Exhibit (d)(3)

GOLUB CAPITAL BDC, INC.

NOTICE TO STOCKHOLDERS WHO ARE RECORD HOLDERS

Up to 33,451,902 Shares of Common Stock Issuable Upon Exercise of Transferable Rights

April 8, 2020

Dear Stockholder:

Enclosed for your consideration is a prospectus supplement, dated April 1, 2020 (as amended and supplemented from time to time, the “Prospectus Supplement”), and the accompanying prospectus, dated June 27, 2019 (together with the Prospectus Supplement, the “Prospectus”), relating to the offering by Golub Capital BDC, Inc. (the “Company”) of transferable subscription rights (the “Rights”) to subscribe for up to an aggregate of 33,451,902 shares of the Company’s common stock, par value $0.001 per share (the “Common Stock”). The Rights will be issued to stockholders of record (the “Record Date Stockholders”) as of 5:00 p.m., New York City time, on April 8, 2020 (the “Record Date”). The Rights are offered on the terms and subject to the conditions described in the Prospectus.

The Rights may be exercised at any time during the subscription period, which commences on April 9, 2020 and ends at 5:00 p.m., New York City time, on May 6, 2020, unless extended by the Company in its sole discretion (the “Expiration Date”). The Rights are transferable and are expected to be listed for trading on the Nasdaq Global Select Market under the symbol “GBDCR.” The Common Stock is listed on the Nasdaq Global Select Market under the symbol “GBDC.”

As described in the Prospectus, each Record Date Stockholder is entitled to receive one Right for each four outstanding shares of Common Stock owned on the Record Date. The Rights entitle the holder thereof to purchase one new share of Common Stock for every one Right held (the “Primary Subscription”) at the subscription price (the “Subscription Price”), to be calculated as described in the Prospectus as of the Expiration Date. Record Date Stockholders who fully exercise their Rights pursuant to the Primary Subscription will be entitled to subscribe, on the terms and subject to the conditions set forth in the Prospectus, including pro-ration, for additional shares that remain unsubscribed as a result of any unexercised Rights. In addition, any non-Record Date Stockholders who exercise Rights are entitled to subscribe for remaining shares that are not otherwise subscribed for by Record Date Stockholders pursuant to their over-subscription privileges, on the terms and subject to the conditions set forth in the Prospectus. We refer to these over-subscription privileges as the “Over-Subscription Privileges.”

The Subscription Price per share will be 92.5% of the volume-weighted average of the sales prices of our shares of Common Stock on the Nasdaq Global Select Market for the five consecutive trading days ending on the Expiration Date. As a result, the Company is requiring that Rights holders who wish to exercise their Rights deliver to the subscription agent (the “Subscription Agent”) the estimated subscription price of $11.45 per share (the “Estimated Subscription Price”) on or before the Expiration Date, except as otherwise indicated in the Section entitled “The Offering—Payment for Shares” in the Prospectus Supplement. In the event that the Subscription Price exceeds the Estimated Subscription Price, an invoice will be sent for any additional amounts due. Payment for such additional amounts, if any, must be made within ten business days after the confirmation date. In the event the Subscription Price is less than the Estimated Subscription Price, the Subscription Agent will mail a refund to exercising Rights holders. No interest will be paid on amounts refunded.

The Rights will be evidenced by subscription rights certificates (the “Subscription Rights Certificates”). No fractional shares of Common Stock will be issued pursuant to the Rights.

Rights holders who exercise their Rights will have no right to rescind their subscription after receipt of their completed Subscription Rights Certificates together with payment for shares or a notice of guaranteed delivery by the Subscription Agent.

Enclosed are copies of the following documents:

1. Prospectus;

2. Letter to the Company’s Stockholders, dated April 1, 2020;

3. Subscription Rights Certificate;

4. Instructions as to the Use of Golub Capital BDC, Inc. Subscription Rights Certificates;

5. Notice of Guaranteed Delivery; and

6. A return envelope, addressed to Broadridge Corporate Issuer Solutions, Inc.

Your prompt attention is requested. To exercise Rights, you should complete and sign the Subscription Rights Certificate and forward it, with payment of the Estimated Subscription Price in full for each share of Common Stock subscribed for pursuant to the Primary Subscription and the Over-Subscription Privilege to the Subscription Agent, as indicated on the Subscription Rights Certificate and as indicated in the Prospectus. The Subscription Agent must receive the properly completed and duly executed Subscription Rights Certificate and full payment of the Estimated Subscription Price on or prior to the Expiration Date, except as otherwise indicated in the Section entitled “The Offering—Payment for Shares” in the Prospectus Supplement.

ANY QUESTIONS OR REQUESTS FOR ASSISTANCE CONCERNING THE RIGHTS OFFERING SHOULD BE DIRECTED TO BROADRIDGE CORPORATE ISSUER SOLUTIONS, INC., THE INFORMATION AGENT, TOLL-FREE AT THE FOLLOWING TELEPHONE NUMBER: (888) 789-8409 SOLUTIONS, INC., THE INFORMATION or via email at shareholder@broadridge.com.