# 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.

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### Show that two SDE with similar drifts have similar stationary distribution

Is it true? Suppose $$dX_t = \nabla V_1(X_t) + \sqrt{2} dB_t \,,$$ $$dY_t = \nabla V_2(Y_t) + \sqrt{2} dB_t \,.$$ If $V_1$ and $V_2$ are close", would the stationary distributions of $X_t$ ...
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### Extending Schilling's proof of Ito process approximation by simple processes for one-dimensional case to multivariate case

Below is the proof of Lemma 18.5 from Rene Schilling's Brownian motion which states that an Ito process can be approximated uniformly in probability by a simple Ito process. Now it is stated in the ...
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### Expectation of integral of log ito process

Let $dX_t = \mu_t^Xdt + \sigma_t^X dZ_t$. Compute the quantity $E_0[\int_0^{\infty}e^{-\rho t} log X_t dt]$. Here's what I have so far. $log X_t = log X_0 + \int_0^{t} dlog X_s ds$. Also, by Ito's ...
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### SDE of conditional expectation

Assume I have an SDE of the form $$dX_t = a(X_t, t) dt + b(X_t, t) dW_t, X_0 = x_0.$$ Let now with $T>t$ denote $f(X_t)$ the conditional expectation of $X_T$, $$f(X_t) = E[X_T|X_t]$$ My ...
### j-th order derivative of function $f(x) = 1 - \sqrt{1 - x}$ [closed]
Can you please help me with this part of book (Stochastic Calculus for Finance I: The Binomial Asset Pricing Model; Shreve; p.140): Define $f(x) = 1 - \sqrt{1 - x}$ so that f^{’}(x)=\frac{1}{2}(1-x)^...
I am looking at lecture notes in mathematical finance. The prince $C$ of a European call option with strike $K$ and expiry time $T$ in a one period binomial model is derived via no arbitrage principle....