Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...
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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 ...
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Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation. Below is the ...
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Filtration and measure change

I asked this question in math stackexchange but to no avail. So i'm trying the luck here. I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand ...
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What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?

I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process. $$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$ Is anyone able to calculate the mean ...
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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 ...
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Probability density function of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
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Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
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Transformation of Volatility - BS

I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev \sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}} \end{...
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Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$

Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
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How do practitioners use the Malliavin calculus (if at all)?

This question is inspired by the remark due to Vladimir Piterbarg made in a related thread on Wilmott back in 2004: Not to be a party-pooper, but Malliavin calculus is essentially useless in ...
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Why does the price of a derivative not depend on the derivative with which you hedge volatility risk?

I'm trying to derive the valuation equation under a general stochastic volatility model. What one can read in the literature is the following reasoning: One considers a replicating self-financing ...
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Change of measure discrete time

Suppose I have a random walk $X_{n+1} = X_n+A_n$ where $A_n$ is an iid sequence, $\mathsf EA_n = A>0$. How to construct a martingale measure for this case?
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Derivation of Ito's Lemma

My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
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Anticipating stochastic integral $\int_0^T W_T dW_t$

Using basic techniques from Malliavin calculus it can be shown that $$\int_0^T W_T dW_t = W_T^2 - T$$ As can be seen the above integral is a non-adapted stochastic integral. We also know using Ito ...
Consider the vector of $n$ Ito processes $$d \mathbf{X}_t = \mathbf{\mu}(\mathbf{X}_t,t)dt + \Sigma(\mathbf{X}_t,t)d\mathbf{W}_t$$ where $\mathbf{\mu} \in \mathbb{R}^n$ and \$\Sigma \in \mathbb{R}^{...