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Multiplicative Noise in Stochastic Differential Equations (SDEs)

In the realm of stochastic differential equations (SDEs), multiplicative noise is used to model systems in which the amplitude of stochastic fluctuations internal to the system depend on the state of said system. One of the most prominent examples of multiplicative noise in SDEs is Geometric Brownian motion (GBM). GBM is widely used in finance to model stock prices, currency exchange rates, and other assets. The Geometric Brownian Motion (GBM) model is widely used in financial mathematics to describe the evolution of asset prices. It assumes that the proportional returns of the asset follow a normal distribution over infinitesimal time intervals. The GBM stochastic differential equation is given by: d X t = μ X t d t + σ X t d W t , {\displaystyle dX_{t}=\mu X_{t}\,dt+\sigma X_{t}\,dW_{t},} where:

• X t {\displaystyle X_{t}} is the asset price at time t {\displaystyle t} ,
• μ {\displaystyle \mu } is the expected return (drift rate),
• σ {\displaystyle \sigma } is the volatility of returns,
• W t {\displaystyle W_{t}} is a standard Brownian motion (Wiener process).

Theorem (Itô's formula). Let X t {\displaystyle X_{t}} be given by: d X t = b ( t , ω ) d t + σ ( t , ω ) d W t . {\displaystyle dX_{t}=b(t,\omega )\,dt+\sigma (t,\omega )\,dW_{t}.} Let f ( t , x ) {\displaystyle f(t,x)} be a C 1 , 2 {\displaystyle C^{1,2}} function (i.e., C 1 {\displaystyle C^{1}} in time, C 2 {\displaystyle C^{2}} in space). Then the process Y t = f ( X t ) {\displaystyle Y_{t}=f(X_{t})} satisfies d f ( X t ) = ( ∂ t f ( t , X t ) + ∂ x f ( t , X t ) b ( t , ω ) + 1 2 ∂ x , x f ( t , X t ) σ 2 ( t , ω ) ) d t + ∂ x f ( t , X t ) σ ( t , ω ) d W t . {\displaystyle df(X_{t})=\left(\partial _{t}f(t,X_{t})+\partial _{x}f(t,X_{t})b(t,\omega )+{\frac {1}{2}}\partial _{x,x}f(t,X_{t})\sigma ^{2}(t,\omega )\right)dt+\partial _{x}f(t,X_{t})\sigma (t,\omega )dW_{t}.}

Set f ( t , x ) = log ⁡ x . {\displaystyle f(t,x)=\log x.}

Applying Itô's formula to Y t = f ( t , x ) {\displaystyle Y_{t}=f(t,x)} , we compute: d ( log ⁡ X t ) = d ( Y t ) = [ 1 X t ( μ X t ) + 1 2 ( − 1 X t 2 ) ( σ 2 X t 2 ) ] d t + 1 X t ( σ X t ) d W t . {\displaystyle d(\log X_{t})=d(Y_{t})=\left[{\frac {1}{X_{t}}}(\mu X_{t})+{\frac {1}{2}}\left(-{\frac {1}{X_{t}^{2}}}\right)(\sigma ^{2}X_{t}^{2})\right]dt+{\frac {1}{X_{t}}}(\sigma X_{t})dW_{t}.}

Simplifying each term: d ( log ⁡ X t ) = d ( Y t ) = ( μ − 1 2 σ 2 ) d t + σ d W t . {\displaystyle d(\log X_{t})=d(Y_{t})=\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)dt+\sigma dW_{t}.}

Integrating in time, we have: log ⁡ X t = Y t = Y 0 + ( μ − 1 2 σ 2 ) t + σ W t , where Y 0 = log ⁡ X 0 . {\displaystyle \log X_{t}=Y_{t}=Y_{0}+\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t},\quad {\text{where}}\quad Y_{0}=\log X_{0}.}

Exponentiating both sides gives the solution for X t {\displaystyle X_{t}} : X t = exp ⁡ ( Y t ) = X 0 exp ⁡ ( ( μ − 1 2 σ 2 ) t + σ W t ) . {\displaystyle X_{t}=\exp(Y_{t})=X_{0}\exp \left(\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t}\right).}

The solution to this SDE can be explicitly written as: X t = X 0 exp ⁡ ( ( μ − 1 2 σ 2 ) t + σ W t ) , {\displaystyle X_{t}=X_{0}\exp \left(\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t}\right),} where X 0 {\displaystyle X_{0}} is the initial asset price.

The key properties of the GBM model include:

• Log-normal distribution: For any fixed t > 0 {\displaystyle t>0} , X t {\displaystyle X_{t}} follows a log-normal distribution.
• Non-negativity: X t > 0 {\displaystyle X_{t}>0} almost surely for all t {\displaystyle t} , ensuring realistic modeling of asset prices that cannot become negative.
• Multiplicative noise: The random fluctuation term σ X t d W t {\displaystyle \sigma X_{t}\,dW_{t}} is proportional to X t {\displaystyle X_{t}} , reflecting the empirical fact that larger asset prices tend to exhibit larger absolute fluctuations.

The GBM model forms the basis for the Black–Scholes model used to derive closed-form solutions for European option pricing.

In financial mathematics, the presence of multiplicative noise reflects the empirical observation that the magnitude of fluctuations in asset prices tends to scale with the asset's value. This property is crucial in the derivation of models such as the Black–Scholes model for option pricing.

The Cox–Ingersoll–Ross (CIR) model is described by the stochastic differential equation: d r t = a ( b − r t ) d t + σ r t d W t , {\displaystyle dr_{t}=a(b-r_{t})\,dt+\sigma {\sqrt {r_{t}}}\,dW_{t},} where:

• r t {\displaystyle r_{t}} is the short-term interest rate,
• a > 0 {\displaystyle a>0} is the speed of mean reversion,
• b > 0 {\displaystyle b>0} is the long-term mean level,
• σ > 0 {\displaystyle \sigma >0} is the volatility parameter,
• W t {\displaystyle W_{t}} is a standard Brownian motion.

The CIR process does not have a simple closed-form solution in terms of r t {\displaystyle r_{t}} and W t {\displaystyle W_{t}} . However, its conditional distribution is known: for fixed initial value r 0 {\displaystyle r_{0}} , the variable r t {\displaystyle r_{t}} follows a scaled noncentral chi-squared distribution.

For numerical simulation, the Euler–Maruyama method can be applied, discretizing time with step size Δ t {\displaystyle \Delta t} : r n + 1 = r n + a ( b − r n ) Δ t + σ r n Δ W n , {\displaystyle r_{n+1}=r_{n}+a(b-r_{n})\Delta t+\sigma {\sqrt {r_{n}}}\,\Delta W_{n},} where Δ W n {\displaystyle \Delta W_{n}} are independent normal increments with Δ W n ∼ N ( 0 , Δ t ) . {\displaystyle \Delta W_{n}\sim {\mathcal {N}}(0,\Delta t).}

Because of the square-root diffusion term, care must be taken to ensure r n ≥ 0 {\displaystyle r_{n}\geq 0} during simulation. Several methods are used to address this:

• Full truncation scheme: setting negative values to zero.
• Reflection scheme: reflecting negative values back to positive.
• Semi-explicit scheme:

r n + 1 = ( r n + a ( b − r n ) Δ t + σ 2 Δ W n ) 2 , {\displaystyle r_{n+1}=\left({\sqrt {r_{n}+a(b-r_{n})\Delta t}}+{\frac {\sigma }{2}}\Delta W_{n}\right)^{2},} which better preserves positivity and improves numerical stability.

Alternatively, r t {\displaystyle r_{t}} can be exactly sampled by generating a random variable from the appropriate noncentral chi-squared distribution.

The Heston model is a stochastic volatility model used in mathematical finance to describe the evolution of asset prices and their volatility. It extends the Black–Scholes framework by allowing the volatility to change randomly over time.

The Heston model is defined by the following system of stochastic differential equations: d S t = μ S t d t + v t S t d W t ( 1 ) , d v t = κ ( θ − v t ) d t + ξ v t d W t ( 2 ) , {\displaystyle {\begin{aligned}dS_{t}&=\mu S_{t}\,dt+{\sqrt {v_{t}}}S_{t}\,dW_{t}^{(1)},\\dv_{t}&=\kappa (\theta -v_{t})\,dt+\xi {\sqrt {v_{t}}}\,dW_{t}^{(2)},\end{aligned}}} where:

• S t {\displaystyle S_{t}} is the asset price at time t {\displaystyle t} ,
• v t {\displaystyle v_{t}} is the instantaneous variance (i.e., square of volatility),
• μ {\displaystyle \mu } is the drift rate of the asset,
• κ {\displaystyle \kappa } is the rate at which v t {\displaystyle v_{t}} reverts to its long-term mean θ {\displaystyle \theta } ,
• ξ {\displaystyle \xi } is the volatility of volatility,
• W t ( 1 ) {\displaystyle W_{t}^{(1)}} and W t ( 2 ) {\displaystyle W_{t}^{(2)}} are standard Brownian motions with correlation ρ {\displaystyle \rho }

The key feature of the Heston model is that the volatility v t {\displaystyle {\sqrt {v_{t}}}} is itself a random process driven by a square-root diffusion (similar to the CIR process). This allows the model to capture important empirical features of financial markets, such as:

• Volatility clustering,
• Leverage effect (via ρ < 0 {\displaystyle \rho <0} ),
• Implied volatility smiles and skews.

The Heston model admits a closed-form solution for European option prices using characteristic functions and Fourier transform methods, which makes it both tractable and flexible for calibration to market data.

General Mathematical Form of Multiplicative Noise

In general, a stochastic differential equation with multiplicative noise can be written as: d X t = f ( X t , t ) d t + g ( X t , t ) d W t , {\displaystyle dX_{t}=f(X_{t},t)\,dt+g(X_{t},t)\,dW_{t},} where:

• f ( X t , t ) {\displaystyle f(X_{t},t)} is the drift term,
• g ( X t , t ) {\displaystyle g(X_{t},t)} is the diffusion coefficient,
• W t {\displaystyle W_{t}} is a standard Brownian motion.

When the diffusion coefficient g {\displaystyle g} depends explicitly on the state variable X t {\displaystyle X_{t}} , the noise is said to be multiplicative. This contrasts with additive noise, where g {\displaystyle g} is independent of X t {\displaystyle X_{t}} . Multiplicative noise introduces complexities in both analytical and numerical treatments of SDEs, including the need to carefully choose between interpretations such as the Itô calculus and the Stratonovich calculus.

In particular, under the Itô interpretation, the presence of state-dependent noise can induce additional drift terms when transforming variables, a phenomenon known as the Itô correction.

References

1. Maria Petrou, Costas Petrou (2010) Image Processing: The Fundamentals. John Wiley & Sons. 818 pages. ISBN 9780470745861 https://books.google.com/books?id=w3BpSIxN9ZYC&pg=PA325 ↩