# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### Characteristic function and distribution of a random variable

This is exercise 4.3 in Bjork, Arbitrage Theory in Continous Time. $$X_t = \int^t_0 \sigma(s)dW_s$$ $\sigma$ is a deterministic function and $W_t$ is brownian motion. I am asked to find the ...
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### For the Brownian motion integrate

I want to calculate $$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$ I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to ...
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### Statistical estimation vs Stochastic calibration of models

I have never been able to deduce the precise differences between model building from the statistical perspective and the stochastic processes/calibration perspective. I can only infer that these are ...
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Define time increment $\mu:=t_{k+1}-t_{k}$. Consider the signal $x(\mu)-\mathbb{E}[x(\mu)]$ defined as x(\mu)-\mathbb{E}[x(\mu)]=\frac{1}{\mu}\int_{t_{k}}^{t_{k+1}}\int_{0}^{\tau}e^{A(\tau-\delta)}... 2answers 1k views ### Black and Normal Model for Caplet using Python I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ? I am valuing caplet that caps interest rate on ... 1answer 406 views ### How to calculate the covariance between two stochastic integrals? How to calculate the covariance between the integral of a Brownian motion at different times: $$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$ I know the answer is: $$\... 0answers 56 views ### Model of asset substitution/risk shifting in continuous time Consider a firm with cash flows X_t, which under a risk-neutral probability measure, follows a geometric brownian motion:$$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]where r>0 is the risk-free ... 0answers 66 views ### Euler discretization with jumps There is a process B_t = B_0\prod_{i=1}^{N_t}(1-Z_n), where Z_n=e^{-ξ_n} for i.i.d exponentially distributed random variables (ξn)_{n≥1} with rate ρ=20. {N_t} is a counting process ... 0answers 21 views ### From one period to multi period risk neutral pricing For a one period economy, we have the price of an asset as: p_0 = E^Q [p_1 * \frac {B0}{B1}] where B0 = e^{-r_0} = time 0 price of risk free bond maturing at time =1 and r_0 is known at t0. ... 0answers 104 views ### Correlated GBM and OU processes I want to model two different stochastic processes, such that: X_t , V_t are correlated with coefficient \rho. Where: \frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t} and dV_t=\theta(\mu_2-V_t)dt+\... 3answers 119 views ### Need help to interpret the definition of a diffusion process https://studentportalen.uu.se/uusp-filearea-tool/download.action?nodeId=1134155&toolAttachmentId=218130 In these lecture notes at page 15 and 16 I am looking at the definition of diffusion ... 0answers 60 views ### Price of a stochastic game between an agent and the market In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as \begin{align} V(x,C) = \dfrac{1}{\... 2answers 2k 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 117 views ### The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure Q with respect to the ... 1answer 110 views ### How to deduce the formula of the wealth process of a stochastic volatility model? I am reading the paper Solution of the HJB Equations Involved in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu. The authors consider the utility function U: \mathbb{R} \to \... 1answer 54 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 ... 0answers 234 views ### Pricing caplet with Bachelier (normal dynamic) using forward measure I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution. Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing : C_t(T,T+d) ... 2answers 369 views ### Ho Lee model in Baxter&Rennie I am currentyl reading Baxter&Rennie and I have a difficulty with understanding a derivation of formula for one function, g(x,t,T) (this can be found on page 152 in the book). I know that there ... 0answers 41 views ### If you have normally distributed returns, shouldn't you have the same adjustment factor as lognormally distributed? We know that when using lognormal returns, the number you need to plug in is not the apparent return, but \mu-\sigma^2/2 because what you really have is, in essence, (1) a deterministic growth of \... 1answer 78 views ### What does \int dS \phi (S - K) mean in Gatheral's book? In Gatheral's book on stochastic volatility, he writes the price of an option as\int_K^\infty dS \phi (S - K)$$where \phi is a density. Where does this come from? I have multiple questions: ... 0answers 72 views ### Extension of HJM to multiple factors The HJM model calibrates the entire forward curve using the existing yield curve data and this results in the following expression for its instantaneous forward rate-$$df(t,T)=\sigma(t,T)\int_0^T\... 1answer 159 views ### Application of Vibrato Montecarlo methods Ciao, I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain. In short the main idea of the method is the ... 1answer 201 views ### Discretizing a Continuous Time Stochastic Volatility Model How does the discrete time stochastic volatility model arise from the continuous time one? Also, forgive me for cross-posting. I have the following continuous time SDE for a stochastic volatility ... 0answers 72 views ### Prove the given stochastic integral are equally distributed LetW^i_t$and$W_t$be pairwise independent Brownian motions for$i \in \{1, \dots , d\}$. Let$X_t^i$be$d$independent Ornstein–Uhlenbeck processes for$i \in \{1, \dots , d\}$, i.e. each$X_t^i$... 1answer 184 views ### Two papers - two different solutions of the Ornstein-Uhlenbeck process Bernal 2016 says that the solution of $$dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1)$$ equals $$r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)... 0answers 130 views ### Bond prices at future times under Vasick one-factor model In Vasicek one-factor model (and in other affine models), the price of a zero-coupon bond at time t conditional on the information at this time is$$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-... 1answer 618 views ### Calculating the stochastic integral of$\exp(-rt)S_t$I am currently reading lecture notes which aim to show that if $$S_t = S_0 \exp (\mu t + \sigma W_t)$$ then, under the probability measure$\tilde{\mathbb{P}}$with density $$\gamma_T = \exp (c W_T ... 0answers 366 views ### Applying Ito's formula to complex functions Within my lecture notes, the following definition is given: We say that the stochastic process X_t has stochastic differential$$ dX_t = b_t dt + \sigma_t dW_t $$if and only if$$ X_t = ... 1answer 162 views ### Girsanov's Theorem for Multiple Risky Assets Girsanov's theorem provides the measure transformation from probability measure P to Q such that-$dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$is a martingale under the P measure where$\xi_t=e^{-\...
Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...