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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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5
votes
2answers
2k views

Geometric brownian motion vs. Ornstein Uhlenbeck

I'm looking at the SDE of Geometric brownian motion(*): $$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$ (with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$) and the SDE of ...
2
votes
2answers
426 views

Question about the martingale property of stochastic integral

Let $W_{t}$ be a Wiener process, and let $$X_{t} = \int^{t}_{0}W_{\tau}d\tau$$ Is $X_{t}$ a martingale? We can rewrite in differential form as $$dX_{t} = W_{t}dt$$ ,which means $X_{t}$ is a diffusion ...
2
votes
2answers
295 views

Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1 $ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
2
votes
2answers
153 views

Problem with deriving the dynamics of a process

I'm trying to solve the following problem. Given a process $X_t$ and a process $Z_t$, with the dynamics of $X_t$ as $$ dX_t = (\alpha + \beta X_t)dt + (\gamma + \sigma X_t)dW_t $$ and $Z_t$ defined ...
6
votes
0answers
156 views

Expectation over Markov Process and discrete Ito integral (discrete stochastic calculus)

I am doing a research on communication protocol design. A file of $n$ blocks is transferred in several rounds and $R_i$ denotes the number of blocks received in the $i$-th round. The sender sends $n-...
2
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1answer
449 views

Methods of SDE Calibration

There is somewhere summary of methods that can be used to estimate parameters of SDE? I currently using MLE and regression due to linear dependence between samples. I searching for something ...
6
votes
1answer
750 views

How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?

Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
0
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1answer
63 views

Please help me with this problem of double exponential distribution

please help me with this problem of double exponential distribution
1
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1answer
272 views

Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$

Let $T > 0$. Let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \sigma(W_u, u \in [0,t])$ where $W_t$ is standard Brownian ...
1
vote
2answers
82 views

zero coupon problem calculus

I encounter a problem: do we have the following equality : $B(0,T_{i})e^{\int_{0}^{t}r_{s}ds}=B(t,T_{i})$ and if yes why because I am stuck with this ... I try to use that : $B(t,T_{i}) = B(0,T_{i})e^...
2
votes
1answer
167 views

Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale

We are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $\mathbb ...
1
vote
1answer
151 views

Prove uniqueness, and prove $Y_t$ is a martingale by considering $dZ_t$ and $dL_t$

Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $...
3
votes
1answer
133 views

How do one solve $ \int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u $? Double integral with general deterministic function $\delta(t)$

How do one solve $ \int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u $ ? $\delta(t)$ is a general deterministic function. $r$ is constant.
0
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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 ...
1
vote
1answer
417 views

How to change to risk neutral measure in a mean reversion process?

For example, in the Ornstein-Uhlenbeck process do I just replace the drift term with the risk free rate, like in the GBM case?
1
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2answers
119 views

Stochastic process theory question

*S follows a process $dS= mSdt + oSdz$ where m and o are constant. What is the probability followed by $ Y=(Se)^{(r-t)} $. If S follows a process $ dS= k (b-S) dt + oSdz $ where k, b, o are ...
3
votes
2answers
144 views

$ \mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace $ ? Exp. of an indicator funct and a diffusion with non-proportional vol

How to compute $ \mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace $ ? where $ dS_t = S_t r dt + \sigma dW_t $ and $ 1_{S_T > K} $ is the indicator function being one when ...
2
votes
1answer
211 views

Brownian motion. Solve stoc. integral by using Ito's lemma

I want to show that following statement is true by using Ito's lemma to solve stochastic integrals: I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2] Then df=(b^...
0
votes
1answer
132 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.
4
votes
1answer
2k views

How to apply the Feynman-Kac formula?

I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...
6
votes
1answer
3k views

Extended Hull White Interest Rate Model for Zero Coupon Bond

Let's take the following three SDEs: $$dr=u(r,t)dt + w(r,t)dX$$ $$u(r,t)=a(t)-br$$ $$w(r,t)=c$$ where $b$ and $c$ are constants and $a(t)$ an arbitrary function of time $t$. If Zero Coupon Bond $Z(...
4
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0answers
438 views

How to compute the stochastic integral of log-normal process?

How do you compute the following integral: $$\int_0^t e^{\mu s + \sigma W_s} ds$$ or $$\int_0^t e^{\mu s + \sigma W_s} dW_s$$ ? Are those integrals stochastic processes of some well-know type (...
1
vote
1answer
413 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 ...
2
votes
3answers
176 views

Is a bond expiring at $T$ clean or dirty price a martingale under the $T$-Forward measure?

When we say Bond prices are martingale under T-Forward measure, do we mean their Clean Price is a martingale or is it their dirty price. I guess it should be dirty price, as clean price is just a ...
7
votes
1answer
163 views

Proof that the stopping time for a Brownian Motion is finite for given target levels

Given a standard brownian motion $W_t$ and defining $\tau$ as: $\tau :=inf\{t\geq0:W_t=1$ or $W_t=-2\}$ The proof below shows that the stopping time is finite: $P(\tau < t) \geq (|W_t|>2)\\$ ...
3
votes
1answer
207 views

What is the correlation between these two functions of GBMs?

Let's say that I have two correlated GBMs: $$dA_t = A_t \sigma^A dW^A_t$$ $$dR_t = R_t \sigma^R dW^R_t$$ $$dW^R_t dW^A_t = \rho dt$$ I am trying to price a derivative which payoff at time $T$ is: $$...
6
votes
1answer
185 views

Why is GARCH more often applied in risk analysis than stochastics?

I am trying to look out for something I can engage in for my final year project (M.Sc) but my interests lie more in risk analysis (specifically credit risk). I have tried searching the web but really ...
1
vote
0answers
64 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 ...
8
votes
3answers
899 views

Is the average of independent Brownian Motions still a Brownian Motion?

If $W$ and $B$ are independent Brownian Motions (BM thereafter), then the average of $W$ and $B$ is $X_t=\frac{1}{2}(W_t+B_t)$. Where do I begin to show that indeed it is still a BM? Also, if both ...
2
votes
1answer
260 views

Difference between stochastic calculus and newton calculus

As I am not a student of hard core mathematics,I just want to know how stochastic calculus is different from newton calculus. What make stochastic calculus different from simple newton calculus ?
2
votes
1answer
227 views

Probability of Stock breaching barrier

If a stock has a process: $dS(t) = sigma*dB(t)$, where $B(t)$ is a standard Brownian motion, and current stock price is $S(0)$. There is a barrier $H>S(0)$. What is the probability that the stock ...
1
vote
0answers
228 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 ...
3
votes
1answer
219 views

Derivation using Ito's Lemma of price process

Define $q(t)$ as the log price minus a linear trend $$ q(t) = \ln P(t) - \mu t $$ Assume the log price process = Equation 1: $$ dq(t) = - \Theta q(t) dt + \sigma dW(t) $$ Can you show that the ...
3
votes
1answer
159 views

About the boundary conditions of the Black-Scholes-Merton PDE

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
4
votes
1answer
159 views

Computation of Expectation

This question has so long preoccupied my mind.Please help me to solve it. Question: Assume $X_t$ described by the following stochastic differential equation $$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} ...
8
votes
2answers
626 views

how we can derive $PIDE$ of double exponential Jump-diffusion model (we know as kou model)?

I'm working in double exponential Jump-diffusion model (we know as kou model) with following form , under the physical probability measure $P$: \begin{equation} ‎\frac{dS(t)}{S(t-)}=\mu‎‏ ‎dt+\sigma ‎...
4
votes
1answer
333 views

Closed form solution of PDE of Option Price

Let $V=V(S_t,t)$ be the option price and \begin{align} V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\ V(S_T,T)=\ln (S_T)^{2}. \end{align} My question: How can I obtain a closed form solution of ...
7
votes
2answers
2k views

Distribution of stochastic integral

Suppose that $f(t)$ is a deterministic square integrable function. I want to show $$\int_{0}^{t}f(\tau)dW_{\tau}\sim N(0,\int_{0}^{t}|f(\tau)|^{2}d\tau)$$. I want to know if the following approach is ...
8
votes
3answers
480 views

Stochastic Calculus Rescale Exercise

I have the following system of SDE's $ dA_t = \kappa_A(\bar{A}-A_t)dt + \sigma_A \sqrt{B_t}dW^A_t \\ dB_t = \kappa_B(\bar{B} - B_t)dt + \sigma_B \sqrt{B_t}dW^B_t $ If $\sigma_B > \sigma_A$ I ...
5
votes
1answer
120 views

Why $W_{t}^3$ is not a martigale?(by Definition)

If $W_t$ be a wiener process then,how can i show that $W_{t}^{3}$ is not a martingale by definition?
4
votes
1answer
264 views

How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$

How can I calculate? \begin{align} Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right) \end{align} Thank you for your attention.
5
votes
1answer
319 views

Lipschitz condition in mathematical finance

I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ...
3
votes
1answer
272 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
1
vote
1answer
113 views

Stochastic calculus: what am I doing wrong?

it is just the computation of a second moment but however is creating debate !!... Can someone spot the error?
2
votes
2answers
642 views

Is the Brownian motion multiplication rule a definition or is it a theorem?

Is the Brownian motion multiplication rule a definition or is it a theorem? Refer to the highlight part of http://i.stack.imgur.com/doQuT.png where $dw_1(t)dw_1(t)=dt$
2
votes
0answers
384 views

Multivariate Itô's lemma

Hey guys I'm looking for worked examples who show how to apply Itô's lemma in several variables, starting from the very basics. Thank you in advance!
6
votes
2answers
2k views

How to compute the variance of this stochastic integral?

I'm new to stochastic calculus and I did an exercise but I don't know if it is correct, so I need somebody with more experience to check if it is true. I am trying to compute the variance of the ...
8
votes
1answer
4k views

Multidimensional Ito's Lemma for Vector-Valued functions

Consider the vector of $n$ Ito processes $$ d \mathbf{X}_t = \mathbf{\mu}(\mathbf{X}_t,t)dt + \Sigma(\mathbf{X}_t,t)d\mathbf{W}_t $$ where $\mathbf{\mu} \in \mathbb{R}^n$ and $\Sigma \in \mathbb{R}^{...
8
votes
1answer
233 views

pdf of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
1
vote
1answer
571 views

Derivation of HJB equation

I am trying to derive the HJB equation in a stochastic setting. Let me exemplify my problem with the simplest case where there is no control, just one state variable. Assume the payoff is given by $$ ...