What is Stochastic calculus: Definition and 28 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. cianfa72

    A Karhunen–Loève theorem expansion random variables

    Hi, in the Karhunen–Loève theorem's statement the random variables in the expansion are given by $$Z_k = \int_a^b X_te_k(t) \: dt$$ ##X_t## is a zero-mean square-integrable stochastic process defined over a probability space ##(\Omega, F, P)## and indexed over a closed and bounded interval ##[a...
  2. C

    A SDE valuation equation (stochastic calculus)

    I read from a text: "suppose a stock with price ##S## and variance ##v## satisfies the SDE $$dS_t = u_tS_tdt+\sqrt{v_t}S_tdZ_1$$$$dv_t = \alpha dt+\eta\beta\sqrt{v_t}dZ_2$$ with $$\langle dZ_1 dZ_2\rangle = \rho dt$$ where ##\mu_t## is the drift of stock price returns, ##\eta## the volatility of...
  3. WMDhamnekar

    I Why Does Yₙ Converge to Zero for q < 1/2 in a Random Walk?

    Let ##X_1, X_2, \dots ##be independent, identically distributed random variables with ##P{X_j = 1} = q, P{X_j = −1} = 1 − q.## Let ##S_0 = 0 ##and for n ≥ 1, ##S_n = X_1 + X_2 + \dots + X_n.## Let ##Y_n = e^{S_n}## Let ##Y_{\infty} = \lim\limits_{n\to\infty} Y_n.## Explain why ##Y_{\infty} =...
  4. WMDhamnekar

    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...
  5. tworitdash

    A Analytical form of a summation

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  6. C

    Courses Graduate level Mathematics courses of interest for Biological Physics

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  7. E

    I Definition of a diffusion coefficient

    Consider this Ito proces: $$dX_t = \mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ with W_t being a wiener process. My question: What is the diffusion coefficient of X? My motivation for asking: A lot if financial literature refer to "diffusion coefficient" and I haven't understood by googling it, because...
  8. J

    Expectation Value of a Stochastic Quantity

    Homework Statement I'm working on a process similar to geometric brownian motion (a process with multiplicative noise), and I need to calculate the following expectation/mean; \langle y \rangle=\langle e^{\int_{0}^{x}\xi(t)dt}\rangle Where \xi(t) is delta-correlated so that...
  9. E

    A Simulation from a process given by "complicated" SDE

    Actually this is more of a simulation question but since PF doesn't have Simulation category I ask here. I need to simulate a path from a proces given by this Stochastic DE: $$ dX_t = -a(X_t-1)dt+b\sqrt{X_t}dB_t $$ where ##B_t## is wiener process/brownian motion and a and b are just some...
  10. T

    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...
  11. T

    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...
  12. E

    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...
  13. 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...
  14. S

    Basic Stochastic Calculus Question, why does dB^2 = dt?

    As the title says, while using Stochastic Calculus, can someone explain some of the properties of differentials? Why does dB_t dB_t=dt Also, why does dt dt=0 and dB_t dt=0 I don't really get why these work?
  15. O

    Strange right-arrow symbol ([itex]\mapsto[/itex]) in stochastic calculus.

    What does the right arrow mean in this context: " ...Then the process t \mapsto \int_{0}^t \phi_s dM_s are well-defined continuous local martingales, whose quadratic variations are given by ... " Is this supposed to mean "the process X that is the mapping X: t \mapsto \int_{0}^t \phi_s dM_s"
  16. S

    Applying Ito's Lemma: Solving a Stochastic Differential Equation

    Homework Statement I'm trying to figure out how to use Ito's Lemma, but all I have are notes and proofs. It would help if someone could go through one actual example with me: Use Ito's Lemma to solve the stochastic differential equation: X_t=2+\int_{0}^{t}(15-9X_s)ds+7\int_{0}^{t}dB_s and...
  17. D

    Stochastic Calculus - Limit Law

    [PLAIN]http://img17.imageshack.us/img17/1061/stochcalcq4.png I am currently taking a class in quantitative finance, part of which includes an introduction to stochastic calculus. This is the first time i have encountered stochastic differential equations, so it is all quite new to me. I am...
  18. V

    Stochastic calculus in mathematician's vs physicist's view

    Hello, I've studied physics at a university previously and actually earned a degree in theoretical physics, but then switched over to mathematics, where I focused on stochastic analysis/calculus/processes (I'll just call it stochastics). Now, I remember taking a course on stochastics while...
  19. A

    Stochastic Calculus for Beginners: Literature Suggestions

    Hi to everybody, I'm going to apply for the theme mentioned in the title during my study and further by writing scientific works. Also I'm very excited with it because of its applications. Couldn't anyone suggest some literature for beginners in Stochastic Calculus? P.S. I also have some...
  20. T

    A stochastic calculus question

    In the attached equations, for the second last step to the last step why dSdS=sigma2S2dt ?
  21. T

    A question for stochastic calculus

    The attachment is from Shreve's stochastic calculus book In the attachment what does the symbol ^ mean? Thanks
  22. H

    How Do You Tackle Complex Problems in Stochastic Calculus?

    Urgent Stochastic Calculus help! Hi, I am new to stochastic calculus and finding some difficulty in understanding things. How to approach the solutions for problem under the topics like martingale, linear diffusion SDEs, expectation of martingale, Ito stochastic integral formulas...
  23. I

    Conditional Expectations (Stochastic Calculus)

    Homework Statement Let (X_n; for all counting number n) be a sequence of independent random variables. We focus on the random walk S_n := X_1 + . . . + X_n and set F_n = 'sigma-algebra' of (S_1, . . . , S_n). 1. Compute E[S_(n+1) \ F_n] 2. For any z belonging to the complex plane C...
  24. A

    How do you get from calculus to stochastic calculus?

    What is the path of study to understand stochastic calculus? I bought the book "Elementary Stochastic Calculus with Finance in View" (Mikosch) because it was touted as a non rigorous introduction to stochastic calculus, and I spent three days trying to decipher the first two pages. :(
  25. C

    Understanding Stochastic Calculus and Expected Value Formulas

    Hello all If you throw a head I give you $1. If you throw a tail you give me $1. If R_i is the random amount ($1 or -$1) you make on the ith toss then why is: E[R_i] = 0, E[R^2_i]=1, E[R_iR_j] = 0 ? If S_i = \sum^i_{j=1} R_j which represents the total amount of money you have won up to...