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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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Black-Scholes N(d1) and N(-d1)

I think my question is how can I calculate N(-d1) if I know N(d1)? I got a question (with solution): So I know how to calculate the value of the call, but how should I get the value of N(-d1) or N(-d2) given the value of N(d1) or N(d2)?
Betty
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Derivative of Function of Brownian motion

I have a smooth function that takes as input a Brownian motion $B_t$. My question is how does one find the time derivative of the expectation? In other words, how do you calculate $\frac{d}{dt} \mathbb{E} f(B_t)$.
JohnKnoxV
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Local Martingale on $[0,T]$

What is the definition of a local martingale on $[0,T]?$ I guess the definition should be: $M = (M_t)_{t \in [0,T]}$ is a local martingale if there exists a localizing sequence $\tau_n$ such that for all $n$ the process $M^{\tau_n}$ is a…
White
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Perturbation theory of SDEs

I am looking for some kind of perturbation theory for approximating solutions to stochastic differential equations like \begin{equation} dX_t =\mu(X_t,t) dt + \sigma(X_t,dt) dW_t \end{equation} with respect to a Wiener process $dW_t$, but I haven't…
OscarNieves
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linearity of stochastic differential equations

If $dX(t) = a_1(t)dt + b_1(t)dW(t)$ $dY(t) = a_2(t)dt + b_2(t)dW(t)$ Can we say that $d(X(t)+Y(t)) = (a_1(t)+a_2(t))dt + (b_1(t)+b_2(t))dW(t)$ ?
Markoff Chainz
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Solving Forward Equation

I've currently started reading 'Lectures on Partial Differential Equations' by Faris. Page 44 he states the following forward equation: $$J=a(y)p-\frac{1}{2}\frac{\partial \sigma(y)^2p}{\partial y} = 0$$ I understand how to solve this and obtain the…
John Smith
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Time change and stochastic differentials

If $X$ is a $C$-continuous semi-martingale then $$ \int_0^{C_t} H_s\, dX_s = \int_0^t H_{C_s}\,dX_{C_s}. $$ As far as I'm aware, this and its consequences are the only relation between stochastic integrals and time-changes. If $f$ is increasing,…
ItsNotMeItsYou
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Continuous Everywhere

If a stochastic process $X_{t}$ $\sim N(0,t^3/3)$ and $Y_{t}$ is defined as follows: $Y_{t} = X_{t}/t$, if $t>0$ and $ Y_{t} = 0$, if $t=0$, then how can I show that $Y_{t}$ is continuous in $t>=0$ almost everywhere? I started as follows: $X_{t}$…
user60041
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Brownian motions identical distributions

Let $(B_t)_t$ be a standard Brownian motion, and $$ A = \sup\{t\leq 1\mid B_t =0 \},\qquad B = \inf\{ t\geq 1\mid B_t =0 \}. $$ I would like to show that $A$ and $B^{-1}$ are identically distributed and find their distribution. Could you please help…
eugen1806
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Quick question regarding $E^Q(Y) = E^P(YZ_t)$

Let : $P$ and $Q$ two equivalent probability measures and $Z$ the Radon Nikodym derivative : $Z = \frac{dQ}{dP}$ . $Z(t)$ is the expectation of the Radon Nikodym derivative $Z(t) = E^P \left[Z|F(t) \right]$, and $Y$ is an $F(t)-$ measurable…
user30614
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Proving $E^{x}[|B_s-B_t|^4]=n(n+2)|t-s|^2$

Prove that: $$E^{x}\left[ \left| B_{t}-B_{s}\right| ^{4}\right] =n\left( n+2\right) \left| t-s\right| ^{2},$$ where $B$ is a brownian motion. Now there are two methods I have tried, but I am clearly missing something vital as my attempts are…
ALEXANDER
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Brownian Motion questions.

Questions regarding Brownian Motion: Question 1a: Prove that $E\left(exp\left( iuB_{t}\right)\right)=exp\left(-\dfrac{1}{2}u^{2}t\right)$ Let $B_t$ be a Brownian Motion on $\mathbb{R}$, $B_0$ = 0 and $E=E^{0}$ What we know: $B_t$ is a Gaussian…
ALEXANDER
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How to differentiate dY(t,B)/dt in the following?

I understand that result should be 0, but I don't understand how to get to it Thanks a lot in advance (ADDinfo) I am actually trying to prove that the following process can be written as stochastic integrals with respect to B: The solutions then…
Alex
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Is the semimartingale decomposition of an exponential process just itself?

Let $L$ be the stochastic exponential of the process $\int_0^t\frac1{X_(s)}dM(s)$, an Ito integral with respect to $M$. Is the semi martingale decomposition of $L$ just $L$? I think so, because the Ito integral is a local martingale, so therefore…
Jack M
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The conditional expectation of exp(XY) given Y

$ X \sim N(0;\sigma^2) $ and Y is a random variable that is independent of X. How do I compute $ \mathbb{E}[e^{XY}|Y] $ ?
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