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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.

The main flavors 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 , such as semimartingale processes, but the related Stratonovich integral is frequently useful in problem formulation.

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 Ito'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 $\mathbb{R}^n$.

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Quick question about proof of Levy's theorem

The following question arise from the proof of Levy's theorem in Richard Bass - Stochastic processes (can be seen via Google books, its on page 77). So we have $(M_t)_{t\geq 0}$ a continuous local martingale, $M_0=0$ adapted to $\{\mathcal{F}_t\}$…
htd
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Does $Y\int_a^b f(s,\cdot )dW_s=\int_a^b Yf(s,\cdot )dW_s$ ? whenever $Y$ is a random variable?

Does $$Y\int_a^b f(s,\cdot )dW_s=\int_a^b Yf(s,\cdot )dW_s \ \ ?$$ whenever $Y$ is a bounded random variable ? I could prove it whenever $Y$ is $\mathcal F_a$ as follow : If $f$ is predictable, the solution is obvious. If $f$ is not predictable…
joshua
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How to solve $dY_t = dt + 2 \sqrt{Y_t} dW_t$?

I am trying to solve: $ dY_t = dt + 2 \sqrt{Y_t} dW_t$ with $Y_0 = y_0 > 0$, where $W$ is one dimentional standard Brownian motion. Using the clue in this linked question, I use the substitution, $Z_t=\exp(-\sqrt{Y_t})$. I derive the derivatives as…
Bravo
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Expectation and variance of (this) yet another stochastic process

(Personal note: there is an incredible amount of questions whose title consists in some permutation of the words I chose for the title, and I guess someone may have already answered. I swear I looked many of them up to see if someone did. If I…
marco
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Stopping theorem needed?

Let $(B_t)$ a standard Brownian motion, $\mu \in \mathbb{R}$, $a \in \mathbb{R_+}$ and ${\tilde{B}}_t = B_t + \mu t$. Now we define the stopping time $\tau_a = \inf(t\geq0,{\tilde{B}}_t\geq0) $. I have to show that $$E[\exp(\lambda…
Kadir
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Does $\int_a^b X_s^2dW_s$ independent of $\mathcal F_a$?

In the book Stochastic calculus of Baldi, there is the following lemma : For the last equality, they made a quite long and complicate proof. But, don't we simply have that $$\int_a^b X_sdW_s\quad \text{and}\quad \int_a^bX_s^2ds$$ are independent of…
Walace
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Does the domain of the generator of an Ito diffusion contain $C^2$ or just $C_c^2$?

Does the domain of the generator of an Ito diffusion contain $C^2$ or just $C_c^2$? From Wikipedia (For the generator $A$) One can show that $C_c^2$, i.e. any compactly-supported $C^2$ (twice differentiable with continuous second derivative)…
Tim
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Step verification in derivation of Ito formula.

At the page 125 (see 4.90) we consider function $u(t,W_t)=f(t,at+bW_t)$ and $a(\omega), b(\omega)$ are random functions. Why is it enough to consider function $u(t,x) = f(t,at+bx)$? What is unclear: f(t,at+bx) is random function, because a and b…
Anton Sorokovskiy
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Cumulative probability with changing p

Assuming following situation: I have a 10 sided dice (1-10). I´m allowed to roll the dice 10 times. For the first roll the proability to hit each number is the same 10%. For each roll consecutive roll the chance to roll a 5 is increased by 2%. Now i…
LordCatGod
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The fact that $(dB_t)^2=dt$ is it a convention notation, or it can be proved rigorously?

The fact that $(dB_t)^2=dt$ is it a convention notation, or it can be proved rigorously ? Or more generaly, if $$dY_t=a(t)dt+b(t)dB_t,$$ does the equatity $$(dY_t)^2=b(t)^2dt$$ can by justify rigorously ? If yes, how ? because, I'm not so sure how…
Todd
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Stocahstic Growth Model: Confirm derivation

I need to verify that a particular statement from this paper is correct https://www.sciencedirect.com/science/article/pii/S0895717703901476 Lets assume that a species growth is given by the following deterministic differential equation…
kangkan Dc
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Stochastic Differential of a particular Jumping Process

I am quite new with Stochastic Calculus, and I was wondering if it is possible to (formally, i.e. mathematically rigorously) if it's possible to get the differential of the stochastic process that I will define below. So I have the initial…
CA-Math
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Conversion from Ito equations to Stratonovich

I have the stochastic equations $$ dx = pdt + \beta (x^2 + p)dV \\ dp = xdt - \gamma (x^3 + p^2)dW \\ $$ where $dV$ and $dW$ are mutually independent Wiener processes. I am asked to calculate the corresponding Stratonovich equations. I know how to…
Tyler D
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Itô integral multiplied with random Riemann integral

I have seen the following equation I can't follow: $$ \mathbb{E}\left[\int_0^Tf(s,X_s)ds B_T\right] = \int_0^T\mathbb{E}(f(s,X_s)B_s)ds$$ where $(B)_t$ is a standard Brownian motion and $(X)_t$ is an Itô-process driven by $B$. I am pretty sure, the…
gringer
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Showing that transformation is a transition function

I need to show that $P_{t}f(x) = e^{-\frac{t}{x}}f(x) + \int_{x}^{\infty}ty^{-2}e^{-\frac{t}{y}}f(y)dy$ is a transition function on $(\mathbb{R}_{+},\mathcal{B}(\mathbb{R}_{+}))$. How do I do it? Any tips?
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