochastic differential equations comes equipped with an inherent probability space and a natural probability measure μ. Our algorithm repeatedly and randomly perturbs the probability measure of the Brownian motion in the model which, in turn, changes the underlying measure in an effort to expose rare behaviors. These changes can be characterized using Girsanov's Theorem.

A stochastic differential equation model is naturally associated with a probability measure μ. Our non-i.i.d. sampling strategy can be thought of as the assignment of a set of probability measures μ1, μ2,... to . Each unique sample σ i is associated with an implied probability measure μ i and is generated from under μ i in an i.i.d. manner. Our proofs require that all the implied probability measures are equivalent. That is, an event is possible (resp. impossible) under a probability measure if and only if it is possible (resp. impossible) under the original probability measure μ.

Finally, we present our Statistical Verification algorithm (See Figure 2) in terms of a generic non-i.i.d. testing procedure sampling with random "implied" change of measures. Our algorithm is relatively simple and generalizes our previous Bayesian Statistical verification algorithm  to non-i.i.d. samples using change of measures. The algorithm draws non-i.i.d. samples from the stochastic differential equation under randomly chosen probability measures. The algorithm ensures that the implied change of measure is bounded so as to make the testing approach fair. The variable n denotes the number of samples obtained so far and x denoted the number of samples that satisfy the AFM specification ϕ. Based upon the samples observed, we compute the Bayes Factor under the new probability measures. We know that the Bayes Factor so computed is within a factor of the original Bayes Factor under the natural probability measure. Hence, the algorithm divides the Bayes Factor by the factor η2nif the Bayes Factor is larger than one. If the Bayes Factor is less than one, the algorithm multiplies the Bayes Factor by the factor η2n.

The present paper only considers SDEs with independent Brownian noise. We believe that these results can be extended to handle SDEs with certain kinds of correlated noise. Another interesting direction for future work is the extension of these method to stochastic partial differential equations, which are used to model spatially inhomogeneous processes. Such analysis methods could be used, for example, to investigate properties concerning spatial properties of tumors, the propagation of electrical waves in cardiac tissue, or more generally, to the diffusion processes observed in nature.

Recent history is rich with microstructure studies of financial markets and with associations of specific families of probability distributions to financial stochastic processes. For good reviews of the microstructure literature see these works respectively [1,2]. For associations of probability distributions such as the widely applied Gaussian, normal inverse Gaussian, and more inclusively the generalized hyperbolic, see these studies [3,4]. In many instances such inquiries assume at the outset various forms of stochastic processes, as defined by stochastic differential equations, and then set forth to estimate parameters. Popular choices are Itô diffusions and Ornstein-Uhlenbeck processes, with and without the superposition of pure jump Lévy processes. 350c69d7ab