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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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Is $X_t = e^{\int_0^t B_sdB_s - \frac{1}{2}\int_0^t B_s^2 dB_s}$ a martingale?

Question: Is $X_t = e^{\int_0^t B_sdB_s - \frac{1}{2}\int_0^t B_s^2 dB_s}$ a martingale with respect to the filtration generated by $B_t$? In order to determine whether the above expression is a martingale, I thought that it'd be a good first step…
Jess
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Time scaled polynomial Brownian Motion

I want to choose constants $a$ and $b$ such that the process $$X_t = t^aP\left(\frac{B_t}{t^b}\right)$$ is a martingale, where $B_t$ is a Brownian Motion and $P(y)$ is a polynomial of degree n. Thus far, I have used Ito's lemma to determine that the…
Jess
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Stochastic Calculus - Ito decomposition

I have got one question about Ito decomposition. Suppose $W_t$ is a Brownian Motion: $X_t = W_t^2 + \int_0^t(W_t^3-1)du$ How to get $dX_t$? I am quited comfused by the integral. Should we calculate the integral first since it is just a normal…
Brain Zhang
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Ito's Product Formula?

I'm asked to consider three Ito processes $(X(t), t \ge 0)$, $(Y(t), t \ge 0)$, and $(Z(t), t \ge 0)$. I am asked to show: $$d(X(t)Y(t)Z(t)) = X(t)Y(t)dZ(t) + X(t)Z(t)dY(t) + Y(t)Z(t)dX(t) + X(t)dY(t)dZ(t) + Y(t)dX(t)dZ(t) + Z(t)dX(t)dY(t)$$ My…
Eddie
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Theorem in finite case

I have found the following theorem in a book: Let $s \in S$ be any state of an irreducible Markov chain on state space $S=\{0,1,2,...\}$. The chain is recurrent if there exists a solution $\{ y_j : j \not= s \}$ to the inequalities $$ y_i \geq…
mr_T
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Black Scholes Pricing of a claim

Question: Let H(x)=1/x be the payoff function for a European style derivative security. Find a closed form expression for the price: $$ u(t,x)=e^{-r(t-t)}E[H(S_T)|S_t=x] $$ for this claim using Black Scholes dynamics, with r and sigma constants. My…
WeakLearner
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Ito's Lemma for Integral

Let $S$ follow GBM with $dS=(r-q)S\,dt+\sigma S\,dW$ where $W$ is a standard Brownian motion. Define $I_t=\int_0^t qe^{r(t-u)}S_u \,du$, then how can I determine $dI_t$? The answer should be $dI_t=(rI_t+qS_t)\,dt$. (Oh and this is not homework, I…
drawar
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Log normal stock prices - Steps after Ito

When we specify a GBM stock price: $$dS = \mu S dt + \sigma S dW$$ And then we change it to: $$\frac{dS}{S} = \mu dt + \sigma dW$$ The we assume: let $Z_t = f(S_t) = \log S_t$, where $f(x) = \log x$. Then by the Ito's formula, we have: $$ Z_t -…
BlueTrin
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Ito's lemma for a boolean

If I have a stochastic process defined as usual by $dx=f(x,t)dt+g(t,x)dW$, how can I compute the Ito's formula for a process $n=\phi(t,x):=(x/t>a)$, i.e., $dn = (\ldots)dt + _\ldots$ ? I have relaxed $n$ as $\tilde{n}:=\frac{1}{1+e^{-2k\xi}}$,…
asciatopo
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Stochastic Differential Equation calculating E(X-1(t))

If I have the following ODE: $$\frac{dEX(Y)}{dt} = 1 + (\sigma^2-1)E(Y) $$ where $ Y(t) = \frac{1}{X(t)}$, $dX = X(1-X)dt +\sigma Ydw$, $dY = (1-Y+\sigma^2Y)dt - \sigma Ydw $ How do I find $E(X^-1(t))$? I have solution where I first taking the…
Cathy
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Why is this the expectation of this cross product?

Question: Show that the average of $B$ i.d. (identically distributed, but not necessarily independent) random variables, each having a variance of $\sigma^2$, with positive pairwise correlation $\rho$ has a variance of $\rho…
Tim
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Proof of the existence and uniqueness solution of a BSDE

Consider the following BSDE: $$\begin{cases} dY_t &= -f(t, \boldsymbol{x}_t, Y_t, \boldsymbol{z}_t) ~ dt + \boldsymbol{z}_t^\intercal ~ d\boldsymbol{w}_t,\hspace{0.64cm} t\in[0, T] \\ Y_T &= \max \left( …
poglhar
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Throw 3 dices and pick 2, is it realley the same as throwing 2 dices?

I calculated the propabilites to get numbers 2 to 12 by throwing 3 dices and picken only 2 out of them. If someone throws 3 times a 1 the event is [1, 1, 1]. There are 3 possible combinations to get a 2 out of the thrown dice. I summarized all…
Aaron
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Question on Protter's book about quadratic variation

On protter's book Stochastic Integration and Differential Equations, there is some confusion for me about the decomposition about quadratic variation: Since the process $[X, X]$ is non-decreasing with right continuous paths, and since $\Delta [X,…
George
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Given a stochastic differential equation, what is the quadratic variation?

Given a stochastic differential equation, what is the quadratic variation? e.g, assume $dX_t = X_t \cos{B_t}dB_t - \frac{1}{2}\sin^2{B_t}dt.$ Is this correct? $$d\langle X \rangle_t = (dX_t)^2 = X_t^2 \cos^2{B_t}(dB_tdB_t) +…
Oskar
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