# Stochastic calculus Definition and 11 Discussions

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly. This field was started and created by Kiyoshi Ito in the midst of World War II.
The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.
The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of processes, but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines). The Stratonovich integral can readily be expressed in terms of the Itô integral. The main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Itô's lemma. This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than Rn.
The dominated convergence theorem does not hold for the Stratonovich integral; consequently it is very difficult to prove results without re-expressing the integrals in Itô form.

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1. ### I Martingale, Optional sampling theorem

In this exercise, we consider simple, nonsymmetric random walk. Suppose 1/2 < q < 1 and ##X_1, X_2, \dots## are independent random variables with ##\mathbb{P}\{X_j = 1\} = 1 − \mathbb{P}\{X_j = −1\} = q.## Let ##S_0 = 0## and ##S_n = X_1 +\dots +X_n.## Let ##F_n## denote the information...

8. ### Stochastic calculus:Laplace transformation of a Wiener process

Homework Statement I am asked to show that E[\exp(a*W_t)]=\exp(\frac{a^2t}{2}) Let's define: Z_t = \exp(a*W_t) W_t is a wiener process Homework Equations W_t \sim N(0,\sqrt{t}) The Attempt at a Solution I want to use the following formula. if Y has density f_Y and there's a ral function g...
9. ### I About stochastic differential equation and probability density

I have two questions about the use of stochastic differential equation and probability density function in physics, especially in statistical mechanics. a) I wonder if stochastic differential equation and PDF is an approximation to the actual random process or is it a law like Newton's second...
10. ### A Smoothness of a value function with discontinuous parameters

Let ##\mu: \mathbb{R}\to \mathbb{R}##, ##f: \mathbb{R}\to \mathbb{R}##, and ##r: \mathbb{R}\to [1, \infty)## be bounded measurable functions (which may be discontinuous). I'm interested in the function ##v:\mathbb{R}\to\mathbb{R}## given by ##v(x) = \mathbb E \left[ \int_0^\infty e^{-\int_0^t...
11. J

### Teaching finance with Monte Carlo simulation

Some of the social sciences suffer from "physics envy". This malady causes educators to inject an unnecssary amount of mathematics into the curriculum as a way of gaining scientific letgitimacy. Sadly for most undergrads, the math actually gets in the way. I wrote a paper in which I describe the...