Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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81 views

Question about Stochastic Calculus,(change of measure)?

Can any one give some hint for this question? Let $\{S_t\}_{t=0}^\infty$ be an asset price process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Assume that the log-return of $...
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28 views

Variance of integrated dynamical system

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)}...
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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. ...
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106 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+\...
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3answers
126 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 ...
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43 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 $\...
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74 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\...
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227 views

Pricing a structured note instrument

I am trying to work out the following fixed income problem, where I am asked to price a structured note in Excel, which seems to me to be a reverse collar. My purpose was replicating this structured ...
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121 views

Geometric Brownian Motion: Drawdown as a function of time

Suppose I have a strategy (model it as the usual geometric Brownian motion with a drift). Question is, how does max drawdown grow as a function of duration?
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1answer
443 views

Stochastic integrals wrt to independent Wiener processes are uncorrelated, but potentially dependent?

In Proof of Proposition 1.2.20 in the following lectures notes http://math.uni-heidelberg.de/studinfo/reiss/sode-lecture.pdf I found following quote " stochastic integrals with respect to ...
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57 views

Is the 'constant weight in the risky asset' portfolio-strategy self-financing?

My question concerns a topic in quantitative finance that I feel is often brushed under the table: is a given strategy self-financing. We have two assets, one risky and one riskless, defined by the ...
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1answer
55 views

Discretizing the conditional variance in the Arbitrage Free Dynamic Nelson Siegel model

for my thesis I am trying to fit the correlated factor arbitrage free dynamic Nelson Siegel model to yield data. I use the Kalman filter to model this but since the model is in continuous time, I need ...
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146 views

Characteristic function of SDE with coefficients depending upon second coupled SDE

Say we have the following two SDEs driven by the same single Brownian: $$ dx_t = -0.5\sigma^2g(\psi)^2dt + \sigma g(\psi)dW_t \quad\quad d\psi_t = -(H\psi_t+0.5\sigma^2)dt + \sigma dW_t$$ where $...
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59 views

Flow Variable and Stock Variable

I am new to stochastic control and I need your help! Suppose that we are a trader and we are trading based two sources of signal. One comes from the stock's flow of dividends as well as another trader'...
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401 views

Different definitions of arbitrage

Consider the following setup: Let $S=\left(S_1,\ldots,S_n\right)$ be a $n$-dimensional price process and denote by $V$ its value process defined by $V_t=\phi_t\dot\ S_t$ for $t=0,\ldots,T$. In "...
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63 views

Using malliavin derivative to find the worst Delta-positive hedge?

Background: I've heard that Malliavin Calculus can be used to show the explicit form of a delta-neutral hedge (given an SDE driven market model). For example, here is a sketch here on page 21 on how ...
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578 views

SDE for a portfolio of two correlated assets $ Y_{t} = 2 S^{1}_{t} S^{2}_{t}$

I am analysing a problem where I have two correlated stocks described by Brownian motions $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$ \frac{dS^{2}_{t}}{S^{...
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1answer
471 views

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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67 views

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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245 views

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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116 views

Term Structure and short rates

If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + \sigma^2t(t-\...
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1answer
83 views

Complete Multiperiod Binomial model

I have the following deifnition of a Complete multiperiod binomial model: A multi period binomial model can be called complete if every derivative security can be replicated by trading in the ...
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2answers
236 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it fully,...
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105 views

Intensity Function of Stochastic Processes

I'm fitting some financial data to a model based on a stochastic process and evaluating the fit of it by looking at the compensator. However, I cannot understand well what does it mean to take the ...
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2answers
647 views

How to find the mean and variance of this stochastic process?

$ I_t = \int_0^t e^{i W_s} dWs $ where $W_s$ is the standard brownian motion and $i$ is the complex number. Any help will be appreciated!
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467 views

Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire [duplicate]

Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value: \begin{align*} \mathbb{E}(S_T\log(S_T)). \end{align*} Using a change of measure we can write ...
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234 views

How to compute the conditional variance of this jump process?

Let $N_t$ be a Poisson process with intensity $\lambda>0$ and $S_t$ follows a pure jump process $$dS_t=S_t(J_t-1)dN_t$$ where $J_t$ is the jump size variable if $N_t$ jumps at time $t$. Also, ...
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1answer
148 views

How to prove $\int_0^t W_s^2dWs = \frac{1}{3}W_s^3 - \int_0^t W_s ds$ using Ito's formula? [closed]

Please help me with this problem.
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1answer
54 views

Let $W_t$ denote a standard Brownian motion. Evaluate this integral [closed]

$$ \int_{0}^{t}d(W_{u}^2) $$ How can I deal with this kind of problem? If there is no function given to apply Itô's formula.
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107 views

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, ...
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378 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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1answer
546 views

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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56 views

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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242 views

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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1answer
3k views

Given $S$ is a Geometric Brownian Motion, how to show that $S^n$ is also a Geometric Brownian Motion?

Suppose that a stock price $S$ follows Geometric Brownian Motion with expected return $\mu$ and volatility $\sigma:$ $$dS = \mu S dt +\sigma S dz$$ How to find out the process followed by variable $...
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1answer
441 views

How to define the $f$ function to apply Ito's lemma?

\begin{equation} Z(t) = \exp (a W(t)) \end{equation} I am asked to find $dZ$. I am pretty sure it can be done using Ito's lemma. But in all my textbook (Bjork) examples Ito's lemma is giving from a $...
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1answer
74 views

Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process

We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
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1answer
118 views

How to replicate the future instantaneous short rate?

Suppose we have an interest rate model $R(t)=\alpha(t)d(t)+\sigma d\tilde{W}(t)$, where the brownian motion is under the risk neutral measure. Suppose $S(t)$ is the price at time $t$ for a contract ...
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1answer
70 views

Trouble understanding Notation in Stochastic Calculus (wedge symbol ∧)

I am a beginner in Stochastic Calculus. I am having trouble understanding the meaning behind a specific notation which appears in the topic of Ito process which in differential notation can be written ...
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254 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}...
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1answer
117 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 $0 < t_1 < ... < t_n \leq T$,...
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157 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}{...
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73 views

What is Variance of delta of brownian motion [closed]

I am new to this. If variance of Brownian motion b is t, what is the variance of db? db is delta of b
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48 views

Compute value of $\mathbb{E}(B_3)$

I wonder would anybody tell me how to calculate $\mathbb{E}(B_3)$ Assuming that $\int_0^{t}r_s\,ds\sim N(0.03t,0.25t)$, then is ===== I have similar problem solved: Assuming that $\int_0^t r_s ds \...
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1answer
46 views

Accumulation Rate of Variance in Random Walk

I am slightly confused with the terminology Shreve (2008), he states: "The variance of the symmetric random walk accumulates at rate one per unit time, so that the variance of the increment over ...
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1answer
86 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 ...
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330 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 ...
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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'). \...
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1answer
57 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 ...
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174 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 ...

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