How to give a solution to any (simple) Ito integral from a stoch. process ?

To find the solution to a simple Ito integral from a stochastic process, you can follow these steps:

Identify the stochastic process and the Ito integral you want to solve. For example, let's say you have the stochastic process dX(t) = μ dt + σ dW(t)and you want to solve the Ito integral:I(t) = ∫[0,t] X(s) dW(s)

Use Ito's lemma to find the stochastic differential equation (SDE) satisfied by the Ito integral. In this case, we havedI(t) = X(t) dW(t)Using Ito's lemma, we can find the SDE satisfied by I(t). Applying Ito's lemma to f(t,X) = X(t)g(t), we have:df(t,X) = g(t) dX(t) + X(t) g'(t) dt + 0.5 X(t) g''(t) σ^2 dtSubstituting X(t) = I(s) and dX(t) = dI(s), we have:d(I(t)g(t)) = g(t) dI(t) + I(t) g'(t) dtSubstituting dI(t) = X(t) dW(t), we get:d(I(t)g(t)) = g(t) X(t) dW(t) + I(t) g'(t) dt

Solve the SDE to find the solution to the Ito integral. In this case, we have:d(I(t)exp(-λt)) = -λ I(t) exp(-λt) dt + exp(-λt) X(t) dW(t)Integrating both sides from 0 to t, we get:I(t) exp(-λt) = ∫[0,t] exp(-λs) X(s) dW(s)Therefore, the solution to the Ito integral is:I(t) = exp(λt) ∫[0,t] exp(-λs) X(s) dW(s)Substituting X(t) = μ dt + σ dW(t), we get:I(t) = exp(λt) ∫[0,t] exp(-λs) [μ ds + σ dW(s)] dW(s)which can be evaluated using standard techniques for integrating stochastic integrals.

Markus Kreer

One of the typical applications encountering your question lies in mathematical finance. That is the Vodoo-science of OTC derivatives which are traded by investment banks and risk managers in corporate and government treasuries.

For the valuation of financial derivatives, e.g. options on a stock or a commoditiy, the underlying price of the stock is modelled by a stochastic process.

In classical theory of Black-Scholes-Merton the logarithm of the (positive) stock price X(t) is described by a certain stochastic differential equation which can be solved using Ito integrals to obtain the well-known log-normal distribution as solution.

However, derivative contracts on X(t) such as e.g. a call option which entitles the purchaser to buy the stock in consideration at a future time T>t for a certain price K, also follow a stochastic process f(X(t)) where it is known that the payoff price at maturity T is f(X(T))=max(0,X(T)-K). The process f(X(.)) is described by a certain stochastic differential equation derived from a no-arbitrage condition from the original stochastic differential equation for X(t). In general, solving this stochastic differential equation using Ito integrals is cumbersome or even impossible. But the task to determine the call option price now, at time t, i.e. f(X(t)) turns out to be related to computing the expectation E(f(X(T)|t) which is a "path integral" over all possible paths. This can be done approximately by a Monte Carlo method.

This principle of handling the stochastic differential equation for the derivative the is called "risk neutral valuation" and has some deep roots in the Levy representation theorem for Wiener martingales. All the essential maths involved is nicely described in the book "Financial Calculus: An Introduction to Derivative Pricing" by Martin Baxter and Andrew Rennie.

Jerzy Kazimierz Filus

Thank you for all the three answers. From these I can see that the field of stochastic equations and integrals is still open to many future research. I rethought the problem in recent hours and now I think that in the case of all the processes realizations being continuous (the Wiener process, by example) the situation is much easier when we integrate (only) the process dX(t) from t to a future epoch T since in this case the fundamental theorem of calculus allows to obtain as the solution the random variable X(T) - X(t). But this does not work in the case of discontinuity (unless the points of the discontinuity are known). Again thank you for your answers. Yours, Jerzy F.

Nicolae Suciu

Besides the financial time series, there are many other examples of Itô integrals (trajectories of diffusion processes) arising in real-life applications. The most known is the Brownian motion introduced by Albert Einstein to explain the irregular movement of particles suspended in a fluid (https://www.maths.usyd.edu.au/u/UG/SM/MATH3075/r/Einstein_1905.pdf).

“Concrete” Itô integrals are obtained in the experimental study of Moroni and Cushman (Physics of Fluids 13, 81, 2001, https://doi.org/10.1063/1.1328076) where trajectories of air bubbles diffused into a porous medium are reconstructed with a fluorescent-particle image velocimetry.

Trajectories obtained from either experimental or numerical experiments can be used to approximate the probability density of the underlying stochastic process. A simple way to do that numerically is to compute weak solutions of the Itô equation with random walks on lattices. The probability density can be readily approximated by counting the number of random walkers at lattice sites. See details on this approach in Section 3.2 of our recent paper (http://dx.doi.org/10.48550/arXiv.2112.10692).

Regards,

Nicolae

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