Quantitative Finance Stack Exchange
Quantitative Finance Stack Exchange

Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

Stochastic calculus 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 flavours of stochastic calculus in Quantitative Finance are the Itô calculus and its variational relative the Malliavin calculus.

Source: https://en.wikipedia.org/wiki/Stochastic_calculus

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Distribution of stochastic integral

Suppose that $f(t)$ is a deterministic square integrable function. I want to show $$\int_{0}^{t}f(\tau)dW_{\tau}\sim N(0,\int_{0}^{t}|f(\tau)|^{2}d\tau)$$. I want to know if the following approach is correct and/or if there's a better…
Kian
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Solving Path Integral Problem in Quantitative Finance using Computer

I've asked this question here at Physics SE, but I figured that some parts would be more appropriate to ask here. So I'm rephrasing the question again. We know that for option value calculation, path integral is one way to solve it. But the solution…
Graviton
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How were these SDE derived?

Can anyone give me a detailed explanation of how below equations (3) and (4) are derived from (1) and (2)? \begin{align*} \frac{dF_{t,T}}{F_{t,T}} &=\sigma e^{-\lambda(T-t)}dB_t, \tag{1}\\ \ln(F_{t,T})&=\ln(F_{0,T})-1/2\int_{0}^{t}\sigma^2…
snowave
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Stochastic Calculus Rescale Exercise

I have the following system of SDE's $ dA_t = \kappa_A(\bar{A}-A_t)dt + \sigma_A \sqrt{B_t}dW^A_t \\ dB_t = \kappa_B(\bar{B} - B_t)dt + \sigma_B \sqrt{B_t}dW^B_t $ If $\sigma_B > \sigma_A$ I would consider the volatility $B_t$ to be more volatile…
Phun
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Girsanov's Theorem - Change of Measure

I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by: $$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$ Now $\hat P$ is a new probability measure. The…
user3001408
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Obtaining characteristics of stochastic model solution

I want to use the following stochastic model $$\frac{\mathrm{d}S_{t}}{ S_{t}} = k(\theta - \ln S_{t}) \mathrm{d}t + \sigma\mathrm{d}W_{t}\quad (1)$$ using the change in variable $Z_t=ln(S_t)$ we obtain the following SDE $$\mathrm{d}Z_{t} = k(\theta…
RockScience
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A question on Ito

If we know the dynamics of $S$, then we can estimate the value of $S$ at a time point, $t$. Here, I have a question concerning how to solve for $S_t$ by Itô because I obtained different results by different approaches. For a geometric Brownian…
Hebe
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Girsanov Theorem for Quanto/Compo adjustment

Assume that I have a foreign asset $$Y_t = Y_0 \exp \left((r_f-\frac{1}{2}\sigma^2_Y)t+\sigma_Y W_t^1\right)$$ and an exchange rate $$X_t = X_0 \exp\left((r_d-r_f-\frac{1}{2}\sigma^2_X)t+\sigma_X W_t^2\right)$$ I would like to compute the…
Jim
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A stochastic differential equation

Consider the following stochastic differential equation (SDE) $$d X_s= \mu (X_s + b)ds + \sigma X_s d w_s $$ where constants $\mu, \sigma, b > 0$ and initial position $X_0$ are given. If $b=0$, then the above equation is a geometric Brownian motion…
S_R
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Stochastic growth model

In this problem we consider a model of stochastic growth. In particular, consider the following system of SDEs: \begin{align} dX_t &= Y_t dt + \sigma_XdZ_{1t}\\ dY_t &= -\lambda Y_t dt + \sigma_Y \rho dZ_{1t} + \sigma_Y\sqrt{1-\rho^2}dZ_{2t}\\ X_0…
user48018
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Intuition for Martingale Representation Theorem

Can you please explain Martingale Representation Theorem in a non-technical way that what is it and why is it required? Most of the stuffs I studied so far are quite technical, and I failed to grasp the underlying intuition.
Daniel
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How do we calcualte $E[W_sW_t|W_s]$

$W_t$ is a Brownian motion. How do we calculate this expectation? there are two cases: $s < t$ $t < s$ Do we have to distinguish the two cases or there is a unified way of calculating it
Peaceful
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For the Brownian motion integrate

I want to calculate $$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$ I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to get the other one and the full answer of their…
Hobong
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Square of Wiener process

In Ito's calculus one often comes $dW^2=dt$. How does this come about? What is it's relation to the Milstein method?
Borun Chowdhury
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For $B_t$ a Brownian motion what is the probability that $B_1>0$ and $B_2<0$?

Let $B_t$ be a Brownian Motion. What's the probability that $B_1>0$ and $B_2<0$?
Antonius Gavin
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