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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 $\mathbb{R}^n$.

5617 questions
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Can I swap limit and expectation?

I know that a random variable $X$ is integrable, that is $\mathbb{E}\left[|X| \right]<+\infty$. Can I apply the dominated convergence theorem and state that $\lim_{a \to +\infty} \mathbb{E} (|X|\ 1_{|X| \geq a})$ is equal to $\mathbb{E}…
Fred G.
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Expected value of time integral of geometric brownian motion

Given that the stochastic process follows, $$ \frac{dS_t}{S_t} = \mu dt + \sigma dW_t $$ How do i calculate the expected value of, $$ \int_0^T S_te^{r(T-t)} dt $$ in terms of T. What I tried was the below, $$ \begin{align} \int_0^T S_te^{r(T-t)}…
Danny
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Black Scholes Differential Form

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that $$ S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu\, dt + \sigma\, dW_t). $$ I feel that basic…
user7348
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variance of $W_te^{W_t}$

I wanted to compute $\mathrm{var}[W_te^{W_t}]$. I had no problem computing the mean, but I'm not able to do the same with the mean of the squared variable, basically the trick of putting $\mathrm{E}(W_te^{W_t})=f(t)$ (and solving the differential…
Vittorio Apicella
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Pricing a claim dependent on two stock processes

QUESTION Consider two stock processes: $$ dS^1_t=S^1_t(r\,dt+\sigma_1\,dW^1_t) $$ $$ dS^1_t=S^2_t(r\,dt+\sigma_2\,dW^2_t) $$ $$ t,S^1_0,S^2_0\ge0 $$ and $$ W^1_t,W^2_t $$ are standard independent brownian motions under a risk neutral measure P. What…
WeakLearner
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Ito integral show $\int_0^t Z X_u dM_u = Z\int_0^tX_u dM_u$

Fix a continuous local martingale $M$ starting at $0$. Suppose $X \in \mathscr{P}^*(M)$, i.e. $X$ is progressively measurable and $\int_0^tX_u^2 d\langle M \rangle_u<\infty$ a.s. Then suppose $Z$ is $\mathscr{F}_0$-measurable (we are NOT assuming…
nullUser
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Joint distribution of $W(t)$ Brownian motion and $B(s)=\int_0^t \operatorname{sgn} ( W(s) ) dW(s)$

Let $(W(t))$ be a brownian motion and $B(s)=\int_0^t \operatorname{sgn} ( W(s) ) dW(s)$. Does one know the joint distribution $(W(s),B(s))$ for a given $s$? I know some related theory like Tanaka's formula and the joint distribution of the absolute…
htd
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Confused by the notation in Steele's Stochastic Calculus

Recently started Steele's Stochastic Calculus text and as I was going through the first chapter. While making a case for why $\tau$ is finite he introduces a super-script of $d$ on the variables without any prior mentions of the same. Anybody who…
Vivaan Gupta
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Ito's lemma on $Z_t = t^2B_t - 2\int^t_0sB_sds$

I'm currently picking up some introductory questions on stochastic calculus, and ran into some issues with one of them. Let $Z_t = t^2B_t - 2\int^t_0sB_sds$, where $B$ is a standard Brownian Motion. Now to apply Ito's Lemma: If we let $Z_t =…
user619755
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Bound on the conditional expectation of a stochastic integral given W_T

Suppose that $f \colon \mathbb R \to \mathbb R$ is a bounded function and let $X$ denote the stochastic integral $$ X = \int_0^T f(W_t) \, d W_t. $$ Is it possible in general to bound conditional expectations of the form: $$ \mathbb E [X | W_T = x]…
Roberto Rastapopoulos
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Determining spot price differential equation from future contract process

In appendix A of Valuing Energy Options in a One Factor Model Fitted to Forward Prices by Les Clewlow and Chris Strickland they develop a stochastic differential equation of spot prices exploiting the relationship between Forwards and Spot prices…
konnoc84
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How to derive the solution for the SDE $dX_t = \theta(t)X_t dB_t$ from Itô's formula?

I want to show that the Itô-process $$ Z_t = \text{exp}( \int_0^t \theta_s dB_s - \int_0^t \frac{1}{2} \theta_s^2 ds) $$ is a solution to the SDE $$ dX_t = \theta(t)X_t dB_t$$ using Itô's formula, $$ dg(t,B_t) = \frac{\partial g}{\partial…
Oskar
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Stochastic process for "stock" given stochastic process for "flow"

I am trying to learn some stochastic calculus, and I have gotten stuck trying to answer what I thought would be a simple question. I think I am misunderstanding something fundamental about stochastic calculus, and I hope that answering this…
Frank
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Step in proof of Ito formula

I am reading a book on stochastic processes. The author proved Ito formula for $f(t,w(t))$ where $w(t)$ is brownian motion with filtration $F_t$. Then he wants to prove Ito formula for $x(t)=a(t)+b(t)w(t)$ where $a(t)=a(0)$ and $b(t)=b(0)$ and…
Random Number
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Power Alpha Exotic Option with Zero Strike

Consider a European option on a stock with price $S$ that at expiration at time $T$ pays $S^\alpha$ where $\alpha$ is some arbitrarily chosen power. Could someone explain how to use stochastic calculus and Ito’s Lemma to derive the formula for the…
Cindy Philip
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