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Conditioned Diffusions which are Brownian Bridges

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  • Itai Benjamini
  • Susan Lee

Abstract

Let X t be a one-dimensional diffusion of the form dX t=dB t+μ(X t)dt. Let Tbe a fixed positive number and let $$\bar X_t $$ be the diffusion process which is X t conditioned so that X 0=X T=x. If the drift is constant, i.e., $$\mu (x) \equiv k$$ , then the conditioned diffusion process $$\bar X_t $$ is a Brownian bridge. In this paper, we show the converse is false. There is a two parameter family of nonlinear drifts with this property.

Suggested Citation

  • Itai Benjamini & Susan Lee, 1997. "Conditioned Diffusions which are Brownian Bridges," Journal of Theoretical Probability, Springer, vol. 10(3), pages 733-736, July.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:3:d:10.1023_a:1022657828923
    DOI: 10.1023/A:1022657828923
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    Cited by:

    1. Giorno, Virginia & Nobile, Amelia G., 2023. "On a time-inhomogeneous diffusion process with discontinuous drift," Applied Mathematics and Computation, Elsevier, vol. 451(C).

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