Questions tagged [stochastic-calculus]

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.

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Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
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What are the prerequisites for stochastic calculus?

I am not a math student, and only kind of picking up something whenever I need it. After emerged in the field of machine learning, probability, measure theory and functional analysis seem to be quite ...
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Intuition for random variable being $\sigma$-algebra measurable?

Is there some sort of intuition or a good ilustrative example for random variables being $\sigma$-algebra measurable? I understand the definition, but when looking at martingales, the meaning of ...
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Translations of Kolmogorov Student Olympiads in Probability Theory

I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward. I ...
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Gradient Descent on Non-Convex Function Works But How?

For Netflix Prize competition on recommendations one method used a stochastic gradient descent, popularized by Simon Funk who used it to solve an SVD approximately. The math is better explained here ...
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Implementing Ornstein–Uhlenbeck in Matlab

I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in ...
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What are some open research problems in Stochastic Processes?

I was wondering, what are some of the open problems in the domain of Stochastic Processes. By Stochastic Processes. Any examples or recent papers or similar would be appreciated. The motivation for ...
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Stochastic calculus book recommendation

I'm a quantitative researcher at a financial company. I have a PhD in math, but I'm an algebraist, so I only took the two required analysis courses in grad school (measure theory for the first, and I ...
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Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
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Itō Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itō integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
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Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
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Where to begin in approaching Stochastic Calculus?

I have experience in Abstract algebra (up to Galois theory), Real Analysis(baby Rudin except for the measure integral) and probability theory up to Brownian motion(non-rigorous treatment). Is there a ...
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Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very similar ...
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How to compute $\mathbb{E}(\exp(\int_0^t W_s ds)|W_t)$?

I am trying to compute the conditional expectation $$\mathbb{E}\left[\exp\left(\int_0^t W_s ds\right)\middle|\, W_t\right]$$ where $W$ is a standard Wiener process and where $s\le t$. To initially ...
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Why do people simulate with Brownian motion instead of “Intuitive Brownian Motion”?

I have just recently begun studying Brownian motion and stochastic calculus at the level of an undergraduate or beginning graduate student of applied mathematics. (Textbooks I've looked at are by ...
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Integral of a Gaussian process

Let $(\Omega,\Sigma,P)$ be a probability space and $X: [0,\infty) \times \Omega \to \mathbb{R}$ be a Gaussian process (i.e. all finite linear combinations $\sum_i a_i X_{t_i}$ are Gaussian random ...
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Why isn't the Ito integral just the Riemann-Stieltjes integral?

Why isn't the Ito integral just the Riemann-Stieltjes integral? What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral: $$\int_0^Tf(t)\;dB(t).$$ ...
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Given a Riemannian manifold $(M,g)$, the paths of a Brownian motion on it can be written as the following stochastic differential equation in local coordinates: $$dX_t = \sqrt{g^{-1}} dB_t - \frac{1}{... 3answers 3k views Physical meaning of Ito integrals I'm having trouble getting my head around the meaning of the stochastic Ito integral. Specifically: the intuitive meaning of "Stochastic Integral" to me is a function that takes a time t and ... 5answers 2k views Why do people write stochastic differential equations in differential form? I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ... 1answer 1k views Calculate \mathbb{E}(W_t^k) for a Brownian motion (W_t)_{t \geq0} using Itô's Lemma Show by using Ito's Lemma, for k \geq 2 the following result hold.$$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$where W(t) = N(0,t) is standard Brownian motion. I think E[W(t)^k]... 4answers 4k views Wiener Process dB^2=dt Why is dB^2=dt? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that dB=\sqrt{dt}Z, but I don't know what ... 2answers 2k views What are some easier books for studying martingale? What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ... 3answers 2k views In stochastic calculus, why do we have (dt)^2=0 and other results? I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": (dt)^2=0 ... 1answer 1k views Covariance of Gaussian stochastic process Could someone help me to figure out solutions of following problems?: Let X = (X_t)_{t \geq 0} be a Gaussian, zero-mean stochastic process starting from 0, i.e. X_0 = 0. Moreover, assume that ... 2answers 578 views Problem 3.24 of “Brownian Motion & Stochastic Processes” by Karatzas and Shreve - Submartingales and stopping times I'm doing the problem 3.24 of Brownian Motion and Stochastic Processes by Karatzas and Shreve. There is two specific parts troubling me, I need some help to see what to do. Here is the problem: ... 1answer 471 views Is an SDE really equal to an integral equation, or is it rather “its integral” that is? Ive been told and been reading in some textbooks on SDE's that an SDE really is an integral equation. In other words, that  dX= \beta dt + \sigma dW \, "really means" \, X_{t}= X_{0} +\int_{0}... 1answer 530 views Higher math and statistics/probability So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ... 1answer 408 views Final Step in calculating option prices under the Heston Stochastic Volatility Model Let:$$ \alpha = -\frac{u^2}{2}-\frac{iu}{2}+jiu\\ \beta = \lambda-\rho \eta i u - j \rho \eta\\ \gamma = \frac{\eta ^2}{2}\\ $$where j \in \{0,1\} and i^2=-1, g=\frac{r_-}{r_+} and r_{\pm}=\... 2answers 2k views Girsanov: Change of drift, that depends on the process Known: If I am looking at an SDE like: dX_t = b(t,\omega) dt + dW_t with W_t a Brownian motion under a measure P. I know that I can change the drift by using Girsanov to dX_t = (b(t,\omega)+... 2answers 644 views How to calculate the PSD of a stochastic process Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ... 1answer 275 views Prove the density of this SDE is not smooth in a parameter Consider the following, 1-dimensional, equation$$X_t^x = x + \int_0^t \mathbb{E} |X_s^x| \, ds + B_t , $$where B is a Brownian motion. This a McKean-Vlasov equation, sometimes called a nonlinear ... 1answer 2k views Hölder Continuity of Fractional Brownian Motion I would like to prove the following theorem: Let H\in (0,1). The fractional Brownian motion B_H admits a version whose sample paths are a.s. Hölder continuous of order strict less than H. ... 1answer 124 views Is a vector of independent Brownian motions a multivariate Brownian motion? Given a filtered probability space (\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P): If B_1, B_2, \dots, B_m  are all real \mathcal{F}_t Brownian motions, jointly independent. Is the resulting ... 3answers 3k views Expected value of average of Brownian motion For a standard one-dimensional Brownian motion W(t), calculate:$$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$Note: I am not able to figure out how to approach this problem. All ... 1answer 698 views Difference between weak ( or martingale ) and strong solutions to SDEs Hi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE :$$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} ... 1answer 2k views What is the difference between “filtration for a Brownian motion” and “filtration generated by a Brownian motion”? I'm reading Shreve's book "Stochastic Calculus for Finance: Vol II". In 5.3.1, after the Theorem 5.3.1 (Martingale representation, one dimension), Shreve explains: "The assumption that the filtration ... 1answer 2k views (Elementary) Markov property of the Brownian motion Let B=(B_t)_{t\ge 0} be a Brownian motion on a probability space (\Omega,\mathcal A,\operatorname{P}), i.e. B is a real-valued stochastic process with B_0=0 almost surely B has independent ... 1answer 935 views Exponentials of stochastic processes and Brownian motions This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given X_t=\int_0^ur_sds for a ... 1answer 2k views Time derivative of white noise It is known that the time derivative of Wiener process W(t) is defined as white noise \xi(t) \begin{align} \xi(t) = \frac{dW}{dt} \end{align} By considering dW/dt as finite difference form \... 1answer 1k views Application of the Burkholder Davis Gundy inequality The proof of the Feynman-Kac formula uses a lemma which I need to prove, but I can not figure it out. The lemma is the folllowing: Let X be a weak solution ofdX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t...
Does the following property for martingales hold? Given a continuous martingale $(X_t)_{t\leq T}$ that is almost surely strictly positive at time T, i.e. $\mathbb{P}(X_T >0)=1$, we have \$P(X_t > ...