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- 1 Suppose that X is a stochastic process with dynamics dXt = µdt +σdWt , where W is a P-Brownian motion. The drift µ and the volatility σ are both constants. Find if there is a measure Q such that the drift of process X under Q is η(∈ R) instead of µ.Consider a continuously differentiable function f : R^2 → R^2 and a differential equation x' = f(x) with an equilibrium x^∗ = 0, and suppose that there exists a homoclinic orbit O(x) connecting to this equilibrium (in both forward and backward time). Which of the following three statements is true? (a) The equilibrium x^∗ = 0 is stable. (b) The equilibrium x^∗ = 0 is unstable. (c) It depends on the right hand side f whether the equilibrium x^∗ = 0 is stable or unstable.(a) Solve the initial value problem given by y' - y/x = 1/x, y(1) = -1 (b) Consider the differential equation y' = 1/y - y Find its critical values, and classify them as attractors, repellers, orsemi-stable. Are any of the non-constant solutions defined on all of R? If so, give an example. Are any of the non-constant solutions not defined on all of R? If so, give an example.
- A fish population in a lake satisfies dB/dt=cB(K-B) Fish are then added at A fish per hour. Find the DE for B with respect to t. Is there a new limiting population? If so, what is the solution?Regular Perturbation Theory: For the initial value problem d^2 (x)/dt^2 + x + (epsilon)x=0 with x(0)=1 and dx/dt (0)=0Assume an asset price S_t follows the geometric Brownian motion, dS_t = µS_tdt + σS_dW_t, where µ and σ are constants and r is the risk-free rate. 1. Using the Ito’s Lemma find the stochastic differential equation satisfied by the process Xt = S_t^n , where n is a constant. 2. Compute E[X_t] and Var[X_t]. 3. Using the Ito’s Lemma find the stochastic differential equation satisfied by the process Y_t = S_tert
- Find the equilibrium solution for P. Furthermore determine whether P is initially increasing faster if the initial population is 120 or 200.Using Fourier series expansion, solve the heat conduction equation in one dimension ∂T∂t=k∂2T∂x2 with the Dirichlet boundary conditions: T=T1 if x=0 and T=T2 if x=L. The initial temperature distribution is given by T(x,0) =f(x).1. Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. a) ?(? − 4)? ′′ + 3?? ′ + 4? = 2, ?(6) = 0, ? ′(6) = −1 b) (? + 1)? ′′ + ?? ′ + ? = sec ?, ?(0) = 2, ? ′(0) = −1 c) (? − 4)? ′′ + 3?? ′ + ln(?) ? = sin ?, ?(1) = −2, ? ′(1) = −1
- B) Let dP/dt =.5P - 50. Find the equilibrium solution for P. Furthermore, determine whether P is intially increasing faster if the initial population is 120 or 200.Weather is notoriously difficult to predict. Models are subject to chaotic motion and must consider the initial conditions. The famous butterfly effect states that if a butterfly flaps its wings in Tahiti, that small event might cause a hurricane to hit Texas. This leads us to the following model: suppose that weather at time t is always between 0 and 1 and is governed by x_(t+1)=〖4x〗_t (1-x_t ). For x_0 = 0.2 and x_0 = 0.2000001, determine x_1; x_2;… x_50. Assume values for x closer to zero represent mild weather, and closer to 1 represent extreme weather. How do your calculations illustrate the butterfly effect? Support your answer with graphs. Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure.Weather is notoriously difficult to predict. Models are subject to chaotic motion and must consider the initial conditions. The famous butterfly effect states that if a butterfly flaps its wings in Tahiti, that small event might cause a hurricane to hit Texas. This leads us to the following model: suppose that weather at time t is always between 0 and 1 and is governed by x_(t+1)=〖4x〗_t (1-x_t ). For x_0 = 0.2 and x_0 = 0.2000001, determine x_1; x_2;… x_50. Assume values for x closer to zero represent mild weather, and closer to 1 represent extreme weather. How do your calculations illustrate the butterfly effect? Support your answer with graphs.