# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### Obtaining the drift of a Wiener process formed from a random walk

I'm trying to understand how the equation for Geometric Brownian Motion is formed from a random walk. I'm following the book 'Statistics of Financial Markets' but I'm struggling to follow how the ...
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### Is there anyone tried to use simultaneous stochastic differential equations?

I am looking for some examples or attempts of using simultaneous stochastic differential equations for financial analysis but there has been none so far. Is it just so nasty to apply such thing in ...
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### stochastic calculus and multidimentional itos lemma

I am considering a number of assets (N) in a portfolio. each asset follows a geometric Brownian motion process therefore the stochastic differential equation is dS(i) = S(i)μdt + S(i)σdX(i). The ...
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### Advantage of continuous time stochastic calculus over discrete version?

I'm new to the stochastic calculus, and I keep converting the continuous stochastic differential equation to its counterpart in discrete time, such as the autoregressive models. I wonder in practice, ...
251 views

### Show a process is Martingale

$$Z(t)=(\frac{S(t)}{H})^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$. How can I show that $Z(t)/Z(0)$ is a postive Q-...
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### Change of numeraire from bank account to Zcb [closed]

Why is there no drift adjustment when numeraire is changed from bank account (risk neutral measure) to zero coupon bond who matures at time of payoff (fwd risk neutral measure) ?
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### Notation clarity on continous proesses [closed]

Can someone clarify differences between $dX_t,\frac{\partial X_t}{\partial t},\int_0^t X_{t'}dt',\int_0^tdX_{t'}$? Does $\int_0^t\frac{\partial X_{t'}}{\partial{t'}}d{t'}=X_t$?
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### Stochastic process and brownian motion

I just read the following and i am having some difficulty to interpret it: We begin our analysis in the standard Black-Scholes world consisting of a bank account process of price denoted by $B_t$, ...
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Let $r$ a random process defined by : $$dr_t=\theta(t)dt + \sigma dW_t$$ $\theta$ is deterministic in $t$ and $W$ a brownian motion. I don't know where my calculation below is going wrong : Let ... 1answer 182 views ### Variance of the Cox-Ingersoll-Ross short rate Shreve II page 151, the Cox-Ingersoll-Ross model is defined as $$dr_t=(\alpha-\beta r_t)dt+\sigma\sqrt{r_t}dW_t$$ By applying Ito's Lemma, we obtain \begin{align} r_t&=r_0e^{-\beta t}+\frac{\alpha}... 1answer 113 views ### Does the partition of time in a simple process depend on the omega in probability space? In Steven Shreve's book "Stochastic Calculus for Finance 2", page 126, a simple process\Delta(t)$is a stochastic process such that there is a partition of time$0 < t_1 < ... < t_n \leq T$,... 1answer 111 views ### stochastic discount factor transformation I have $$\frac{dM_t}{M_t}=-\frac{\mu}{\sigma} dW_t + \gamma_t dB_t, \tag{1}$$ where$B_t$and$W_t$are two independent Brownian Motions, which was further presented as $$M_t=\exp \left( -\frac{\mu}{... 1answer 409 views ### Ho-Lee model - A and B derivation for P(t,T)=e^{-A(t,T)-B(t,T)r_t} I am analyzing the transition of the bond prices in the affine models in the form of P(t,T)=e^{-A(t,T)-B(t,T)r_t} using the property that the diffusion and the drift of an affine model can be ... 2answers 149 views ### Multivariate Ito problem M_t=\frac{X_t}{Y_t} I am analyzing a problem given in the lecture slides published here (Slide 7-8 Example of Multivariate Ito’s Lemma). Can anybody explain how the M_t was calculated out of the Ito formula. I ... 1answer 57 views ### integration of squared brownian motion w.r.t time How to prove \int_0^1 B_s^2ds is a random variable and compute its first two moments? From excercise 1.15 on the book martingales and brownian motion. 1answer 54 views ### Event Occurs Almost Surely Consider an uncountably infinite space, an infinite coin-tossing. Let (\Omega,\mathcal{F},\mathbb{P}) be the probability space. If a set A\in\mathcal{F} satisfies \mathbb{P(A)=1}, then we say ... 2answers 182 views ### Integral of Wiener process over time This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion W_t:$$d(\int W_t dt ) = W_t dt$$I ... 2answers 972 views ### Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate] My question is about a stochastic integral of brownian motion w.r.t time. Let W(t) the Wiener process (or brownian motion). I want to calculate this: \begin{eqnarray} X(t)=\int_{0}^t dt' W(t'). \... 1answer 47 views ### standard brownian vs brownian motion We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons . Here we don't speak about any particular distribution for X t. We only say it is a brownian ... 1answer 142 views ### Zero value of cash flow for future in Shreve's book Here is the statements of future price in Shreve's book Stochastic Calculus for Finance II page 244 to proof the ... 1answer 218 views ### How to calibrate an SDE's by finite difference equation? I would like a general framework for the calibration of the unknown parameters in an arbitrary stochastic differential equation. I have a proposed method that seems reasonable in theory, but is ... 1answer 451 views ### Mean Reverting to its own variance? Good morning all, When trying to decipher some documentation I have come across this stochastic process which seems to me much like a Ornstein-Uhlenbeck (or Vasicek) process.$$dX_t=-\kappa(X_t-\... 1answer 142 views ### Integral with respect of$(dW_s)^n$I know $$\int _0^t dW_s=W_t-W_0=W_t$$ Since$ dW_s dW_s=ds$, so $$\int _0^t( dW_s)^2=\int_0^t ds=t-0=t$$ I Want to know why for$n\ge 3$we have $$\int _0^t (dW_s)^n=0$$ My try $$(dW_s)^2 dW_s (... 1answer 184 views ### Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions Could someone please suggest with detailed steps and/or a reference, 1) How to convert the below discrete time summation to continuous time form and write it as an integral? 2) Any methods to ... 1answer 120 views ### Lebesgue-Stieltjes integration and related topics The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ... 1answer 129 views ### approximating fBm stochastic integral Suppose I have the following stochastic integral:$$\int_a^b f(t)dB_H(t)$$with the term dB_H(t) a fractional brownian motion with associated H parameter. Is it true that for H \in (1/2,1), ... 1answer 55 views ### trading strategy problem - initial capital x buys S over time [0,T] at the constant rate of x/T euros per unit of time I am looking for clarification to the trading strategy problem where the number of stocks is depending on time. In the Market with zero safe rate and stock dynamics defined as$$\frac{dS_t}{S_t}=\... 1answer 63 views ### Please help me with this problem of double exponential distribution please help me with this problem of double exponential distribution 1answer 98 views ### Motivation: Stochastic Interest rate model what is a reason that someone might be interested in a stochastic-interest model such as the Chen model? Also can you provide me with a link to an easy to read motivational paper/part of a paper on ... 0answers 33 views ### Change of numeraire/probability when asset pays dividends So I was looking at Margrabe's formula for exchange call options in the book 'Mathematical Methods for Financial Markets' (Jeanblanc, Chesney, Yor), and I was having trouble justifying their change of ... 1answer 62 views ### Not clear on an SDE solution example on YouTube [closed] This video, from about 6 to 12 minutes: https://youtu.be/qdbkvD4N-us I feel like I’m following him ok, but then at the end his f(t,B(t)) has become an f(t,x) and there is no B(t) in his result, so it ... 0answers 32 views ### Filtrations and the different “kinds” of pre-knowledge I am searching for a reference I think I saw in a book by either Shreve or Oskendahl. I am struggling with a theoretical question. As I recall how it was posed, the idea of no prior information (or ... 1answer 310 views ### Trouble understanding jump part in Kou double exponential jump diffusion model I am trying to work with Kou's double exponential Jump-diffusion model and simulate a price path in a programming language. So the dynamics of the asset price in Kou's model follow: \begin{equation} ... 0answers 37 views ### Variance Equations is missing definition here: https://www.nrc.gov/docs/ML1208/ML12088A329.pdf Campbell, Lo, Mackinlay: The Econometrics of Financial Markets on page 159 i am looking at equation 4.4.9 in the last line, =$I\sigma_{\...
Consider the SDE $$dT(t) = ds(t) + a(s(t) - T(t))dt + \sigma dW(t)$$ where $s(t)$ is a deterministic function that turns out to be the long-term mean (this SDE is used to model daily temperature, so ...