How can I code a stochastic equation with jumps?

I am able to code Brownian motion seperately and Poisson process seperately. But I need to construct a stochastic differential equation with drift coefficient is zero . diffusion coefficient is 1 and jump coefficient is 1. But the SDE with jumps simulation.

I need to plot . X_{t}=x+W_{t} + \int_{E} N(ds,de)

for brownian

randn('state',100)

T=1;

N=500;

dt=T/N;

dW=sqrt(dt)*randn(1,N);

W=cumsum(dW);

plot(0:dt:T,[0,W],'r-')

xlabel('t')

ylabel('W(t)')

title('Position over time')

for poisson

function poisson03fig2

clc

fprintf('\nfunction poisson03fig2 OutPut:')

nfig = 0;

K = 10;

KP = 2*K +1; % Include sample of K jumps only.

p = zeros(KP,1);

kstates = 4;

LT = zeros(KP,kstates);

% Begin Calculation:

for kstate = 1:kstates % Test Multiple Simulated Sample Paths:

LT(1,kstate) = 0;

p(1) = 0; % Set initial scaled jump time

% and jump count.

rng(kstate); % Set initial state for repeatability

% or path change.

DTe = -log(rand(K,1)); % Generate random vector of

% K exponential variates.

for k = 1:K % Simulated sample scaled jump times

% LT(k+1) = lambda*T(k+1):

LT(2*k,kstate) = LT(2*k-1,kstate) + DTe(k);

LT(2*k+1,kstate) = LT(2*k,kstate);

p(2*k) = p(2*k-1);

p(2*k+1) = p(2*k-1) + 1;

end

end

% Begin Plot:

nfig = nfig + 1;

scrsize = get(0,'ScreenSize'); % figure spacing for target screen

ss = [5.0,4.0,3.5]; % figure spacing factors

fprintf('\n\nFigure(%i): Simulated Jump Sample Paths\n',nfig)

figure(nfig)

plot(LT(1:KP,1),p,'k-',LT(1:KP,2),p,'k:',LT(1:KP,3),p,'k-.' ...

,LT(1:KP,4),p,'k--','LineWidth',2);

title('Simulated Simple Jump Sample Paths'...

,'FontWeight','Bold','Fontsize',22);

ylabel('P(t), Poisson State'...

,'FontWeight','Bold','Fontsize',22);

xlabel('\lambda t, Scaled Time'...

,'FontWeight','Bold','Fontsize',22);

hlegend=legend('Sample 1','Sample 2','Sample 3','Sample 4'...

,'Location','Southeast');

set(hlegend,'Fontsize',16,'FontWeight','Bold');

set(gca,'Fontsize',16,'FontWeight','Bold','linewidth',3);

set(gcf,'Color','White','Position' ...

,[scrsize(3)/ss(nfig) 60 scrsize(3)*0.60 scrsize(4)*0.80]); %[l,b,w,h]

% End Code

Answer 1

[Aditya]

Hi Maryam,

I understand that you are facing problem in plotting SDE with jump simulations .Here's how you can simulate and plot the SDE with jumps using the given coefficients:

% Set parameters
T = 1;
N = 500;
dt = T / N;

% Simulate Brownian motion
randn('state', 100);
dW = sqrt(dt) * randn(1, N);
W = [0, cumsum(dW)];

% Simulate Poisson process
lambda = 1; % Jump rate
Njumps = poissrnd(lambda * T); % Number of jumps
Tjumps = sort(rand(1, Njumps) * T); % Jump times

% Construct the SDE with jumps
X = W;
for i = 1:Njumps
    X = X + (Tjumps(i) <= dt:dt:T) - (Tjumps(i) <= dt:dt:T) .* dt;
end

% Plot the SDE with jumps
t = 0:dt:T;
plot(t, X, 'r-')
xlabel('t')
ylabel('X(t)')
title('SDE with Jumps: X(t) = x + W(t) + \int_{E} N(ds, de)')

In this code, poissrnd(dt, 1, N) generates a random vector of Poisson-distributed jumps with a rate of dt. The jumps are then added to the Brownian motion W to construct the SDE solution X. The resulting process X is plotted against time t.

Note that the code assumes you have the Statistics and Machine Learning Toolbox installed for the poissrnd function, which generates random numbers from a Poisson distribution.

Make sure to adjust the parameters and customize the code as needed for your specific requirements.

Hope this helps.

Thanks and Regards,

Aditya Kaloji