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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Why do trajectories differ when introducing a second Brownian motion in SDE Simulation?

I am working on simulating a SDE using both the direct approach and a transformed variable approach. My SDE is: dX_t = \kappa X_t \, dt + \sigma \sqrt{X_t} \, dB_{1,t} - \gamma \sqrt{...
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Convergence of the martingale $M_t = \frac{1}{\sqrt{1-t}}e^{-\frac{B_t^2}{2(1-t)}}$ to zero, as $t \to 1^-$

I am self-learning applied stochastic calculus from the text: A first course in Stochastic Calculus, by L.P. Arguin. Exercise problem 5.18 in my text, asks to prove the almost sure convergence of a ...
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A cadlag Feller process for $\mathcal F$ is Markov w.r.t $\mathcal F_+$ (Th. 46, Chap. 1, Stochastic Integration - Protter)

In page 35 of the book Stochastic Integration by P. Protter, he defines a Feller process as follows: Then he states the following theorem. In the proof, he used the following strategy: Next, he ...
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Prove $(g_i ◦ X_i) , i ∈ \{1, . . . , n\}$, are independet random variables (Check my proof please)

Let $(Ω, \mathcal A, \Bbb P)$ a probability space, $n ∈ \Bbb N$ and let $(R_i, \mathcal R_i), (S_i, \mathcal S_i), i ∈ \{1, . . . , n\}$,measuring spaces. Let $X_i: Ω → R_i, i ∈ \{1, . . . , n\}$, ...
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Existence of Fokker-Planck equation under Cauchy boundary condition.

A 1D Fokker-Planck equation within a constrain region is uniquely characterized by three functions and a boundary conditon: A drifting term $\mu(x,t)$. A diffusion term $\sigma(x,t)$. An initial ...
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Change of variable in functional Fokker-Planck equation

Starting from the following (steady-state) functional Fokker-Planck equation for the probability of a path $\{\phi(t)\}_{t=0}^T$ 0 = \frac{\delta}{\delta{\phi}} \left\{ \frac{\...
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