The SDE dX = b(X)dt + σ(X)dW has a unique strong solution under which conditions on b and σ?
Ab and σ must both be bounded
Bb and σ must be globally Lipschitz continuous and satisfy a linear growth condition
Cb must be continuous and σ must be constant
Db and σ must be analytic functions
The standard existence-uniqueness theorem for SDEs requires two conditions: (1) Lipschitz continuity: |b(x)-b(y)| + |σ(x)-σ(y)| ≤ K|x-y|, ensuring the solution doesn't branch; and (2) linear growth: |b(x)| + |σ(x)| ≤ K(1+|x|), preventing explosion in finite time. These parallel the Picard-Lindelöf conditions for ODEs. Under these conditions, the Picard iteration X_{n+1}(t) = X(0) + ∫₀ᵗ b(X_n)ds + ∫₀ᵗ σ(X_n)dW converges to the unique strong solution.
Question 2 Multiple Choice
An SDE solution X(t) is called a 'strong solution' if it is adapted to the filtration generated by the driving Brownian motion W. A 'weak solution' relaxes this by allowing the Brownian motion itself to be part of the solution. Which is more general?
AStrong solutions — they allow more flexibility in the choice of probability space
BWeak solutions — every strong solution is a weak solution, but some SDEs have weak solutions without having strong solutions
CNeither — strong and weak solutions always coincide
DThey are incomparable — each applies in different settings with no inclusion
A strong solution is a measurable function of the driving Brownian motion — it is 'built from' W. A weak solution only requires that some probability space exists supporting a Brownian motion and a process satisfying the SDE. The Tanaka SDE dX = sgn(X)dW has a weak solution (|W(t)|, which is a reflected Brownian motion) but no strong solution — the solution cannot be constructed as a measurable function of W. Weak existence is the more general concept.
Question 3 Short Answer
Explain the relationship between the Picard iteration method for ODEs and the construction of SDE solutions.
Think about your answer, then reveal below.
Model answer: Both use fixed-point iteration. For the ODE dx/dt = f(x), Picard iteration defines x_{n+1}(t) = x₀ + ∫₀ᵗ f(x_n(s))ds and shows convergence using the contraction mapping theorem under Lipschitz conditions. For the SDE dX = b(X)dt + σ(X)dW, the same idea applies: X_{n+1}(t) = X₀ + ∫₀ᵗ b(X_n)ds + ∫₀ᵗ σ(X_n)dW. Convergence is proved in L² using the Itô isometry (which converts the stochastic integral's L² norm to ∫E[σ²(X_n)]ds) and Gronwall's inequality. Lipschitz ensures the map is a contraction; linear growth prevents finite-time explosion.
The parallel is precise: ODE theory uses Picard iteration with the sup-norm and the Banach fixed-point theorem. SDE theory uses Picard iteration with the L² norm and the Itô isometry. The Itô isometry is the key tool that makes the stochastic integral behave like a deterministic integral for norm estimation purposes.
Question 4 True / False
The SDE dX = −X dt + dW always has bounded solutions because the drift −X pulls the process toward zero.
TTrue
FFalse
Answer: False
The drift −X does pull the process toward zero (this is the Ornstein-Uhlenbeck process), and the process is mean-reverting with a stationary distribution N(0, 1/2). However, the solution X(t) is a Gaussian process that can take arbitrarily large values — it is unbounded almost surely at any finite time. The mean-reversion means the process spends most of its time near zero and returns after large excursions, but it does not stay bounded. Boundedness of solutions (as in the ODE dx/dt = -x) is fundamentally different from stationarity of the distribution.